Tristan Roussillon
email: tristan.roussillon@liris.cnrs.fr

Isabelle Sivignon
email: isabelle.sivignon@gipsa-lab.grenoble-inp.fr

Reversible Polygon that Faithfully Represents the Convex and Concave Parts of a Digital Curve

Keywords: digital curve, polygonal representation, convex and concave parts

From results about digital convexity, we define a reversible polygon that faithfully represents the maximal convex and concave parts of a digital curve. Such a polygon always exists and is unique in the general case. It is computed from a given digital curve in linear-time using well-known routines: adding a point at the front of a digital straight segment and removing a point from the back of a digital straight segment. It may helps to extract perceptually meaningful parts of shape outlines or lines.

Introduction

In [START_REF] Dorksen-Reiter | Shape Representations of Digital Sets based on Convexity Properties[END_REF][START_REF] Eckhardt | Polygonal representations of digital sets[END_REF][START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF][START_REF] Dorksen-Reiter | A linear algorithm for polygonal representations of digital sets[END_REF], Eckhardt and Dorksen-Reiter looked for a reversible polygon that faithfully represents the convex and concave parts of the boundary of a connected digital set. We call such a polygonal representation faithful polygon (FP for short) in this paper. Because of the chosen digitization scheme (Gauss digitization), they failed to properly define the FP of any open digital curve and show some digital curves for which it does not exist [START_REF] Dorksen-Reiter | Shape Representations of Digital Sets based on Convexity Properties[END_REF]. Therefore, the problem of defining and computing the FP of a digital curve remained open. Though, this problem is closely related to digital convexity and a lot of progress has been made on this topic in the last decade [START_REF] Debled-Rennesson | Detection of the Discrete Convexity of Polyominoes[END_REF][START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF][START_REF] Dorksen-Reiter | A linear algorithm for polygonal representations of digital sets[END_REF][START_REF] Brlek | Lyndon + christoffel = digitally convex[END_REF][START_REF] Roussillon | What does digital straightness tell about digital convexity[END_REF][START_REF] Provencal | Two linear-time algorithms for computing the minimum length polygon of a digital contour[END_REF]. Two recent algorithms [START_REF] Roussillon | What does digital straightness tell about digital convexity[END_REF][START_REF] Provencal | Two linear-time algorithms for computing the minimum length polygon of a digital contour[END_REF] retrieve some kinds of locally extremal points, but the authors did not notice that their algorithm is also able to compute a polygonal representation close to the FP.

The FP of a digital curve may lead to many applications. One of them consists in decomposing digital objects into perceptually meaningful parts, which is of great interest in shape recognition. The concept of convexity plays an important role because a visual part, which is not necessarily convex, is assumed to be convex at a given scale [START_REF] Latecki | Convexity rule for shape decomposition based on discrete contour evolution[END_REF]. Finding a good scale with respect to a given purpose or studying the convexity at various scales may solve the problem.

In addition, the convex and concave parts of an object straightforwardly give the sign of the curvature of its boundary, which is known to be a relevant piece of information about its shape. For instance, curvature zero-crossings at various scales lead to an interesting shape representation, known as the curvature scale space representation [START_REF] Mokhtarian | A theory of multi-scale, curvaturebased shape representation for planar curves[END_REF]. Moreover, at a given scale, the endpoints of maximal convex (resp. concave) parts correspond to points of minimal negative (resp. maximal positive) curvature [START_REF] Latecki | Convexity rule for shape decomposition based on discrete contour evolution[END_REF].

In Euclidean geometry, a given region R is said to be convex if and only if for any pair of points p, q ∈ R the line segment [pq] is included in R. However, in digital image processing, when each pixel is viewed as a point of Z 2 , the only convex regions (in the Euclidean sense) are isolated points, which is not satisfactory at all. Many authors defined the convexity of digital sets, i.e. sets of points of Z 2 (see for instance, Sklansky [START_REF] Sklansky | Recognition of convex blobs[END_REF], Kim [START_REF] Kim | Digital convexity, straightness, and convex polygons[END_REF][START_REF] Kim | Digital straightness and convexity[END_REF], Kim and Rosenfeld [START_REF] Kim | On the cellular convexity of complexes[END_REF][START_REF] Kim | Digital straight lines and convexity of digital regions[END_REF], Kim and Sklansky [START_REF] Kim | Digital and cellular convexity[END_REF], Chassery [START_REF] Chassery | Discrete convexity: Definition, parametrization, and compatibility with continuous convexity[END_REF] and Ronse [START_REF] Ronse | Definition of convexity and convex hulls in digital images[END_REF]). Most of these definitions may be proved to be equivalent for simply connected sets [START_REF] Kim | Digital straightness and convexity[END_REF][START_REF] Kim | Digital convexity, straightness, and convex polygons[END_REF][START_REF] Ronse | Definition of convexity and convex hulls in digital images[END_REF][START_REF] Eckhardt | Digital lines and digital convexity[END_REF]. However, they failed to properly define the convex and concave parts of open digital curves (because a convex open digital curve may also be considered as a digital set that is not convex). In many applications such as line drawings processing, dealing with open digital curves is important. That is why we define in this paper convex and concave parts by means of the slope of the maximal digital straight segments, i.e. digital straight segments of a digital curve that cannot be extended neither at the front nor at the back (see [START_REF] Klette | Digital straitghness -a review[END_REF] for a review about digital straightness). This definition was first proposed in [START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF] and is also used in [START_REF] Feschet | Canonical representations of discrete curves[END_REF][START_REF] Provencal | Two linear-time algorithms for computing the minimum length polygon of a digital contour[END_REF]. In a practical point of view, this definition enables us to deal with any digital curve, which may correspond to the boundary of a given digital set or not, like digital spirals (Fig. 2). In a theoretical point of view, this definition appears to be quite natural since convexity is closely related to straightness: convex sets are defined by means of line segments and line segments are convex sets.

In section 2, we recall some definitions related to digital straightness and digital convexity. In section 3, we extend some important previous results about digital convexity and present new ones in order to precisely define the FP of a digital curve in section 4. We prove its main properties (existence and uniqueness, reversibility, faithful representation of the convex and concave parts) before concluding in section 5.

Definitions and Preliminaries

Two points P, Q ∈ Z 2 are 4-neighbors (resp. 8 neighbors) if and only if --→ P Q 1 = 1 (resp.

--→ P Q ∞ = 1).

For all k ∈ {4, 8}, a k-connected digital curve (also called k-arc), denoted by C n-1 and C 1 .

In the sequel, we define convex and concave digital curves by means of the slope of the maximal digital straight segments. That's why we need to define digital straightness first.

Digital Straightness

The digital straight line (DSL for short) D(a, b, µ) of slope a b and intercept µ is the set of points (x, y) ∈ Z 2 verifying µ ≤ axby < µ + ω with a, b, µ, ω integer and gcd(a, b) = 1 [START_REF] Reveillès | Géométrie Discrète, calculs en nombres entiers et algorithmique[END_REF]. The DSL is a 4-connected sequence of points if ω = |a| + |b| and a 8-connected sequence of points if ω = max (|a|, |b|).

We choose to work in this paper with the concept of 4-neighborhood, so that ω = |a| + |b| and k = 4. The results derived in the rest of the paper are applicable for 8-neighborhood but with few changes.

The points lying on the upper (resp. lower ) leaning line verifying ax-by = µ (resp. axby = µ + ω -1) are called the upper (resp. lower ) leaning points.

In the sequence, upper leaning points alternate with lower ones.

A digital straight segment (DSS for short) S(a, b, µ, ω) of slope a b and intercept µ is a connected part of the DSL D(a, b, µ, ω) containing at least three consecutive leaning points in a clockwise orientation when upper leaning points (in sequence order) are taken before the lower leaning points (in sequence order too). As shown in Fig. 1, the same set of points does not have the same parameters when scanning forward or backward. Actually, the upper (resp. lower) leaning line always lies on the left (resp. right) side of a little man walking along the DSS. As we will see in the next sections, this invariant is key to compute the FP of a digital curve from the leaning points of DSSs.

A upper (resp. lower) digital edge (also called pattern and reversed pattern in [START_REF] De Vieilleville | Revisiting digital straight segment recognition[END_REF]) is a DSS whose first and last points are upper (resp. lower) leaning points.

The DSS of Fig. 1 is actually a upper digital edge when scanning forward (a), but a lower digital edge when scanning backward (b).

Due to its definition, any part of a DSS is obviously a DSS. As a corollary, a part that is not a DSS cannot be contained in longer parts that are DSSs. Therefore, any DSS S of C can be defined as maximal iff all the parts C ′ of C containing S, i.e. such that S ⊂ C ′ , are not DSSs.

The set of all maximal DSSs (MSs for short, also called fundamental segments in [START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF]) that lie on a given digital curve is unique. The first point of any two distinct MSs cannot be identical because if it is, the shortest MS is necessarily contained in the longest one and is thus not maximal. Consequently, the MSs can be ordered without any ambiguity according to the position of their first point on the contour. Let us then denote by (S i ) i∈1,...,m the sequence of the MSs lying on C. For the sake of clarity the indices are taken modulo m for closed digital curves.

There exists an elegant algorithm that computes the set of MSs of a digital curve in linear time [START_REF] Feschet | Optimal time computation of the tangent of a discrete curve: Application to the curvature[END_REF][START_REF] Lachaud | Fast, accurate and convergent tangent estimation on digital contours[END_REF]. An illustration of the output of this algorithm is depicted in Fig. 2. The mechanism can be coarsely described as follows: given a MS, the next one is computed first by removing points from the back of the segment until it can be extended at the front and then by adding points at the front of the segment until it is maximal.

The key tasks are adding [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF][START_REF] Kovalevsky | New definition and fast recognition of digital straight segments and arcs[END_REF] and removing [START_REF] Lachaud | Fast, accurate and convergent tangent estimation on digital contours[END_REF] a point at one extremity of a DSS in constant time and space. In order to extend this definition to arbitrary digital curves, the usual approach consists in decomposing the digital curve into monotonic pieces (as done for instance in [START_REF] Provencal | Two linear-time algorithms for computing the minimum length polygon of a digital contour[END_REF]). However, this approach does not address special cases occuring in junctions. That is why we propose below a local definition based on the slope of the MSs. It extends the definition of [START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF]Definition 8.6] only stated for digital curves having points of increasing x-coordinates.

Definition 2 (Digital convexity from MSs). Two consecutive MSs

S i (a, b, µ, ω) and S i+1 (a ′ , b ′ , µ ′ , ω ′ ) make a convex turn iff ab ′ -a ′ b > 0, but make a concave turn iff ab ′ -a ′ b < 0.
A digital curve C is locally convex (resp. concave) everywhere if and only if any two consecutive MSs make a convex (resp. concave) turn. This last definition is valid for any open digital curves. For instance, the spiral-shaped digital curve of Fig. 2 is locally convex everywhere because the slope of any two consecutive MSs make a convex turn.

For sake of clarity, we merely say that a digital curve is convex (resp. concave) when it is locally convex (resp. concave) everywhere.

Note that a convex digital curve C is concave if the points are scanned in the reverse way and conversely. Definition 1 and Definition 2 are perfectly equivalent for digital curves having points of increasing x-coordinates as proven in [START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF]Theorem 8.1]. Due to the importance of this equivalence in the next sections, we recall this result in the following lemma:

Lemma 1 ([8, Theorem 8.1]). Let the points C 1 , C 2 , . . . , C n ∈ Z 2 of a digital curve C have increasing x-coordinates. Any two consecutive MSs of C make a convex (resp. concave) turn if and only if C is upper (resp. lower) convex.
In the rest of the paper, we refer to this lemma several times, because we always consider small parts of digital curves having no more than two MSs. Lemma 2 is intuitive and its proof straightforwardly stems from Definition 1 for the class of monotonic digital curves. Writing a proof for any digital curve is however not trivial and the simplest way to achieve this requires using Lemma 1.

Two consecutive MSs intersect in at least two distinct points

P, Q ∈ Z 2 so that |x P -x Q | = 1 or |y P -y Q | = 1.
Lemma 2. Any part of a convex (resp. concave) digital curve is convex (resp. concave).

Proof

Let C be a convex digital curve of n points (n > 1). The concave case is symmetric. If we can prove that the sequence of points C 1,n-1 is a convex digital curve, we can similarly prove that the sequence of points C 2,n is a convex digital curve. By induction, we can thus prove that any part of a convex digital curve is convex.

Let us now focus on C 1,n-1 . Its sequence of MSs is hardly the same as the one of C 1,n . More precisely, all the MSs, except the last one, are equal. As a consequence, we can focus on the part C k,n covered by the second-to-last and the last MS of C 1,n . If C k,n-1 contains only one MS, we are done, otherwise C k,n-1 contains only two MSs and it remains to show that they make a convex turn. This step is not trivial because the slope of the last MS of C k,n-1 may be smaller of greater than the second-to-last MS of C k,n .

Let us assume without loss of generality that the points of C k,n have increasing x-coordinates. The two MSs of C k,n make a convex turn because C 1,n is assumed to be convex. Due to Lemma 1, C k,n is thus upper convex. It is then obvious from Definition 1 that C k,n-1 is also upper convex, which means that

C k,n-1 is convex due to Lemma 1.
Due to the previous lemma, any part of a convex (resp. concave) digital curve is convex (resp. concave). As a corollary, a part that is not convex (resp. concave) cannot be contained in a convex (resp. concave) part. Therefore, as for DSSs, we can define any convex (resp. concave) part P of C as maximal if and only if all the parts C ′ of C containing P , i.e. such that P ⊂ C ′ , are not convex (resp. concave). The set of all maximal convex or concave parts that lie on a given digital curve is unique and provides a decomposition into maximal convex and concave parts that is studied in the following section.

Local convexity properties

In this section, we recall important local convexity properties that may be found in [START_REF] Debled-Rennesson | Detection of the Discrete Convexity of Polyominoes[END_REF]Theorem 9], [22, lemma 1] and [START_REF] Roussillon | What does digital straightness tell about digital convexity[END_REF]Lemma1]. They are crucial for either showing the existence and uniqueness of the FP or designing a linear-time algorithm that computes such a polygon, which is done in the next section.

Local criterion for checking convexity

In order to decide if two consecutive MS S i and S i+1 make a convex (resp. concave) turn it is actually enough to consider the point of S i+1 following the last point of S i , without considering the whole MS S i+1 : Theorem 1. Let S i (a, b, µ, ω) and S i+1 (a ′ , b ′ , µ ′ , ω ′ ) be two consecutive MSs lying along a digital curve C. Let P (x P , y P ) be the point of S i+1 following the last point of S i . The MSs S i and S i+1 make a convex (resp. concave) turn iff ax Pby P > µ + ω (resp. ax Pby P < µ -1).

Proof

Let C k,l be the sequence of points covered by S i and C k ′ ,l ′ be the sequence of points covered by S i+1 . Note that P = C l+1 . Let us assume without loss of generality that the points of C k,l ′ have increasing x-coordinates (see Section 2.2).

⇒ If ax Pby P > µ + ω, C k,l+1 is not lower convex due to the results of [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF].

The whole part C k,l ′ is thus not lower convex either. Due to Lemma 1, S i and S i+1 does not make a concave turn, but necessarily a convex one.

Similarly, if ax Pby P < µ -1, S i and S i+1 make a concave turn.

⇐ If S i and S i+1 make a convex turn, C k,l ′ is convex due to Definition 2.

The part C k,l+1 is thus convex too, due to Lemma 2. Since C k,l+1 is not a DSS, either ax Pby P < µ -1 or ax Pby P > µ + ω [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF]. The first case leads to a contradiction due to the forward implication. As a result, only the second one is true, i.e. ax Pby P > µ + ω.

If S i and S i+1 make a concave turn, we can similarly show that ax Pby P < µ -1.

Close results may be found in [START_REF] Debled-Rennesson | Detection of the Discrete Convexity of Polyominoes[END_REF]Theorem 9] and [22, In practice, the maximal convex and concave parts of a digital curve are retrieved in the course of the MSs computation. Let C k,l be the last MS S(a, b, µ, ω) and let C l+1 = P be the next point on the digital curve. If S belongs to a convex (resp. concave) part and ax P -by P < µ-1 (resp. ax P -by P > µ+ω), then S is the end of a maximal convex (resp. concave) part and the beginning of a concave (resp. convex) part (and would be depicted with a yellow bounding box).

Furthermore, Theorem 1 answers to Eckhardt's question [START_REF] Eckhardt | Polygonal representations of digital sets[END_REF]: how far one can decide whether a part of a digital curve is convex or not by a method that is as local as possible? The answer is actually that the smallest part required for checking convexity is given by a MS, plus at least one of the two points located just before and after this segment. Any smaller part of any digital curve is a DSS, and thus both convex and concave, which is useless.

To sum up, looking at the location of the two points M and P that respectively bound a MS S(a, b, µ, ω) from back and front, is a way of classifying the local geometry of the digital curve:

• if ax Mby M > µ + ω and ax Pby P > µ + ω (resp. ax Mby M < µ -1 and ax Pby P < µ -1 ), S belongs to a convex (resp. concave) part as shown in Fig. 4.a (resp. Fig. 4.b).

• if ax Mby M < µ -1 and ax Pby P > µ + ω (and conversely), S makes the transition between a convex and a concave part (Fig.

Local criterion for extracting dominant points

We go further and consider local digital convexity at a thinner scale in order to retrieve some local dominant points, defined as follows:

Definition 3 (dominant point). Any point C k that cannot be strictly contained in a upper (resp. lower) digital edge C i,j (i < k < j) is a convex (resp.

Proof

Let us assume that C i,j is convex. A similar reasoning leads to the proof about concave digital curves. Let us assume that C k,l is strictly included in C i,j , i.e.

i < k < l < j. The special cases i = k or l = j shorten some parts of the proof given below. Moreover, let us focus on the last upper leaning point of C k,l , denoted by C h , because the same applies for its first upper leaning point.

Due to Definition 3, the point C h is a convex dominant point iff there exists no upper digital edge C p,q strictly containing it (p < h < q). Since C k,l cannot be contained in a longer DSS having the same parameters, the part C k-1,l+1 is not a DSS [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF]. Since C p,q is a DSS, only three cases are possible: both C p and C q belong to S (i), C p belongs to S but not C q (ii), C q belongs to S but not C p (iii). Since the last two cases are similar, let us focus on the first two cases.

⇒ We have to prove that there does not exist any upper digital edge C p,q that strictly contains C h if ax Pby P ≥ µ + ω.

Let us assume without loss of generality that the points of C k,l have increasing x-coordinates. In case (i), the upper leaning point C h is obviously located above the straight segment [C p C q ] because C p and C q are both located strictly below the upper leaning line of S. As a consequence, C p,q is not an upper digital edge. In case (ii), the upper leaning point C h is located below the straight segment [C p C q ] if its slope is greater than the one of S. The point P = C l+1 , which belongs to C p,q , is such that ax Pby P ≥ µ + ω, i.e. is located below the lower leaning line of S. As a result, C p,q is clearly not convex due to Lemma 1 and is therefore not an upper digital edge (Fig. 5.b).

⇐ We have to prove now that ax Pby P ≥ µ + ω if there does not exist any upper digital edge C p,q that strictly contains C h .

If C p and C q both belong to C k,l , there does not exist any upper digital edge C p,q that strictly contains C h , for any value of ax Pby P . Let us assume now that C p belongs to C k,l while C q does not. Since C k,l cannot be contained in a longer DSS of same parameters, ax Pby P ≤ µ -1 or ax Pby P ≥ µ + ω [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF], but the first inequality cannot be true without raising some contradictions. Indeed, if ax Pby P = µ -1, C h is contained in a digital edge [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF], which raises a contradiction. Finally, if ax Pby P < µ -1, C p,q is not convex due to [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF] and Lemma 1, which raises again a contradiction.

A similar result (but not as general as Theorem 2) may be found in [START_REF] Roussillon | What does digital straightness tell about digital convexity[END_REF]Lemma1]. These results show that MSs are not enough to extract convex or concave dominant points of respectively convex or concave digital curves. A smaller part is often required. In convex (resp. concave) digital curves, it is actually enough to consider parts corresponding to DSSs S(a, b, µ, ω) bounded by two points M and P such that ax M -by M ≥ µ+ω (resp. ax M -by M ≤ µ-1)

and ax Pby P ≥ µ + ω (resp. ax Pby P ≤ µ -1):

• common parts of two consecutive MSs that have two bounding points M and P such that ax Mby M = µ + ω (resp. ax Mby M = µ -1) and ax Pby P = µ + ω (resp. ax Pby P = µ -1) [3, lemma 2] (Fig. 6.a).

• DSSs maximal at the back only that have two bounding points M and P such that ax Mby M > µ + ω (resp. ax Mby M < µ -1) and ax Pby P = µ + ω (resp. ax Pby P = µ -1) and conversely for DSSs maximal at the front only (Fig. 6.b).

• MSs that have two bounding points M and P such that ax M -by M > µ+ω (resp. ax M -by M < µ-1) and ax P -by P > µ+ω (resp. ax P -by P < µ-1)

(Fig. 6.c). A similar process is performed when points are removed from the back of the current DSS.

It turns out that either during the adding step or the removing step, any two consecutive retrieved points are the first and last point of a digital edge, which guarantees that the points of C lying between two convex dominant points are not retrieved. Fig. 7 shows what are the convex dominant points of the digital curve of Fig. 2. Note that if the points are scanning in the reverse way, the digital curve would be considered as concave and its concave dominant points would be similarly retrieved from the lower leaning points due to the orientation of the DSSs (see Section 2.1).

Faithful Polygon

In [START_REF] Dorksen-Reiter | Shape Representations of Digital Sets based on Convexity Properties[END_REF][START_REF] Eckhardt | Polygonal representations of digital sets[END_REF][START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF][START_REF] Dorksen-Reiter | A linear algorithm for polygonal representations of digital sets[END_REF], Eckhardt and Dorksen-Reiter looked for a reversible polygon Σ that faithfully represents a connected digital set S.

They looked for polygons such that:

• The vertices of Σ belong to S,

• S is the Gauss digitization of Σ,

• Σ respects the convex and concave parts of S.

It turns out that even if some digital sets admit such polygons (Euclidean convex hull of convex digital sets for instance), it is impossible to meet the three requirements at the same time for all sets S [START_REF] Dorksen-Reiter | Shape Representations of Digital Sets based on Convexity Properties[END_REF]. However, we show in this section that if we replace the second requirement by a weaker one, a faithful polygonal representation always exists. Moreover, we provide an online and linear-time algorithm that computes such a polygon.

Before considering arbitrary digital curves, we first focus on strictly convex or strictly concave digital curves.

Faithful Polygon of Convex or Concave Digital Curves

In a convex (resp. concave) digital curve C, the set of convex (resp. concave) dominant points is unique because the set of configurations involved in Theorem 2 is unique. Moreover, the points can be ordered in a sequence because they all belong to the sequence of points C. Our approach consists in linking these convex (resp. concave) dominant points by straight line segments as shown in Fig. 7. The resulting polygon is interesting because of the following theorem:

Theorem 3. In a convex (resp. concave) digital curve C, the sequence of convex (resp. concave) dominant points C 1 = C k0 , C k1 , . . . , C km = C n defines a sequence of upper (resp. lower) digital edges C k0,k1 , . . . , C km-1,km such that any two consecutive digital edges make a convex (resp. concave) turn.

Proof

Let us assume that C is a convex digital curve. A similar proof can be derived for concave digital curves.

We have to prove that:

(i) for all i ∈ 0, . . . , m -1, the part C ki,ki+1 is a digital edge.

(ii) for all i ∈ 1, . . . , m-1, the parts C ki-1,ki and C ki,ki+1 make a convex turn.

However, (i) straightforwardly comes from the algorithm given at the end of Section 3 for retrieving the convex dominant points of a convex digital curve.

It therefore remains to show (ii).

For all i ∈ 1, . . . , m -1, let us assume without loss of generality that C ki-1,ki lies in the first quadrant, i.e.

x C k i ≥ x C k i-1 and y C k i ≥ y C k i-1 .

Faithful Polygon of Arbitrary Digital Curves

We can now precisely define the FP of a digital curve and prove its main properties.

Definition 4 (Faithful polygon). Let C 1 i,j , C 2 i,j , . . . , C Λ i,j be the sequence of the maximal convex or concave parts of a digital curve C. The faithful polygon (FP) of C is defined by the concatenation of a subsequence of the convex or concave dominant points of the maximal convex or concave parts of C (see Fig. 8 for an illustration):

⊙ λ∈1,...,Λ (C λ kµ , C λ kµ+1 , . . . , C λ kν )
such that:

• If C λ i,j is convex, C λ kµ , C λ kµ+1 , .
. . , C λ kν are the convex dominant points of C λ i,j located between:

-C i if λ = 1 and the last upper leaning point of the first MS of C λ i,j

otherwise.

-C j if λ = Λ and the first upper leaning point of the last MS of C λ i,j otherwise.

• If C λ i,j is concave, C λ kµ , C λ kµ+1 , . . . , C λ kν are the concave dominant points of C λ i,j located between:

-C i if λ = 1 and the last lower leaning point of the first MS of C λ i,j

otherwise.

-C j if λ = Λ and the first lower leaning point of the last MS of C λ i,j otherwise. The FP of the wave-shaped digital curve of Fig. 3 is shown in Fig. 9. The FP of a digital curve has several interesting properties that are listed and proven below.

last MS of C λ i,j C k λ ν C k λ µ C k λ-1 ν C k λ+1 µ = last MS of C λ-1 i,j first MS of C λ i,j = first MS of C λ+1 i,j
Property 1. Any digital curve C that is not a DSS has a unique FP.

Proof

The decomposition into maximal convex and concave parts is unique and in each maximal convex (resp. concave) part, the set of convex (resp. concave) dominant points is also unique.

As shown in Fig. 10, note that a DSS has two different FPs depending on whether the DSS is considered as convex or concave. Let us assume that C λ i,j is a maximal convex part and C λ+1 i,j

is the maximal concave part that follows for some λ ∈ 1, . . . , Λ-1. The converse case is similar.

Moreover, let S(a, b, µ, ω) be the MS of inflection, i.e. the last MS of C λ i,j and the first one of C λ+1 i,j . Without loss of generality, we assume that it has points of increasing x-coordinates. The points C λ kν and C λ+1 kµ are respectively the first upper and last lower leaning point of S (Fig. 11).

C k λ ν C λ+1 kµ (a) C k λ ν C λ+1 kµ C k λ ν+1 C k λ ν-1 C λ+1 kµ+1 C λ+1 kµ-1 (b)

Proof

For all λ ∈ 1, . . . , Λ, for all κ ∈ µ, . . . , ν -1, C λ kκ,kκ+1 is easily retrieved from its two end points because it is a upper digital edge in convex parts and a lower digital edge in concave parts.

Conclusion, discussion and perspectives

In this paper, we have shown that any digital curve C has a reversible faithful polygon (FP) such that:

• Each vertex of the FP of C belongs to C.

• The FP exactly reflects the maximal convex and concave parts of C.

• C may be retrieved from its FP stored as a sequence of points.

An online and linear-time algorithm exists in order to extract the FP of a digital curve. It only uses well-known routines: adding a point to the front of a DSS [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF] and removing a point from the back of a DSS [START_REF] Lachaud | Fast, accurate and convergent tangent estimation on digital contours[END_REF]. It is thus really easy to implement once these routines are available. Moreover, retrieving a digital curve from its FP can also be achieved in linear-time.

A small translation of each vertex of the FP of a digital curve C is enough to compute the well-known minimum-perimeter polygon (MPP) of the dilatation of C by the closed unit square {(x, y) ∈ R 2 | max (|x|, |y|) ≤ 1 2 }. The FP is reversible and exactly reflects the convex and concave parts of the digital curve, whereas the MPP minimizes the number of inflection points required to represent the digital curve and is known to provide good estimators of tangent and length. The MPP derived from the FP of Fig. 9 is depicted in Fig. [START_REF] Feschet | Optimal time computation of the tangent of a discrete curve: Application to the curvature[END_REF]. Further details may be found in [START_REF] Roussillon | Local convexity properties of digital curves[END_REF]Section 5.3].

As a consequence, the proposed arithmetical algorithm (dedicated to the FP computation) can also compute the MPP and conversely, the combinatorial algorithm of Provençal and Lachaud [START_REF] Provencal | Two linear-time algorithms for computing the minimum length polygon of a digital contour[END_REF] (dedicated to the MPP computation)

1 http://liris.cnrs.fr/m2disco/index en.html?softwares can also compute the FP. The combinatorial approach of [START_REF] Brlek | Lyndon + christoffel = digitally convex[END_REF] has been shown to be about ten times faster than the arithmetical one [START_REF] Debled-Rennesson | Detection of the Discrete Convexity of Polyominoes[END_REF] in checking the convexity of a digital curve [1, Fig. 6]. We can thus expect the same difference in the FP or MPP extraction between the two approaches. However, our approach has the advantage of being purely local, contrary to the one of Provençal and Lachaud [START_REF] Provencal | Two linear-time algorithms for computing the minimum length polygon of a digital contour[END_REF]. Thus, it may lead to an algorithm solving the dynamic problem: update the polygonal representation during the displacement of a point of the digital curve.

Eventually, note that the parts of the digital curve highlighted in its FP or MPP does not reflect the visual parts of the original shape if the resolution is to high with respect to the scale of its main features or if some stochastic noise is introduced in the digitization. Several ways of copying with this problem can be followed while keeping an arithmetic approach that leads to fast algorithms with integer-only computations: (i) find a deformation process of the digital curve so that it sticks to the expected shape, like in digital deformable models [START_REF] De Vieilleville | Digital deformable model simulating active contours[END_REF], (ii) find a discrete simplification process of the digital curve in the manner of [START_REF] Latecki | Convexity rule for shape decomposition based on discrete contour evolution[END_REF] or (iii) work on sub-sampled versions of the initial digital curve as done in [START_REF] Kerautret | Multi-scale analysis of discrete contours for unsupervised noise detection[END_REF]. This work and its perspectives lead to think that digital convexity will help to design an efficient and accurate method dedicated to the extraction of perceptually meaningful parts.

Figure 1 :

 1 Figure 1: The same set of points may be viewed as two DSSs of different parameters when scanning forward or backward. The upper leaning line is depicted with blue, whereas the lower one is depicted with green. The three leaning points are always in a clockwise orientation when upper leaning points (in sequence order) are taken before the lower leaning points (in sequence order too).

Figure 2 :

 2 Figure 2: MSs of a digital curve. Each MS is depicted with a red bounding box.

2. 2 .

 2 Digital Convexity Let us focus now on the definition of digital convexity for open digital curves. The usual definition is based on the convex hull of a set of points and as a consequence is only satisfactory for closed digital curves or some classes of open digital curves like the monotonic ones in [8, Definition 8.4]: Definition 1 (Digital convexity from convex hull). Let the points C 1 , C 2 , . . . , C n ∈ Z 2 of a digital curve C have increasing x-coordinates. The digital curve C is upper (resp. lower) convex if and only if there is no point P ∈ Z 2 between the polygonal line linking the points of C and the upper (resp. lower) part of the convex hull of C.

  Since the coordinates of the points of any DSS monotonically increase or decrease, the x-coordinates (resp. y-coordinates) of the points of two consecutive MSs increase or decrease if |x Px Q | = 1 (resp. |y Py Q | = 1). Due to symmetries, we can assume that the points have increasing x-coordinates. As a result, we can arbitrarily use Definition 1 or Definition 2 in such cases due to Lemma 1.

  lemma 1]. A corollary of Theorem 1 is that each maximal convex or concave part of C is exactly covered by a sequence of consecutive MSs of C, i.e. its first and last point are respectively the first and last point of MSs of C.In Fig.3, the MSs of the maximal convex parts are depicted in blue and yellow, whereas the MSs of the maximal concave parts are depicted in green and yellow. The yellow MSs belong to both a maximal convex part and a maximal concave part.

Figure 3 :

 3 Figure 3: Decomposition of a digital curve into maximal convex and concave parts. A MS is depicted by a bounding box: blue if the MS belongs to a convex part only, green if it belongs to a concave part only, yellow if it belongs to both.

Figure 4 :

 4 Figure 4: Three different local configurations: MS in a convex part (a), in a concave part (b), in an inflection part (c).

Figure 5 :

 5 Fig. 5.a.

Figure 6 :

 6 Figure 6: DSSs S(a, b, µ, ω) that necessarily contain at least one convex dominant point: (a) common parts of two consecutive MSs, (b) DSSs maximal at the back or at the front, (c) MSs. The points (x, y) marked by a square are such that axby = µ + ω, whereas those marked by a cross are such that axby > µ + ω

Figure 7 :

 7 Figure 7: Sequence of convex dominant points linked into a red polygonal line.

Figure 8 :

 8 Figure 8: Illustration of the definition of the FP.

Figure 9 :

 9 Figure 9: FP of a digital curve in red.

Figure 10 : 2 .Property 3 .

 1023 Figure 10: Two possible representations of a DSS

Figure 11 :Property 4 .

 114 Figure 11: Illustration of the proof of Property 3. The MS of inflection is depicted with a red bounding box in (a). The points C λ k ν-1 and C λ+1 k µ+1 are located in either side of the dashed line passing through C λ kν and C λ+1 kµ , the first upper and last lower leaning points of the MS of inflection (b).

  Fig. 9 are the outputs of the program.

Figure 13 :

 13 Figure 13: MPP (in red) of a dilated digital curve (in blue).

  The point C ki+1 cannot be located above the straight line passing through C ki-1 and C ki without raising contradictions. Indeed, either C ki-1,ki+1 is a upper digital edge, which implies that C ki is not a convex dominant point, or C ki-1,ki+1 is not convex due to Definition 1 and Lemma 1, which also raises a contradiction. As a consequence, C ki+1 is located below the straight line passing through C ki-1 and C ki , i.e. C ki-1,ki and C ki,ki+1 make a convex turn.

	Any convex (resp. concave) digital curve C admits a convex (resp. concave)
	polygonal line that is obtained by linking the convex (resp. concave) dominant
	points of C by straight line segments (Fig. 7). Furthermore, this polygonal
	representation is reversible because any part of C lying between two consecutive
	convex (resp. concave) dominant points is a upper (resp. lower) digital edge,
	which can be retrieved in the first octant by the floor (resp. ceil) digitization of
	the straight line segment joining the two points.
	If C is not convex, we can independently compute the convex (resp. concave)
	dominant points of each maximal convex (resp. concave) part. Tricky issues
	occur in inflection parts, which belong to both a maximal convex part and a
	maximal concave part. Though, such parts are MSs and are thus arithmetically
	well defined. We will see below that a convex and a concave dominant point of
	two consecutive maximal convex and concave parts can be linked by a straight
	line segment joining the first upper (resp. lower) and last lower (resp. upper)
	leaning points of the MS of inflection.

Let us assume that C λ i,j is a maximal convex part and C λ+1 i,j

is the maximal concave part that follows for some λ ∈ 1, . . . , Λ-1. The converse case is similar.

For the sake of clarity, let us rename C k λ ν and C λ+1 kµ into M and P respectively.

Due to definition 4, M and P are respectively the first upper and the last lower leaning points of the same MS, assumed without loss of generality to have points of increasing x-coordinates (Fig. 12.a). From the algorithm of Debled and Reveilles [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF], we can therefore conclude that S ∪ L is a DSS and even more precisely an upper digital edge, because M , the first point of S, is also the first upper leaning point of S.

As a consequence, the part of C bounded by C λ kν = M and C λ+1 kµ = P can be retrieved from M and P : S is indeed the floor digitization of [M L], but without L and it remains to add P to end the drawing.

In practice, computing the FP of a digital curve C requires to check whether a given convex (resp. concave) part C i,j is maximal or not and to retrieve its sequence of convex (resp. concave) dominant points. This can be done online


Vincent Guedj

AND Boris Kolev

Nader Yeganefar

K Ähler-Einstein Fillings

Keywords: Kähler-Einstein fillings 2010 Mathematics Subject Classification. 32W20, 32Q20 Complex Monge-Ampère equation, Kähler-Einstein metrics, Local Moser-Trudinger inequality, pseudoconvex domains

published or not. The documents may come

The study of Einstein metrics is an important and classical subject in Riemannian geometry, see [START_REF] Besse | Einstein manifolds[END_REF]. The most popular framework is that of complete manifolds, either compact (without boundary) or noncompact. However, Einstein metrics on compact manifolds with boundary have also been investigated more recently, mainly in two directions which we now describe.

The first direction is that of conformally compact manifolds. Here, one starts with a compact manifold M with boundary ∂M . A complete Einstein metric on (the interior of) M is called conformally compact if after a suitable conformal transformation, it can be extended smoothly up to the boundary (think of the ball model of real hyperbolic space, or look at [START_REF] Biquard | Métriques d'Einstein asymptotiquement symétriques[END_REF] for the precise definition). This extension is not unique, but different extensions are easily seen to induce Riemannian metrics on the boundary which are in the same conformal class, called the conformal infinity of the conformally compact metric. One of the basic questions is then: a conformal class being fixed on the boundary, is it possible to find a conformally compact Einstein metric on M whose conformal infinity is the given conformal class? One then hopes to get links between the geometric properties of the inner metric and the conformal properties of the boundary; for more on this very a ctive research area, the reader may consult e.g. [START_REF] Biquard | Métriques d'Einstein asymptotiquement symétriques[END_REF][START_REF] Anderson | Unique continuation results for Ricci curvature and applications[END_REF].

We now come to the second direction, which has been explored far less than the first one and is more closely related to our present work. One starts again with a compact manifold M with boundary, and fixes some geometric structure on the boundary (for example a metric). The problem is then to find an Einstein metric on M which is smooth up to the boundary, and which induces the given geometric structure on ∂M . Assume for example that there is an Einstein metric on M with pinched negative curvature such that the boundary is convex and umbilical and let h 0 be the induced metric on ∂M . If h is a metric on ∂M which is sufficiently close to h 0 , it has been shown in [START_REF] Schlenker | Einstein manifolds with convex boundaries[END_REF] that there is an Einstein metric on M with negative Einstein constant such that the induced metric on ∂M is h. One of the interesting questions, which has not been fully clarified yet, is to know what "right" geometric structure has to be fixed on the boundary. [START_REF] Anderson | On boundary value problems for Einstein metrics[END_REF] considers the Dirichlet problem as in [START_REF] Schlenker | Einstein manifolds with convex boundaries[END_REF] (given a metric h on ∂M , can one find an Einstein metric on M inducing h on ∂M ?), studies the structure of the space of solutions and observes that this Dirichlet problem is not a well-posed elliptic boundary value problem. On the other hand, if one prescribes the metric and the second fundamental form of ∂M , then any Einstein metric on M is essentially unique by [START_REF] Anderson | Unique continuation results for Ricci curvature and applications[END_REF].

The main purpose of this article is to investigate similar questions in the context of compact Kähler manifolds with boundary. Let M be a compact Kähler manifold with strongly pseudoconvex boundary ∂M . The latter is a CR manifold whose geometric properties are encoded by the (conformal class of its) Levi form, a positive definite Hermitian form defined on the Levi distribution T C (∂M ) (the family of maximal complex subspaces within the real tangent bundle). The question we address is the following:

Problem. Can one find a Kähler-Einstein metric ω on M such that its restriction to the Levi distribution is conformal to the Levi form on T C (∂M )?

To simplify we restrict ourselves in the sequel to studying the case of a strongly pseudoconvex bounded open subset Ω of C n . One can then always make a conformal change of the Levi form so that the pseudo-Hermitian Ricci tensor (introduced by Webster) is a scalar multiple of the Levi form, i.e. ∂Ω is pseudo-Einstein (see [START_REF] Lee | Pseudo-Einstein structures on CR manifolds[END_REF]). Our problem is thus intimately related to the Riemannian questions recalled above.

It is well known that finding a Kähler-Einstein metric is equivalent to solving a complex Monge-Ampère equation. More specifically, letting µ denote the Lebesgue measure in C n normalized such that µ(Ω) = 1, we will be interested in the following Dirichlet problem : find a smooth strictly plurisubharmonic function ϕ on Ω which vanishes on the boundary ∂Ω and satisfies

(dd c ϕ) n = e -εϕ µ Ω e -εϕ dµ in Ω,
where ε ∈ {0, ±1} is a fixed constant. If ϕ is a solution of this problem, then it is easy to see that dd c ϕ is a Kähler-Einstein metric with the sign of the Einstein constant given by ε, and moreover its restriction to the Levi distribution is conformal to the Levi form on T C (∂Ω) (see section 2 for more details on this). Actually, if ε = 0, -1, then the Monge-Ampère equation above has always a solution by Theorem 1.1 in [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF], so that we will only consider the positive curvature case corresponding to ε = 1. Our main result is Theorem 1. Let Ω ⊂ C n be a bounded smooth strongly pseudoconvex domain. Then the complex Monge-Ampère problem

(M A) (dd c ϕ) n = e -ϕ µ Ω e -ϕ dµ
in Ω, and ϕ |∂Ω = 0 has a strictly plurisubharmonic solution which is smooth up to the boundary.

By the considerations of section 2, a consequence of this theorem is that our geometrical problem has a solution:

Corollary 2. Let Ω ⊂ C n be a bounded smooth strongly pseudoconvex domain. Then there is a smooth (up to the boundary) Kähler-Einstein metric on Ω with positive Einstein constant such that the restriction of the metric to the Levi distribution of ∂Ω is conformal to the Levi form.

Let us now say a few words about the proof of our main theorem. We will use a Ricci inverse iteration procedure, as described first in the compact Kähler setting by [START_REF] Keller | Ricci iterations on Kähler classes[END_REF] and [START_REF] Rubinstein | Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics[END_REF], whereas related results have recently been obtained in [START_REF] Berman | Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale[END_REF][START_REF]Measures of finite pluricomplex energy[END_REF] by other interesting approaches. More precisely, fix any smooth strictly plurisubharmonic function ϕ 0 on Ω which vanishes on the boundary, and for j ∈ N, let ϕ j be the unique strictly plurisubharmonic solution of the Dirichlet problem

(dd c ϕ j+1 ) n = e -ϕ j µ
whose existence is guaranteed by [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF]. We will then show that (ϕ j ) is bounded in C ∞ ( Ω), so that a subsequence converges in C ∞ ( Ω) to a smooth function which is seen to be a solution of (M A). To prove this boundedness in C ∞ , we proceed in several steps. First, there is a well-known functional F, defined on the space of plurisubharmonic functions, such that a function ϕ solves (M A) if and only if ϕ is a critical point of F (see subsection 3.2). A key result is that this functional is proper in the sense of Proposition 6. This properness result is in turn a consequence of a local Moser-Trudinger inequality (see Theorem 5, and also the recent independent results of [START_REF] Berman | Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale[END_REF][START_REF]Measures of finite pluricomplex energy[END_REF]). Next, we show that the sequence (F(ϕ j )) is bounded, so that by properness, the sequence (ϕ j ) has to live in some compact set. Here, com pactness is for the L 1 -topology in the class of plurisubharmonic functions with finite energy introduced in [START_REF] Benelkourchi | Plurisubharmonic functions with weak singularities, Complex analysis and digital geometry[END_REF]. Standard results from pluripotential theory then show that (ϕ j ) is uniformly bounded. To get boundedness in C ∞ , we will finally prove higher order a priori estimates, along the lines of [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF]. Now, let us deal with the uniqueness problem. For this, we impose some restrictions on Ω. First, we assume that Ω contains the origin and is circled; this means that Ω is invariant by the natural (diagonal) S 1 -action on C n . Next, if ϕ is a S 1 -invariant solution of the Monge-Ampère equation with Dirichlet boundary condition, we will say that Ω is (strictly) ϕ-convex if Ω is (strictly) convex in the Riemannian sense for the metric dd c ϕ. Note that being ϕ-convex has a priori nothing to do with being convex in the usual Euclidean sense in C n . We will prove Theorem 3. Let Ω ⊂ C n be a bounded smooth strongly pseudoconvex domain which is circled. Let ϕ be a smooth S 1 -invariant strictly plurisubharmonic solution of the complex Monge-Ampère problem (M A). If Ω is strictly ϕ-convex, then ϕ is the unique S 1 -invariant solution of (M A).

Observe that a S 1 -invariant solution always exists, as follows from the proof of Theorem 1: it suffices to start with an initial datum ϕ 0 which is S 1invariant, the approximants ϕ j will also be S 1 -invariant (by the uniqueness part of [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF]), hence so is any cluster value.

Remark 1. In the proof of this theorem, we will see that we can replace the ϕ-convexity hypothesis by a spectral assumption. Namely, if the first eigenvalue of the Laplace operator (of the metric ω ϕ = dd c ϕ) with Dirichlet boundary condition is strictly bigger than 1, then (M A) has a unique solution. By Corollary 1.2 in [START_REF] Guedj | A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds[END_REF], the condition on the Ricci curvature of ω ϕ and the strict ϕ-convexity imply this desired spectral estimate. However, [START_REF] Guedj | A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds[END_REF][Proposition 4.1] shows that this estimate may fail if Ω is merely strongly pseudoconvex.

To prove Theorem 3, we follow the approach proposed by Donaldson in the compact (without boundary) setting (see [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF][START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF]). The heuristic point of view is the following. The space of all plurisubharmonic functions on Ω which vanish on the boundary may be seen as an infinite dimensional manifold with a natural Riemannian structure. In the S 1 -invariant case, we may use a convexity result of Berndtsson [Ber06] to show that the functional F is concave along geodesics of this space. As a consequence, we show that S 1 -invariant solutions of (M A) coincide with S 1 -invariant maximizers of the functional F. Now, if ϕ and ψ are two S 1 -invariant solutions of (M A), then there exists a geodesic (Φ t ) 0≤t≤1 in the space of Kähler potentials on Ω vanishing on the boundary which joins ϕ to ψ. Therefore, the function t → F(Φ t ), being concave and attaining its maximum at t = 0 and t = 1, must be constant. In particular, its derivative vanishes, which implies that Φ0 has to satisfy a PDE involving the Laplacian of the metric dd c ϕ (see equation (5.1) below). If Ω is ϕ-convex, or more generally if the spectral hypothesis alluded to above is satisfied, then the only solution of this PDE is zero, so that Φ0 vanishes identically. From this, we may deduce that (Φ t ) is a constant geodesic, hence ϕ = ψ. Note that in the above argument, we have implicitly assumed that (Φ t ) is smooth, which may not be the case. For general continuous geodesics, the proof needs some modifications which will be given in section 6.

This uniqueness result has the following application. In [BB11, Conjecture 7.5], it is conjectured that if B is a ball in C n , then any solution of (M A) has to be radial. Theorem 3 shows that this is the case among S 1 -invariant solutions if the radius of the ball is not too large. Indeed, let B ⊂ C n be the ball of radius R > 0 centered at 0. Consider the radial function

ϕ = n + 1 π log 1 + z 2 -log 1 + R 2 .
In an affine chart, ϕ is the potential of the Fubini-Study metric on complex projective space P n (C), normalized to satisfy (M A) on B. Note that B may also be considered as a ball in P n (C), whose radius R F S with respect to the Fubini Study metric is

R F S = n + 1 π arctan R.
The diameter of P n (C) is then

D F S = π(n + 1)/2.
If R F S < D F S /2, then B is strictly convex in P n (C), that is B is strictly ϕ-convex (this is a well-known result, see for example the proof of [GKY11, Proposition 4.1]). By Theorem 3, ϕ is the unique S 1 -invariant solution of (M A), so that all such solutions are radial. We have thus proved Corollary 4. Let B be a ball in C n of radius 0 < R < 1. Then there is a unique S 1 -invariant solution to (M A) on B, and this solution is radial.

The plan of the paper is as follows. In section 2, we gather some wellknown facts on the geometry of pseudoconvex domains and show how our geometrical problem is related to the analytical problem of solving a complex Monge-Ampère equation with Dirichlet boundary condition. In section 3, we prove a local Moser-Trudinger inequality and use it to prove a properness result for the functional F. In section 4, we deal with the regularity problem of solutions of (M A), by getting higher order a priori estimates. This will allow us to prove Theorem 1 in subsection 4.4. In section 5, we obtain a variational characterization of solutions of (M A) in the S 1 -invariant case. Indeed, we show that S 1 -invariant solutions of (M A) are not only critical points of the functional F, but are exactly maximizers of F. Then we proceed to prove Theorem 3. In section 6, we comment on the difficulty of solving (M A) by the usual continuity method, and finally discuss the optimality of constants in the Moser-Trudinger inequality.

Acknowledgements. It is a pleasure to thank Robert Berman and Bo Berndtsson for stimulating discussions related to their joint work [START_REF] Berman | Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale[END_REF]. We would like also to thank Thierry Gallouët and Marc Herzlich for helpful discussions during the preparation of this paper.

Geometric context

2.1. The conformal class of the Levi form. Let Ω ⊂ C n be a bounded domain with smooth boundary. Fix a defining function ρ : C n → R for the boundary ∂Ω, i.e. ρ is a smooth function satisfying

Ω = {ρ < 0} , ∂Ω = {ρ = 0} ,
and dρ does not vanish on ∂Ω. Such a function ρ is not unique, but if ρ is another defining function for the boundary, then there is a smooth positive function u such that ρ = uρ.

Let now x ∈ ∂Ω be a fixed point, and denote by H x the maximal complex subspace of the tangent space T x ∂Ω. If J denotes the complex structure on C n (which is just multiplication by √ -1), then we have

H x = {v ∈ T x ∂Ω; Jv ∈ T x ∂Ω} .
The subspace H x has real dimension 2n -2, and as x varies, we get a distribution H ⊂ T ∂Ω, called the Levi distribution. If (z 1 , . . . , z n ) are the coordinates on C n , then it is easy to see that

H x = v = (v 1 , . . . , v n ) ∈ C n ; n i=1 ∂ρ ∂z i (x)v i = 0 . (2.1)
The Levi form is the Hermitian form defined for v, w ∈ H x by

L x (v, w) = n i,j ∂ 2 ρ ∂z i ∂ zj (x)v i wj .
It is clear from this expression that the Levi form actually depends on ρ, so talking about the Levi form is a slight abuse. However, if ρ = uρ is another defining function for the boundary (with u a smooth positive function), then we have

∂ 2 ρ ∂z i ∂ zj = u ∂ 2 ρ ∂z i ∂ zj + ∂u ∂z i ∂ρ ∂ zj + ∂u ∂ zj ∂ρ ∂z i + ρ ∂ 2 u ∂z i ∂ zj .
Moreover, by using the characterization (2.1) of H and the fact that ρ = 0 on ∂Ω, we infer, denoting by L the Levi form corresponding to ρ, that L = uL.

In other words, the Levi forms corresponding to different defining functions for the boundary differ only by a conformal factor. Thus, the geometrically interesting object on the boundary is the conformal class of the Levi form. We say that Ω is strongly pseudoconvex if the Levi form is a positive definite Hermitian form at each point of ∂Ω. Our previous discussion shows that this notion does not depend on the choice of a defining function for the boundary. Note also that by changing ρ to e cρ -1, where c > 0 is a large enough positive constant, we may assume that the Levi form is positive definite in a neighborhood of Ω, and not only on the Levi distribution.

2.2. Kähler metrics. We give here a brief review of Kähler metrics, mainly to set up some notations and conventions. For more details and proofs, the reader may consult e.g. [START_REF] Moroianu | Lectures on Kähler geometry[END_REF]. Although we will be dealing with domains in C n in the sequel, we consider a general complex manifold X of complex dimension n, and denote by J its complex structure.

2.2.1. Kähler form. A Riemannian metric g on X is called Hermitian if it is J-invariant, i.e. g(J•, J•) = g(•,
•). The C-bilinear extension of g to the complexified tangent bundle T X ⊗C will also be denoted by the same symbol g. The fundamental form associated to g is the real (1, 1)-form ω defined by

ω(•, •) = g(J•, •).
The metric g is called a Kähler metric if ω is a closed differential form; ω is then referred to as the Kähler form of g. It can be shown that g being a Kähler metric is equivalent to the complex structure J being parallel with respect to the Levi-Civita connection of g.

Let (z 1 , . . . , z n ) be local complex coordinates, and let

z 1 = x 1 + √ -1y 1 , . . . , z n = x n + √ -1y n
be the decomposition giving the corresponding real coordinates. As usual, for i = 1, . . . , n, we set

∂ ∂z i = 1 2 ( ∂ ∂x i - √ -1 ∂ ∂y i ), ∂ ∂ zi = 1 2 ( ∂ ∂x i + √ -1 ∂ ∂y i ), dz i = dx i + √ -1dy i , dz i = dx i - √ -1dy i ,
and for i, j = 1, . . . , n,

g i j = g( ∂ ∂z i , ∂ ∂ zj ).
Then the Kähler form is given locally by

ω = √ -1 n i,j=1 g i j dz i ∧ dz j .
Note that on C n , we have g i j = δ ij /2 for the canonical Euclidean metric.

2.2.2. Ricci curvature form. We denote by r the Ricci tensor of X as a Riemannian manifold. The Ricci form of X, to be denoted by Ric (ω) or simply Ric, is the (1, 1)-form associated to r, i.e.

Ric (ω)(•, •) = r(J•, •).
In local holomorphic coordinates, it can be shown that

Ric (ω) = - √ -1∂ ∂ log det g i j .
There follows that the Ricci form is a closed form. Moreover, its cohomology class is equal to 2πc 1 (X) , where c 1 (X) is the first Chern class of X. A Kähler metric ω on X is called Kähler-Einstein if for some constant λ ∈ R, we have Ric (ω) = λω.

2.2.3. Normalization of d c . We set

d c = 1 2π √ -1 (∂ -∂), so that √ -1∂ ∂ = πdd c
. This normalization is of common use in complex analytic geometry, having the following advantages: the positive current T = dd c log z has then Lelong number 1 at the origin in C n ; moreover the Fubini-Study form ω F S writes, in some affine chart C n ,

ω F S = dd c log 1 + z 2 .
Its cohomology class thus coincides with that of a hyperplane (as it should), having total volume

P n ω n F S = C n dd c log 1 + z 2 n = 1.
Note finally that Ric (ω F S ) = (n + 1)πω F S . Likewise, the Laplacian ∆ associated to a Kähler metric ω is defined as

∆ = tr (dd c ),
where tr denotes the trace with respect to ω. Hence, we have

∆ = - 1 π ∂ * ∂.
2.3. Kähler-Einstein metrics on strongly pseudoconvex domains.

Fix Ω ⊂ C n a bounded strongly pseudoconvex domain.

2.3.1. Associated complex Monge-Ampère equations. In this section, we show that finding Kähler-Einstein metrics is equivalent to solving a complex Monge-Ampère equation. We assume first that Ω is endowed with a Kähler metric ω which is smooth up to the boundary, and which satisfies the following normalized Einstein condition:

Ric (ω) = επω, where ε ∈ {0, ±1} (the somewhat unusual π factor is due to our normalization convention for the d c operator). We choose a smooth potential ϕ for ω, so that ω = dd c ϕ. Such a potential is unique up to the addition of a pluriharmonic function on Ω. We are going to see that ϕ satisfies a complex Monge-Ampère equation. As recalled in the previous section, if we denote by (g i j ) the components of the metric in coordinates, then the Ricci form is given by Ric (ω) = -πdd c log (det g i j ). Letting V 0 be the canonical volume form on C n , it is easily checked that ω n is equal to det (g i j )V 0 , up to a multiplicative constant. Therefore, we have the following intrinsic formula for the Ricci form:

Ric (ω) = -πdd c log ω n V 0 .
The Einstein condition on ω can then be written

dd c log (dd c ϕ) n V 0 + εϕ = 0.
Thus, there is a pluriharmonic function h such that

log (dd c ϕ) n V 0 + εϕ = h,
which we may write as a complex Monge-Ampère equation

(dd c ϕ) n = e -εϕ e h V 0 . (2.2)
Conversely, if ϕ is a smooth function satisfying the previous equation for some given pluriharmonic function h, and if ω = dd c ϕ is positive definite, we let the reader verify that ω is a Kähler-Einstein metric with Einstein constant επ.

Boundary conditions.

Let ρ be a boundary defining function for Ω, as described in section 2.1. Recall that L is the Levi form associated to ρ. The (1, 1)-form associated to L, that is L(J•, •), is equal to πdd c ρ with our normalization conventions. Let now ϕ be a smooth real valued function defined on Ω. On a collar neighborhood [-δ, 0] × ∂Ω of ∂Ω (where δ > 0 is fixed), we can write the expansion of ϕ in powers of ρ as follows: for all N ∈ N,

ϕ = ϕ 0 + ρϕ 1 + ρ 2 ϕ 2 + • • • + ρ N ϕ n + o(ρ N ).
(2.3) Here, the functions ϕ i are initially defined on {0} × ∂Ω ≃ ∂Ω, but we can view them as functions defined on the collar neighborhood [-δ, 0] × ∂Ω by setting, with obvious notations, ϕ i (ρ, x) = ϕ i (0, x). Thus, we have for example ϕ 0 = 0 if ϕ| ∂Ω = 0. From the expansion (2.3), we get

dd c ϕ = dd c ϕ 0 + ϕ 1 dd c ρ + dρ ∧ d c ϕ 1 + (dϕ 1 + 2ϕ 2 dρ) ∧ d c ρ + O(ρ).
Using the fact that dρ = d c ρ = 0 on the Levi distribution H (see the characterization (2.1) of H), the previous expansion implies

dd c ϕ| H = dd c ϕ 0 | H + ϕ 1 dd c ρ.
In particular, if ϕ 0 = 0, or more generally if dd c ϕ 0 = 0, then dd c ϕ| H is conformal to the Levi form.

Consider now the following geometrical problem: find a Kähler-Einstein metric ω on Ω such that its restriction to the Levi distribution is conformal to the Levi form. Our previous discussion shows that in order to solve this problem, it is enough to solve the following analytical problem: find a function ϕ such that

(1) dd c ϕ is positive definite, (2) ϕ satisfies the Monge-Ampère equation (2.2), (3) ϕ satisfies the Dirichlet boundary condition on ∂Ω, i.e. ϕ| ∂Ω = 0.

Indeed, the form ω = dd c ϕ is then a solution to the geometrical problem. Note that in the case of nonpositive Ricci curvature, which corresponds to ε = 0 or -1 in equation (2.2), the geometrical problem always has a solution by [CKNS85, Theorem 1.1]. We will therefore consider only the positive curvature case (ε = 1).

2.4. The strategy. In the sequel we let Ω = {ρ < 0} ⊂ C n be a bounded strongly pseudoconvex domain and µ denote the euclidean Lebesgue volume form in C n , normalized so that

µ(Ω) = 1.
We consider the following Dirichlet problem

(M A) (dd c ϕ) n = e -ϕ µ Ω e -ϕ dµ in Ω with ϕ |∂Ω = 0,
where ϕ is strictly plurisubharmonic and C ∞ -smooth up to the boundary of Ω.

We are going to solve (M A) by the an iterative process, solving for each j ∈ N the Dirichlet problem

(M A) j (dd c ϕ j+1 ) n = e -ϕ j µ Ω e -ϕ j dµ in Ω with ϕ j+1 |∂Ω = 0,
where ϕ 0 = ρ (we could actually start from any smooth plurisubharmonic initial data ϕ 0 with zero boundary values).

It follows from the work of Cafarelli-Kohn-Nirenberg-Spruck [CKNS85] that the Dirichlet problem (M A) j admits a unique plurisubharmonic solution ϕ j which is smooth up to the boundary. We are going to show that a subsequence of the sequence (ϕ j ) converges in C ∞ ( Ω) towards a solution ϕ of (M A).

In a compact setting this approach coincides with the time-one discretization of the Kähler-Ricci flow and was first considered by Keller [Kel09] and Rubinstein [START_REF] Rubinstein | Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics[END_REF] (see also [START_REF] Berman | Ricci iteration and Kähler-Ricci flow on log-Fano varieties[END_REF]).

Remark 2. As the proof will show, our result actually holds for any (normalized) volume form µ and with more general boundary values.

Energy estimates

We now move on to showing that the sequence (ϕ j ) is relatively compact in C ∞ ( Ω). The proof reduces to establishing a priori estimates. We first show that one has a uniform a priori control on the energy of the solutions.

Local Moser-Trudinger inequality. The following local Moser-Trudinger type inequality is of independent interest. 1

Theorem 5. There exists 0 < β n < 1 and C > 0 such that for all smooth plurisubharmonic functions ϕ in Ω with ϕ |∂Ω = 0,

Ω e -ϕ dµ ≤ C exp (β n |E(ϕ)|) ,
1 While we were finishing the writing of this paper, two preprints appeared [START_REF] Berman | Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale[END_REF][START_REF]Measures of finite pluricomplex energy[END_REF] which propose similar inequalities with different and interesting proofs.

where E(ϕ) = 1

n+1 Ω ϕ (dd c ϕ) n . We refer the reader to [Mos71, Ono82, Tia97, Tia00, CL04, PSSW08] for related results both in a local and global context. The proof we propose is new and relies on pluripotential techniques, as developed in [BT82, Kol98, Ceg98, Zer01, GZ05, BGZ09].

Proof. Recall that the Monge-Ampère capacity has been introduced by Bedford and Taylor in [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]. By definition the capacity of a compact subset

K ⊂ Ω is Cap(K) := sup K (dd c u) n ; u plurisubharmonic in Ω with 0 ≤ u ≤ 1 .
We will use the following useful inequalities. For any γ < 2 there exists C γ > 0 such that for all K ⊂ Ω,

µ(K) ≤ C γ exp - γ Cap(K) 1/n (3.1)
(see e.g. [START_REF] Zeriahi | Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions[END_REF]). For all smooth plurisubharmonic functions ϕ in Ω with zero boundary values, for all t > 0,

Cap(ϕ < -t) ≤ (n + 1) |E(ϕ)| t n+1 , where E(ϕ) := 1 n + 1 Ω ϕ(dd c ϕ) n .
For the latter inequality, we refer the reader to Lemma 2.2 in [START_REF] Ahag | Partial pluricomplex energy and integrability exponents of plurisubharmonic functions[END_REF].

We infer

Ω e -ϕ dµ = -1 + +∞ 0 e t µ(ϕ < -t)dt ≤ C +∞ 0 exp(t -λt 1+1/n )dt,
where λ := γ (n + 1) 1/n |E(ϕ)| 1/n . We let the reader check that the function h(t) = t -λt 1+1/n attains its maximum value at point

t c = λ -n (1 + 1/n) -n . Moreover h(t) ≤ -t for t ≥ 4 n t c . This shows that +∞ 0 exp(t -λt 1+1/n )dt ≤ 4 n t c exp(h(t c )) + +∞ 4 n tc exp(-t)dt ≤ 4 n t c exp t c n + 1 + 1.
Using the definition of λ and the formula defining t c , we arrive at

+∞ 0 exp(t -λt 1+1/n )dt ≤ c n |E(ϕ)| exp β ′ n |E(ϕ)| + 1, where β ′ n = 1 γ n (1 + 1/n) n .
We can fix e.g. γ = 1 so that β ′ n < 1 for all n ≥ 1. Moreover the desired inequality is obtained by choosing β n so that β ′ n < β n < 1 and enlarging the constant C. Remark 3. Note for later use that the same proof yields an inequality

Ω e -Aϕ dµ ≤ C A exp(β A |E(ϕ)|), (3.2)
where

β A := A n+1 γ n (1 + 1/n) n
is smaller than 1 only if A = A n is not too large. When n = 1, the critical value is A = 2. This is related to a theorem of Bishop as we shall see in section 4.4.

It follows from the recent work [START_REF] Ahag | Partial pluricomplex energy and integrability exponents of plurisubharmonic functions[END_REF] that the optimal exponent γ is actually 2n, improving the bound 2 obtained in [START_REF] Zeriahi | Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions[END_REF], hence also enlarging the allowed constant A n above, when n > 1.

3.2. Properness. We let

E(ϕ) := 1 n + 1 Ω ϕ(dd c ϕ) n
denote the energy of a plurisubharmonic function ϕ and set

F(ϕ) := E(ϕ) + log Ω e -ϕ dµ .
Recall that the energy functional is a primitive of the complex Monge-Ampère operator, namely if ψ s is a curve of plurisubharmonic functions with zero boundary values, then Inspired by techniques from the calculus of variations, it is thus natural to try and maximize the functional F so as to build a critical point. This usually requires the functional to be proper in order to be able to restrict to compact subsets of the space of functions involved. It follows from the Moser-Trudinger inequality (Theorem 5) that the functional F is indeed proper, in the following strong sense: Proposition 6. There exists a > 0, b ∈ R such that for all smooth plurisubharmonic function ψ in Ω, with zero boundary values,

F(ψ) ≤ aE(ψ) + b.
T : ϕ ∈ T → ψ ∈ T
denote the operator such that ψ = T (ϕ) is the unique solution of ( * ). Observe that solving (M A) is equivalent to finding a fixed point of T .

The key to the dynamical construction of solutions to (M A) lies in the following monotonicity property: Proposition 7. For all ϕ ∈ T ,

F(T ϕ) ≥ F(ϕ)
with strict inequality unless T ϕ = ϕ.

Proof. Fix ϕ ∈ T and set ψ := T ϕ. Recall that

F(ϕ) = E(ϕ) + log Ω e -ϕ dµ and E(ψ) -E(ϕ) = 1 n + 1 n j=0 Ω (ψ -ϕ)(dd c ψ) j ∧ (dd c ϕ) n-j .
It follows from Stokes theorem that for all j,

(ψ -ϕ)(dd c ψ) j ∧ (dd c ϕ) n-j = (ψ -ϕ)(dd c ψ) n + d(ψ -ϕ) ∧ d c (ψ -ϕ) ∧ S,
where S is a positive closed form of bidegree (n -1, n -1). Thus

E(ψ) -E(ϕ) ≥ 1 n + 1 Ω (ψ -ϕ)(dd c ψ) n .
We now set

φ := ϕ + log[ e -ϕ dµ], ψ := ψ + log[ e -ψ dµ],
and µ ϕ := e -φµ, µ ψ := e -ψ µ.

Note that the latter are probability measures in Ω with (dd c ψ) n = µ ϕ . It follows from the definition of F and our last inequality that

F(ψ) -F(ϕ) ≥ Ω ( ψ -φ)dµ ϕ = Ω F log F dµ ψ ,
where F = e ψ-φ, hence the latter quantity denotes the relative entropy of the probability measures µ ϕ , µ ψ . It follows from the convexity of -log that

Ω -log[F -1 ] F dµ ψ ≥ -log Ω F -1 F dµ ψ = 0,
with strict inequality unless F = 1 almost everywhere, i.e. φ = ψ.

Observe finally that since ψ and ϕ both have zero boundary values, the equality φ = ψ can only occur when ϕ ≡ ψ, i.e. when ϕ = T ϕ is a fixed point of T , as claimed.

We infer that the energies E(ϕ j ) of the solutions ϕ j of (M A) j-1 are uniformly bounded:

Corollary 8. The sequence (F(T j ϕ 0 )) j is bounded, hence so is (E(T j ϕ 0 ) j .

Proof. Fix ϕ 0 ∈ T (for example ϕ 0 = ρ) and set ϕ j = T j ϕ 0 . Observe that E(ϕ j ) ≤ 0 since ϕ j ≤ 0, hence it suffices to establish a bound from below. The previous proposition insures that the sequence F(T j ϕ 0 )) j is increasing. It follows from Proposition 6 that F(ϕ 0 ) ≤ F(T j ϕ 0 ) ≤ aE(T j ϕ 0 ) + b ≤ b so that the energies E(T j ϕ 0 ) are uniformly bounded.

Higher order estimates

4.1. Uniform a priori estimates. Recall that ϕ j is a smooth plurisubharmonic solution of (M A) j-1 . Its Monge-Ampère measure thus satisfies

(dd c ϕ j ) n = f j µ, with f j = e -ϕ j-1 µ Ω e -ϕ j-1 dµ .
It follows from the previous section that the ϕ ′ j s have uniformly bounded energy. Thus they form a relatively compact family (for the L 1 -topology) in the class E 1 (Ω) of plurisubharmonic functions with finite energy (see [START_REF] Benelkourchi | Plurisubharmonic functions with weak singularities, Complex analysis and digital geometry[END_REF]). When the complex dimension is n = 1, the latter is the class of negative plurisubharmonic functions with zero boundary values and whose gradient is in L 2 ; since (normalized) plurisubharmonic functions are uniformly L 2 , the family (ϕ j ) is thus included in a finite ball of the Sobolev space W 1,2 . In higher dimension, the class E 1 (Ω) is a convenient substitute for the Sobolev spaces, we refer the reader to [START_REF] Benelkourchi | Plurisubharmonic functions with weak singularities, Complex analysis and digital geometry[END_REF] for more details.

We simply recall here that functions in E 1 (Ω) have zero Lelong numbers. For such a function ψ, Skoda's integrability theorem [START_REF] Skoda | Sous-ensembles analytiques d'ordre fini ou infini dans C n[END_REF] ensures that e -ψ is in L q for all q > 1. Since the family (ϕ j ) is moreover relatively compact, Skoda's uniform integrability theorem [START_REF] Zeriahi | Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions[END_REF] insures that the densities f j 's satisfy

Ω f 2 j dµ ≤ C
for some uniform constant C > 0.

Recall now the following fundamental result due to Kolodziej [Kol98]: if ψ is a smooth plurisubharmonic function in Ω with zero boundary values and such that (dd c ψ) n = f dµ

where f ∈ L 2 (µ), then ψ L ∞ (Ω) ≤ C f ,
where the constant C f only depends on Ω and f L 2 . Applying this to ψ = ϕ j yields:

Lemma 9. For all j ∈ N,

-C 0 ≤ ϕ j ≤ 0 (4.1)
for some uniform constant C 0 > 0.

4.2. C 2 -a priori estimates. The goal of this section is to establish the following a priori estimates on the Laplacian of the solutions to (M A) j-1 .

Theorem 10. There exists C > 0 such that for all j ∈ N,

sup Ω |∆ϕ j | ≤ C.
These estimates are "almost" contained in [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF], however hypothesis (1.3) on p. 213 is not satisfied, hence neither [CKNS85, Theorem 1.1] nor [CKNS85, Theorem 1.2] can be applied to our situation.

We nevertheless follow their proof as organized by S. Boucksom [START_REF] Boucksom | Monge-Ampère equations on complex manifolds with boundary[END_REF], explaining some of the necessary adjustments. It will be a consequence of the following series of lemmas.

Lemma 11. There exists

C 1 > 0 such that sup ∂Ω |∇ϕ j | ≤ C 1 .
Proof. It follows from the order zero uniform estimates (4.1) that

(dd c ϕ j ) n ≤ e C 0 µ in Ω.
Let u denote the unique smooth plurisubharmonic function in Ω such that (dd c u) n = e C 0 µ in Ω with u |∂Ω ≡ 0.

The latter exists by [CKNS85, Theorem 1.1]. It follows from the comparison principle that

u ≤ ϕ j ≤ 0 in Ω. This yields the desired control of ∇ϕ j on ∂Ω.

Lemma 12. There exists C 2 > 0 such that

sup Ω |∆ϕ j | ≤ C 2 (1 + sup ∂Ω |∆ϕ j |).
Proof. We let ∆ j denote the Laplace operator with respect to the Kähler form ω j = dd c ϕ j , while ∆ denotes the euclidean Laplace operator. We claim that for all j ≥ 1, ∆ j {log ∆ϕ j + ϕ j-1 } ≥ 0. (4.2) Assuming this for the moment we show how to derive the desired control on ∆ϕ j . Let z j ∈ Ω be a point which realizes the maximum of the function

h j := ϕ j + ϕ j-1 + log ∆ϕ j .
It follows from (4.2) that z j ∈ ∂Ω, otherwise ∆ j h j (z j ) ≤ 0 contradicting ∆ j h j ≥ ∆ j ϕ j > 0.

We infer from Lemma 9 that for all w ∈ Ω, log ∆ϕ j (w) ≤ 2C 0 + h j (z j ) ≤ 2C 0 + log sup ∂Ω ∆ϕ j , which yields the desired upper bound.

It remains to establish (4.2). We shall need the following local differential inequality which goes back to the works of Aubin and Yau: if ω is an arbitrary Kähler form and β = dd c z 2 denotes the euclidean Kähler form, then

∆ ω log tr β (ω) ≥ - tr β (Ricω) tr β (ω) . (4.3)
We apply this inequality to ω = ω j = dd c ϕ j . Observe that Ric(ω j ) = ω j-1 since (dd c ϕ j ) n = e -ϕ j-1 e c j dV. Observe that tr β (ω j-1 ) tr β (ω j ) = ∆ β (ϕ j-1 ) tr β (ω j ) ≤ ∆ j (ϕ j-1 ).

Combined with (4.3), this yields ∆ j log tr β (ω j ) ≥ -∆ j (ϕ j-1 ),

whence (4.2).

Lemma 13. There exists

C 3 > 0 such that sup ∂Ω D 2 ϕ j ≤ C 3 (1 + sup Ω |∇ϕ j | 2 ).
Proof. This follows from a long series of estimates which are the same as those of [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF], up to minor modifications. We only sketch these out, following the proof of [Bou11, Lemma 7.17]. To fit in with the notations of [START_REF] Boucksom | Monge-Ampère equations on complex manifolds with boundary[END_REF], we set ψ = ϕ j -ρ and η = dd c ρ so that ψ is a η-psh function (still) with zero boundary values on ∂Ω such that

(η + dd c ψ) n = e -ψ e F η n
where F is some smooth density. Our problem is thus equivalent to showing an a priori estimate sup

∂Ω D 2 ψ ≤ C 3 (1 + sup Ω |∇ψ| 2 ),
where C 3 is under control. Fix p ∈ Ω. It is classical that one can choose complex coordinates (z j ) 1≤j≤n so that p = 0 and

ρ = -x n + ℜ   n j,k=1 a jk z j zk   + O(|z| 3 )
where z j = x j + iy j . We set for convenience

t 1 = x 1 , t 2 = x 1 , . . . , t 2n-1 = y n , t 2n = x 2n .
Let (D j ) be the dual basis of dt 1 , . . . , dt 2n-1 , -dρ so that for j < 2n,

D j = ∂ ∂t j - ρ t j ρ xn ∂ ∂x n and D 2n = - 1 ρ xn ∂ ∂x n
Step 0: bounding the tangent-tangent derivatives. Observe that the D j 's commute and are tangent to ∂Ω for j < 2n, we thus have a trivial control on the tangent-tangent derivatives at p = 0,

D i D j ψ(0) = 0, for 1 ≤ i, j < 2n.
Step 1: bounding the normal-tangent derivatives. Set K = sup ∂Ω |∇ψ|. We claim that for all 1 ≤ i < 2n

|D i D 2n ψ(0)| ≤ C(1 + K),
for some uniform constant C > 0.

Let h be the smooth function in Ω with zero boundary values such that

∆ η h := n dd c h ∧ η n-1 η n = -n in Ω.
The proof requires the construction of a barrier b = ψ + εh -µρ 2 such that

0 ≤ b and ∆ ψ b := n dd c b ∧ (η + dd c ψ) n-1 (η + dd c ψ) n ≤ - 1 2 tr ψ (η) in B,
where B is a half ball centered at p = 0 of positive radius and ε, µ > 0 are under control. This can be done exactly as in [Bou11, Lemma 7.17, Step 1], as the only information needed is that (η + dd c ψ) n is uniformly bounded from above by Cη n , which follows here from our C 0 -estimate.

One then shows the existence of uniform constants µ 1 , µ 2 > 0 such that the functions v

± := K(µ 1 + µ 2 |z| 2 ) ± D j ψ both satisfy 0 ≤ v ± on B and ∆ ψ v ± ≤ 0 in B.
It follows then from the maximum principle that v ± ≥ 0 in B so that

D 2n v ± (0) ≥ 0 since v ± (0) = 0. Thus |D 2nj ψ(0)| ≤ CK(1 + D 2n b(0)) ≤ C ′ (1 + K), as claimed.
Step 2: bounding the normal-normal derivatives. This is somehow the most delicate estimate. Set again K = sup ∂Ω |∇ψ|. We want to show that D 2 2n ψ(0) ≤ C(1 + K 2 ) for some uniform constant C > 0. Using previous estimates on D i D j ψ(0), it suffices to show that

|ψ zn zn (0)| ≤ C(1 + K 2 ). Recall that det ρ z i zj (0) + ψ z i zj (0) 1≤i,j≤n = e -ψ(0)+F (0)
is bounded from above, and for i < n,

|ψ z i zn (0)| ≤ C(1 + K).
Expanding the determinant with respect to the last row thus yields the expected upper bound, provided we can bound from below the (n -1, n -1)minor det ρ z i zj (0) + ψ z i zj (0) 1≤i,j≤n-1 .

A (by now) classical barrier argument shows that dd c ϕ = η + dd c ψ is uniformly bounded from below by εη on the complex tangent space to ∂Ω (see [START_REF] Boucksom | Monge-Ampère equations on complex manifolds with boundary[END_REF]Lemma 7.16] which can be used since ϕ j is uniformly bounded).

Lemma 14. There exists C 4 > 0 such that

sup Ω |∇ϕ j | ≤ C 4 .
Proof. It follows from previous estimates that sup

Ω ∆ϕ j ≤ C 1 + sup Ω |∇ϕ j | 2 .
Assume that sup Ω |∇ϕ j | is unbounded. Up to extracting and relabelling, this means that

M j := |∇ϕ j (x j )| = sup Ω |∇ϕ j | → +∞ where x j ∈ Ω converges to a ∈ Ω. We set ψ j (z) := ϕ j (x j + M -1 j z).
This is a sequence of uniformly bounded plurisubharmonic functions which are well defined (at least) in a half ball B around zero and satisfy

|∇ψ j (0)| = 1 and sup B ∆ψ j ≤ C.
We infer that the sequence (ψ j ) is relatively compact in C 1 , hence we can assume that (up to relabeling) ψ j → ψ ∈ C 1 (B) where ψ is plurisubharmonic and satisfies ∇ψ(0) = 1.

If a ∈ ∂Ω, it follows from the proof of Lemma 11 that ψ ≡ 0, contradicting ∇ψ(0) = 1. Therefore a ∈ Ω, so we can actually assume that B is a ball of arbitrary size, hence ψ can be extended as a plurisubharmonic function on the whole of C n . Since ϕ j is uniformly bounded, so are ψ j and ψ. Thus ψ has to be constant, contradicting ∇ψ(0) = 1. 4.3. Evans-Krylov theory. It follows from Schauder's theory for linear elliptic equations with variable coefficients that it suffices to obtain a priori estimates

ϕ j 2,α ≤ C (4.4)
for some positive exponent α > 0, in order to obtain a priori estimates

ϕ j k+2,α ≤ C k (4.5) at all orders k ∈ N.
Here

h k,α := k j=0 sup Ω D j h + sup z,w∈Ω,z =w D k h(z) -D k h(w) |z -w| α
denotes the norm associated to the Hölder space of functions h which are ktimes differentiable on Ω with k th -derivative Hölder-continuous of exponent α > 0. Moreover the Evans-Krylov theory (as simplified by Trudinger) can be adapted to the case of complex Monge-Ampère equations, showing that the a priori estimates (4.4) follow directly from Theorem 10. We refer the reader to [START_REF] Blocki | The complex Monge-Ampère equation on compact Kähler manifolds[END_REF] for a detailed presentation of this material.

Conclusion.

It follows from the previous sections that the sequence (ϕ j ) is relatively compact in C ∞ ( Ω). We let K denote the set of its cluster values. We infer from Proposition 7 that the functional F is constant on K: for all ψ ∈ K, F(ψ) = lim j→+∞ ր F(T j ϕ 0 ). Now K is clearly T -invariant, hence F(T ψ) = F(ψ) for all ψ ∈ K. Thus Proposition 7 again insures that T ψ = ψ, i.e. ψ is a solution of (M A).

As explained earlier, this is equivalent to saying that there exists a Kähler-Einstein metric ω = dd c ϕ with Ric(ω) = πω and prescribed values on the boundary of Ω, hence we have solved our geometrical problem.

Uniqueness

Recall that (M A) is the Euler-Lagrange equation of the functional

F(ϕ) := E(ϕ) + log Ω e -ϕ dµ .
If a smooth strictly plurisubharmonic function ϕ with zero boundary values maximizes F, then it is a critical point of F hence ϕ is a solution of (M A). Indeed for any smooth function v

d dt F(ϕ + tv) |t=0 = Ω v(dd c ϕ) n -Ω ve -ϕ dµ Ω e -ϕ dµ = 0, thus (dd c ϕ) n = e -ϕ µ/ Ω e -ϕ dµ .
Our purpose here is to show that the converse holds true when Ω satisfies an additional symmetry property. 5.1. Continuous geodesics. In the setting of compact Kähler manifolds, Mabuchi [START_REF] Mabuchi | Some symplectic geometry on compact Kähler manifolds. I[END_REF], Semmes [START_REF] Semmes | Complex Monge-Ampère and symplectic manifolds[END_REF] and Donaldson [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF] have shown that the set of all Kähler metrics in a fixed cohomology class has the structure of an infinite Riemannian manifold with non negative curvature. The notion of geodesic joining two Kähler metrics plays an important role there and we refer the reader to [START_REF] Chen | The space of Kähler metrics[END_REF] for more information on this.

Our purpose here is to consider similar objects for pseudoconvex domains in order to study the uniqueness of solutions to (M A). Let A denote the annulus A = {ζ ∈ C / 1 < |ζ| < e} and fix two functions φ 0 , φ 1 which are plurisubharmonic in Ω, continuous up to the boundary, with zero boundary values. We let G denote the set of all plurisubharmonic functions Ψ on Ω×A which are continuous on Ω × Ā and such that Proof. The invariance by rotations (i) follows from the corresponding invariance property of the family G. The continuity and boundary properties (ii) follow standard arguments which go back to Bremermann [START_REF] Bremermann | On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of Šilov boundaries[END_REF] and Walsh [START_REF] Walsh | Continuity of envelopes of plurisubharmonic functions[END_REF].

Ψ |∂Ω×A ≡ 0 and Ψ |Ω×∂A ≤ φ,
The maximality property (iii) is a consequence of Bedford-Taylor's solution to the homogeneous complex Monge-Ampère equation on balls, through a balayage procedure: by Choquet's lemma, the sup can be achieved along an increasing sequence which is maximal on an arbitrary ball B ⊂ Ω × A, one then concludes by using the continuity property of the complex Monge-Ampère operator along increasing sequences [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF].

Definition 16. Set Φ t (z) = Φ(z, e t ). The continuous family (Φ t ) 0≤t≤1 is called the geodesic joining φ 0 to φ 1 .

Recall that

E(ϕ) := 1 n + 1 Ω ϕ(dd c ϕ) n
denotes the energy of a plurisubharmonic function ϕ.

Lemma 17. Let (Φ t ) 0≤t≤1 be a continuous geodesic. Then t → E(Φ t ) is affine.

Proof. We let the reader verify that if (z,

ζ) → Φ(z, ζ) is a continuous plurisubharmonic function in Ω × A, then dd c ζ E • Φ = π * (dd c z,ζ Φ) n+1
, where π : Ω × A → A denotes the projection onto the second factor.

It thus follows from Proposition 15 that ζ ∈ A → E •Φ(ζ) ∈ R is harmonic in ζ. The same proposition insures that it is also invariant by rotation, hence it is affine in t = log |ζ|.

Variational characterization.

We now make an additional hypothesis of S 1 -invariance in order to use an important result by Berndtsson [START_REF] Berndtsson | Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains[END_REF]. Namely we assume here below that Ω is circled, i.e. Ω contains the origin and is invariant under the rotations z → e iθ z, and φ 0 , φ 1 are S 1 -invariant, i.e. φ i (e iθ z) = φ i (z).

Under this assumption, it follows from

[Ber06, Theorem 1.2] that t → -log Ω e -Φt dµ is a convex function of t if (Φ t ) is a continuous geodesic.
Proposition 18. Assume Ω is circled and let ϕ be a S 1 -invariant solution of (M A). Then F(ϕ) ≥ F(ψ), for all S 1 -invariant plurisubharmonic functions ψ in Ω which are continuous up to the boundary, with zero boundary values.

Proof. Let (Φ t ) 0≤t≤1 denote the geodesic joining φ 0 := ϕ to φ 1 := ψ. It follows from the above mentioned work of Berndtsson [Ber06] that t → -log e -Φt dµ is convex, while we have just observed that

t → E(Φ t )
is affine, thus t → F(Φ t ) is concave. It therefore suffices to show that the derivative of F(Φ t ) at t = 0 is non positive to conclude that F(ϕ) = F(Φ 0 ) ≥ F(Φ t ) for all t, in particular at t = 1 where it yields F(ϕ) ≥ F(ψ). When t → Φ t is smooth, a direct computation yields, for t = 0,

d dt (F(Φ t )) = Ω Φt (dd c Φ t ) n -e -Φt µ/( e -Φt dµ) = 0 since Φ 0 = ϕ is a solution of (M A).
For the general case, one can argue as in the proof of Theorem 6.6 in [START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF].

Corollary 19. A smooth S 1 -invariant plurisubharmonic function ϕ : Ω → R with zero boundary values is a solution of (M A), i.e. satisfies

(dd c ϕ) n = e -ϕ µ Ω e -ϕ dµ
in Ω

if and only if it maximizes the functional F.

Uniqueness of solutions.

The purpose of this section is to establish a uniqueness result for (M A). Recall that if ϕ is a solution of (M A), we say that Ω is strictly ϕ-convex if Ω is strictly convex for the metric dd c ϕ.

Theorem 20. Assume that Ω is circled and strictly ϕ-convex, where ϕ is a S 1 -invariant solution of (M A). Then ϕ is the only S 1 -invariant solution to (M A).

Proof. Assume we are given ϕ, ψ two S 1 -invariant solutions of (M A). Let (Φ t ) 0≤t≤1 denote the continuous geodesic joining φ 0 = ϕ to φ 1 = ψ. Since the functional F is concave along this geodesic and attains its maximum both at φ 0 and φ 1 , it is actually constant, hence each Φ t is a S 1 -invariant solution to (M A) by Corollary 19, so that

(dd c Φ t ) n = e -Φt µ Ω e -Φt dµ in Ω.
Assume that the mapping (z, t) ∈ Ω × A → Φ t (z) ∈ R is smooth. Taking derivatives with respect to t, we infer

n dd c Φt ∧ (dd c Φ t ) n-1 = -Φt + Ω Φt (dd c Φ t ) n (dd c Φ t ) n ,
so that 1 is an eigenvalue with eigenvector Φt -Ω Φt (dd c Φ t ) n for the Laplacian ∆ t associated to the Kähler form dd c Φ t . Without the regularity assumption, we can take derivatives in the sense of distributions to insure that at t = 0,

n dd c Φ0 ∧ (dd c Φ 0 ) n-1 = -Φ0 + Ω Φ0 (dd c Φ 0 ) n (dd c Φ 0 ) n ,
as in the proof of Theorem 6.8 in [START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF]. Note that Φ 0 = ϕ is smooth.

In particular Φ0 is solution of

-△ψ = ψ -c(ψ) in Ω with ψ |∂Ω = 0, (5.1)
where

c(ψ) = Ω ψ(dd c ϕ) n .
We are going to show that any solution of equation (5.1) has to vanish identically if Ω is strictly ϕ-convex. Namely, assume first that c(ψ) ≥ 0. Write ψ = ψ + -ψ -, where ψ + = max {ψ, 0} and ψ -= max {-ψ, 0}. Multiplying equation (5.1) by ψ + and integrating by parts, we get

Ω dψ + 2 (dd c ϕ) n = Ω (ψ + ) 2 (dd c ϕ) n -c(ψ) Ω ψ + (dd c ϕ) n ≤ Ω (ψ + ) 2 (dd c ϕ) n .
By the variational characterization of the first eigenvalue of the Laplacian, if ψ + doesn't vanish identically, then the last inequality means that the first eigenvalue of ∆ with Dirichlet boundary condition is at most 1. However, by [START_REF] Guedj | A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds[END_REF][Corollary 1.2], we know that this eigenvalue is strictly bigger than 1 because of the strict convexity condition2 . This shows that ψ + = 0 and therefore ψ = 0 because c(ψ) ≥ 0. If c(ψ) ≤ 0, the reasoning is similar and ψ = 0 as well.

As a conclusion, we see that Φ0 = 0 on Ω. Therefore, since the energy

t → E(Φ t )
is affine along the geodesic, and its derivative at t = 0 vanishes, it is constant on the interval [0, 1]. Now, along the geodesic, the derivative of F vanishes and since

F(Φ t ) = E(Φ t ) + log e -Φt dµ ,
we obtain finally that Φt e -Φt dµ = 0.

But Φt ≥ 0 since t → Φ t is convex (by subharmonicity and S 1 -invariance) and therefore Φt = 0 almost everywhere. This leads to Φ 0 = Φ 1 . It follows from the work of Cafarelli-Kohn-Nirenberg-Spruck [CKNS85] that 0 ∈ I, hence the latter is non empty (see the discussion in section 2.3.2).

The a priori estimates derived in section 4 can be adapted to show that I is closed. This is in general the most difficult part of the method. It however turns out here that proving the openness is a delicate issue. Indeed, to do so, we need to show that the linearized (M A) t equation has a trivial kernel. More precisely, we have to prove that if ϕ t is a solution of (M A) t , then every solution3 of -△ψ -tψ + tc(ψ) = 0 in Ω with ψ |∂Ω = 0, (

where c(ψ) := ψ(dd c ϕ t ) n must vanish. Let's introduce the differential operator D : C ∞ (Λ 0,1 Ω) → C ∞ (Λ 0,1 Ω ⊗ Λ 0,1 Ω) defined by Dα := ∇ 0,1 α. We have have then a Bochner formula (up to an inessential multiplicative π factor which we omit for brevity4 ) -△α = D * Dα + Ric(α), α ∈ C ∞ (Λ 0,1 Ω). (6.2) Applying (6.2) to ∂ψ where ψ is a solution of (6.1), we get where ρ is a boundary defining function for ∂Ω, n is the outward unit normal vector field on ∂Ω and L ρ is the Levi form corresponding to ρ (see section 2.1).

If Ω is a strictly pseudoconvex domain then tr L ρ is positive at each point of ∂Ω, however we do not have a priori any control on Hess ρ(Jn, Jn). So, contrary to what happens on a closed manifold where we do not have to deal with this disturbing boundary term, we cannot conclude here.

Remark 4. In the same spirit, we have shown in [START_REF] Guedj | A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds[END_REF] that a ball of sufficiently large radius in complex projective space provides an example of a strongly pseudoconvex domain which is not convex, and for which the Lichnerowicz estimate fails. It should be noticed that one can not expect to solve (M A) t for big values of t, as follows from Bishop's volume comparison theorem. Indeed, let B denote the unit ball in C n . If we can find a solution ϕ of (M A) t on B, this means that we can find a Kähler-Einstein metric ω = dd c ϕ on B satisfying Ric (ω) = tπω. Moreover, the volume V of this metric is

V = B (dd c ϕ) n n! = 1 n! .
But by the Bishop volume comparison theorem, the volume has to be less than or equal to the volume of the 2n-real dimensional sphere endowed with a metric of constant curvature k, with k = (tπ)/(2n -1). This implies that 1 n! ≤ (4π) n (n -1)! k n (2n -1)! , so that t ≤ 4(2n -1) (n -1)!n! (2n -1)! 1/n .

The interested reader will find in [START_REF] Berman | Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale[END_REF] further motivation and references about (M A) t for large (critical) values of t.

Institut Universitaire de France et Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse cedex 09, France E-mail address: vincent.guedj@math.univ-toulouse.fr LATP, CNRS & Université de Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail address: kolev@cmi.univ-mrs.fr LATP, Université de Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail address: Nader.Yeganefar@cmi.univ-mrs.fr

  dE(ψ s ) ds = Ω ψs (dd c ψ s ) n , as follows from Stokes theorem. A similar computation shows that a function ϕ solves (M A) if and only if it is a critical point of the functional F (in other words (M A) is the Euler-Lagrange equation for F).

Proof.

  Immediate consequence of Theorem 5 with a = 1 -β n and b = log C.

3. 3 .

 3 Ricci inverse iteration. Given ϕ ∈ P SH(Ω) ∩ C ∞ ( Ω) with zero boundary values, it follows from the work of Cafarelli, Kohn, Nirenberg and Spruck [CKNS85] that there exists a unique function ψ ∈ P SH(Ω) ∩ C ∞ ( Ω) with zero boundary values such that (dd c ψ) n = e -ϕ µ Ω e -ϕ dµ in Ω. ( * ) We let T := ϕ ∈ P SH(Ω) ∩ C ∞ ( Ω) | ϕ |∂Ω = 0 denote the space of test functions and

  where φ(z, ζ) = φ 0 (z) for |ζ| = 1 and φ(z, ζ) = φ 1 (z) for |ζ| = e. We set Φ(z, ζ) := sup {Ψ(z, ζ) / Ψ ∈ G} . Proposition 15. The function Φ is plurisubharmonic in Ω × A, continuous on Ω × Ā and satisfies (i) Φ(z, e iθ ζ) = Φ(z, ζ) for all (z, ζ, θ) ∈ Ω × A × R; (ii) Φ(z, 1) = φ 0 (z) and Φ(z, e) = φ 1 (z) for all z ∈ Ω; (iii) (dd c z,ζ Φ) n+1 ≡ 0 in Ω × A.

6. Concluding remarks 6. 1 .

 1 The continuity method. A classical strategy to solve (M A) is to use the continuity method, looking at a continuous family of similar Dirichlet problems,(M A) t (dd c ϕ t ) n = e -tϕt µ Ω e -tϕt dµin Ω with ϕ t|∂Ω = 0, where the parameter t runs from 0 to 1. One setsI := {t ∈ [0, 1] / (M A) tadmits a (smooth plurisubharmonic) solution} and then tries to show that I is non empty, open and closed, so that I = [0, 1]. Observe that 1 ∈ I is equivalent to solving the Dirichlet problem (M A) = (M A) 1 .

-

  △ ∂ψ = t ∂ψ = D * D ∂ψ + t ∂ψ because △ and ∂ commute and Ric(α) = tα. Therefore D * D ∂ψ = 0. (6.3)Then, taking the L 2 inner product of D * D ∂ψ and ∂ψ and integrating by parts, without neglecting boundary terms (see[START_REF] Guedj | A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds[END_REF] for details) and using the fact that on the boundary we have △ψ = tc(ψ), ψ) 2 tr L ρ + Hess ρ(Jn, Jn) σ. (6.4)

6. 2 .

 2 Optimal constants. It is natural to wonder whether it is possible to solve(M A) t (dd c ϕ t ) n = e -tϕt µ Ω e -tϕt dµin Ω with ϕ t|∂Ω = 0, for bigger values of t > 1. As noticed in Remark 3, our Moser-Trudinger inequality allows us to get control on slightly larger values of t, with a maximal value depending on n, namely t < (2n) 1+1/n (1 + 1/n) (1+1/n) .

Due to our normalization convention for d c , there is a π factor difference between the definition of ∆ in our present work and the one in[START_REF] Guedj | A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds[END_REF].

In the following, covariant derivative, Ricci tensor and Laplacian referred to the metric defined by ϕt.

In the following computation △ is the ∂-Laplacian.