Renata Grimaldi

Pierre Pansu

Calibrations and isoperimetric profiles *

We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new results on the profile of surfaces of revolution and some hardware know-how.

RESUM É: On construit, sur des surfaces non compactes non simplement connexes, des métriques riemanniennes lisses dont le profil isopérimétrique est lisse, phénomène hautement non générique. Le calcul du profil est basé sur l'idée de calibration, un argument de réarrangement, l'inégalité de Bol-Fiala, un résultat nouveau sur le profil des surfaces de révolution, et des connaissances en plomberie.

1 Introduction

The problem

Let M be a smooth manifold. We are concerned with isoperimetric profiles of metrics on M . Fix a Riemannian metric on M with total volume V . Given v < V , consider all domains (relatively compact open subsets with smooth boundary) in M with volume v. Define I M (v) as the least upper bound of the boundary volumes of such domains. In this way, one gets a function I M : (0, V ) → R + called the isoperimetric profile of M .

For instance, Euclidean space R n has

I R n (v) = const. v n-1
n * Keywords : Isoperimetric inequality, Riemann surface, calibration, Morse theory. Mathematics Subject Classification : 53C20, 49Q20, Work done within the GNSAGA group of INDAM, supported by MURST of Italy and by the Galileo project Asymptotic properties of groups and manifolds.

† Univ Palermo, Dipartimento di Metodi e Modelli Matematici, Palermo, I-90128.

‡1 Univ Paris-Sud, Laboratoire de Mathématiques d'Orsay, Orsay, F-91405 ; 2 CNRS, Orsay, F-91405.

1

for some constant const.(n). Indeed, for each v, round balls of volume v minimize boundary volume among all domains of volume v.

Here is an other example, where the isoperimetric profile is much easier to compute. Let Γ be a lattice of parabolic translations of hyperbolic n-space H n (i.e. Γ has a unique fixed point ζ on the ideal boundary, and acts as a lattice of translations on the complement of ζ, identified with euclidean n -1-space). Let M = H n /Γ. We call such a manifold a complete constant curvature cusp. Then I M (v) = (n -1)v.

This will be proven as Corollary 4 in section 3.

In both examples, the isoperimetric profile I is a smooth function. This is rather exceptional and related with the topological simplicity of the underlying manifolds. Smoothness of the isoperimetric profile is related to uniqueness of minimizers in the following way. For a generic Riemannian manifold, one expects that domains whose boundary has constant mean curvature come in finitely many smooth families. To each such family D i,t , there corresponds a function P i defined by

P i (vol(D i,t )) = vol(∂D i,t )
on some interval. The isoperimetric profile of M is the minimum of these functions P i . If M has nontrivial topology, there must be at least two different families, and it seems very unlikely that the isoperimetric profile be everywhere smooth. This suggests the following question.

Question 1 Does every manifold admit a smooth (resp. real analytic) metric with smooth (resp. real analytic) isoperimetric profile ?

The results

In this note, we construct examples of smooth complete metrics with smooth isoperimetric profiles on topologically non trivial manifolds, mostly in dimension 2.

Theorem 1 Every smooth (resp. real analytic) non compact manifold admits smooth (resp. real analytic) complete metrics of infinite volume whose isoperimetric profile vanishes identically.

For such metrics, there are no extremal domains. That's cheating! Say a Riemannian manifold M has an achieved isoperimetric profile if for each v ∈ (0, vol(M )), there exists a domain D with vol(D) = v and vol(∂D) = I M (v). The isoperimetric profile of a compact Riemannian manifold is always achieved. Here is a simple sufficient condition for non compact surfaces.

Definition 1 Say a non compact 2-dimensional Riemannian manifold M is ultrahyperbolic if

• M is complete;

• its curvature tends to -∞ at infinity;

• its injectivity radius tends to +∞ at infinity.

On such a surface, the isoperimetric profile is achieved, this is proven as Proposition 27 in the appendix.

Theorem 2 Let M be an orientable 2-dimensional manifold. Assume that one of the following properties holds.

• M has at least 4 ends.

• M has 3 ends, at least one of which having infinite genus.

• M has 2 ends and both have infinite genus.

Then M admits a complete smooth Riemannian metric of infinite area with achieved isoperimetric profile I M (a) = a for all a ∈ (0, +∞).

An even larger family of surfaces admit smooth metrics with smooth isoperimetric profiles.

Theorem 3 Let M be an orientable 2-dimensional manifold. Assume that one of the following properties holds.

• M has at least 2 ends.

• M has one end of infinite genus.

Then M admits a complete smooth Riemannian metric of infinite area with achieved isoperimetric profile whose square I 2 M extends to a smooth function on some neighborhood of R + . If furthermore M has finite topological type, one can arrange that M be ultrahyperbolic.

Question 2 Given a manifold M , can one construct metrics with smooth isoperimetric profile in all conformal classes on M ?

Scheme of proof

The proof of Theorem 1 is straightforward : given a function with compact sublevel sets on M , one arranges so that the volume of level sets tends to 0 although the total volume is infinite. The metrics of Theorem 2 are modelled on the 2-dimensional complete constant curvature -1 cusp M 0 , where the solutions of the isoperimetric problem are the sublevels V = {f ≤ t} of a function f such that ∆f = 1. In M 0 , each level set ∂V = {f = t} is calibrated by the unit 1-form ω = * df , i.e.

ω |∂V = vol ∂V .
The calibration method of Harvey and Lawson [START_REF] Harvey | Calibrated geometries[END_REF] can be readily adapted to the isoperimetric problem, see section 3. Given a calibrating 1-form (not quite a calibration since it is not closed), the method yields the value of the isoperimetric profile I(v) at v provided there exists a set with area v and calibrated boundary.

For the surfaces with complicated topology dealt with in Theorem 2, the construction goes in two steps.

First, glue together pieces of complete constant curvature -1 cusps in order to obtain a singular surface with constant curvature -1, equipped with a level function f and a calibrating 1-form * df . This is done in section 4. This kind of surface has plenty of subsets with calibrated boundary. Figure 1 shows an example of a triply punctured sphere made of 3 pieces of a cusp. It has 2 families of calibrated subsets exhibiting different topological types. One family depends on one parameter, the second on two parameters.

Second, smooth away singularities, while maintaining a calibrating 1-form. This is done in section 5.

The resulting surface has less calibrated subsets, since levels passing close to singular points are not calibrated any more. Sometimes, they are still sufficient for the determination of the isoperimetric profile. This not the case for the example of figure 1. Indeed, there is a gap, around the area of the critical sublevel, in the values of the area which are achieved by calibrated subsets.

To overcome this difficulty, the "pipe clearing trick" is applied. Imagine the surface as a pipestry and the calibrated regions as partial fillings of the pipestry with water. View a singular point as an obstacle blocking a pipe A. The traditional plumber's strategy consists in letting the water level raise in a neighboring pipe B, then pushing energeticly the water out of B so that in A water traverses the critical level, see figure 2. This allows, without change in the total amount of water, a change in the topology of the wet region. This hardware knowledge is turned into mathematics in . This trick is related to Pitts' observation that minimax cycles on a surface cannot have too many selfintersections, compare the four-legged star fish in [START_REF] Pitts | Existence and regularity of minimal surfaces on Riemannian manifolds[END_REF] and [START_REF] Calabi | Simple closed geodesics on convex surfaces[END_REF] page 540.

The existence of a neighboring pipe is related to a Morse function (water level) having disconnected level sets. Such functions exist only on surfaces with many ends, as shown in section 6.

The examples of Theorem 3 require two more steps.

When applied to a manifold M with two points removed, Theorem 2 yields a manifold with two constant curvature -1 cusps. Closing the cusps with suitably chosen caps of revolution yields a metric on M whose isoperimetric profile is given by that of the caps, see section 9. The only surfaces of revolution whose isoperimetric profiles were previously known to be smooth, see [START_REF] Benjamini | A new isoperimetric theorem for surfaces of variable curvature[END_REF], [START_REF] Howard | The isoperimetric problem on surfaces of revolution of decreasing gauss curvature[END_REF], [START_REF] Ritoré | Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces[END_REF], [START_REF] Topping | Mean curvature flow and geometric inequalities[END_REF] are those whose curvature is a nonincreasing function of the distance to the pole. For our purposes, this class needs to be slightly enlarged. An argument based on the strict stability of parallels shows that the property that rotationally symmetric domains are extremal is stable under small perturbations, see section 8. This provides us with the suitable caps.

Finally, a rearrangement argument shows that the calibration method extends to forms ω such that dω = u vol where extremal domains are sublevel sets of u. Therefore an extra conformal change produces metrics whose isoperimetric profile equals any prescribed smooth convex function. Such metrics can be arranged to have curvature tending to -∞ and injectivity radius tending to +∞, see section 10.

Acknowledgements

This work owes a lot to V. A. Zorich and V. M. Kesel'man's construction of a singular metric with linear isoperimetric profile in a conformal class, [START_REF] Zorich | On the conformal type of a Riemannian manifold[END_REF]. It arose from an attempt to smooth their metric. Many thanks are due to the anonymous referee for his help in improving the exposition.

Non achieved isoperimetric profiles

We prove Theorem 1.

Let M be a smooth manifold, w : M → R + a smooth Morse function with compact sublevel sets and isolated critical values. Start with any complete smooth metric g on M . Consider the positive continuous function v defined on (0, +∞) by

v(t) = vol({w ≤ t}).
Let u : R+ → (0, +∞) be a smooth function such that uv tends to 0. In the conformal metric g ′ = u(w) 2/(n-1) g, the level set {w = t} has volume uv(t). Let h : R+ → R + be a smooth function

such that +∞ 1 huv(t) dt = +∞. Let g ′′ = g ′ + h(w) 2 dw 2 .
Then the volume of level sets of w still tends to 0 since it does not change. Furthermore, for all 0 < s < t,

vol ′′ ({s < w < t}) = t s ( {w=r} 1 |∇w| ′′ ) dr ≥ t s h(r)u(r)v(r) dr.
Given v > 0 and s > 0, there exists t such that vol ′′ ({s < w < t}) = v. Then vol ′′ (∂{s < w < t}) ≤ vol ′′ ({w = s}) + vol ′′ ({w = t}) = uv(s) + uv(t) tends to 0 as s tends to +∞. This shows that

I (M,g ′′ ) ≡ 0. Since +∞ 1 h(t) dt = +∞, (M, g ′′ ) is complete.
If M is real analytic, the construction is the same, in the real analytic category.

The calibration argument

The following lemma goes back to S.T. Yau, [START_REF] Yau | Isoperimetric constants and the first eigenvalue of a compact manifold[END_REF].

Lemma 2 Let M be a complete n-dimensional Riemannian manifold. Let x ∈ M have injectivity radius ≥ R and let B = B(x, R) be the geodesic ball. Assume that on B, the sectional curvature is less than -ρ 2 . Then I B (v) ≥ (n -1)ρv.

Proof. On B, let r denote the distance to x, and ω = ι ∇r vol. Then |ω| ≤ 1 and dω = h vol where h, the mean curvature of geodesic spheres, is controlled by curvature, h ≥ (n -1)ρ. This form can be used as a "calibration" : given a domain D ⊂ B,

(n -1)ρvol(D) ≤ D h vol = ∂D ω ≤ length(∂D).
In this section, we turn Yau's observation into a systematic tool for computing isoperimetric profiles. Note that, in order to prove isoperimetric inequalities, F. Hélein has made a slightly different use of calibrations, [START_REF] Hélein | Inégalité isopérimétrique et calibration[END_REF]. • there exist compact domains V ǫ such that

vol(V \ V ǫ ) → 0, vol(∂V △∂V ǫ ) → 0
as ǫ tends to 0.

Then I M (v) = c v.
Proof.

Let D be an arbitrary domain. Then

c vol(D) = D c vol = D dω = ∂D ω ≤ vol(∂D) since |ω| ≤ 1. This shows that I M (v) ≥ c v.
Equality holds asymptoticly for sets of the form V ǫ as ǫ tends to 0. Indeed, since ω calibrates ∂V ,

∂V ω = vol(∂V ), whereas lim ǫ→0 vol(∂V ǫ ) c vol(V ǫ ) = 1,
showing that

I M (v) = (n -1)v.
Corollary 4 Let M be a complete constant curvature n-dimensional cusp. Then 4 Constructing a singular metric from a Morse function

I M (v) = (n -1)v.

Standing assumptions on the level function

In this section, M will denote an orientable 2-manifold equipped with a smooth function f satisfying the following.

1. For all a ≤ b ∈ R, f -1 [a, b] is compact.
2. All critical points of f are nondegenerate of index 1.

3. The values of f at distinct critical points are distinct integers.

4. Critical values of f form an interval of Z containing 0 and 1.

In the terminology of the introduction, f is a level function on M . Under assumptions 1 to 4, LG is a trivalent graph. Indeed, f descends to a map f : LG → R which is a covering map over the complement of critical values. The pull-back of a neighborhood of a critical value consists of finitely many arcs on which f is a homeomorphism, and a Y shaped set (3 arcs joined at a the class of a critical point), see figure 4. Edges are oriented by the map f .

Weights

We are about to put a metric on M such that if a is a regular value of f , then f -1 (a) is a disjoint union of constant geodesic curvature curves. Each such curve corresponds to an edge of LG. The total length of f -1 (a) is shared between its connected components in proportion of numbers called the weights of the corresponding edges. In this subsection, we describe how these weights are chosen.

Definition 5 Weights are given recursively to edges of LG in the following way. Assume that f -1 (1/2) consists of N points. Each of the edges on which f vanishes is weighted 1/N . Then as one moves away from f -1 (1/2), following oriented edges, as one encounters a vertex v,

• either v has one incoming edge and two outgoing edges; the weight of the incoming edge has been previously defined; then both outgoing edges are weighted one half of the weight of the incoming edge.

• or v has two incoming edges and one outgoing edge; the weight of the incoming edges have been previously defined; then the outgoing edge is weighted the sum of the weights of the incoming edges. Symmetricly, when proceeding from f -1 (1/2) backwards along oriented edges, the opposite rule is applied : 2 incoming edges share half the weight of 1 outgoing edge, 1 incoming edge adds up the weights of 2 outgoing edges, see figure 5.

Note that for each t ∈ R, the sum of the weights of the edges on which f takes the value t is equal to 1.

Lemma 6 For every oriented edge e of LG originating at p ′ = α(e) and ending at p ′′ = ω(e),

weight(e) ≥ 1 N 2 -|f (p ′′ )|-1 .
Proof. When traversing a vertex of LG, weights decrease at worst by a factor of 2. Furthermore, travelling along

LG from e to an edge on which f takes the value 1/2, at most |f (p ′′ )| + 1 vertices are encountered.

Renormalization of the Morse function

Pick an integer ν ≥ 1 such that N ≤ 2 ν .

Definition 7 Let u = φ • f where φ is a diffeomorphism of R onto (0, +∞) mapping an integer n ∈ Z to 16 n(|n|+ν) .
Lemma 8 For every oriented edge e of LG originating at p ′ = α(e) and ending at p ′′ = ω(e), and every vertex p such that u(p) < u(p ′′ ),

weight(e)u(p ′′ ) ≥ 8u(p).

Proof. One easily checks that for all

n ∈ Z, φ(n) ≥ 2 |n|+4+ν φ(n -1). If f (p ′′ ) = n, then f (p) ≤ n -1, and Lemma 6 gives weight(e) ≥ 1 N 2 -|n|-1 ≥ 2 -|n|-1-ν ≥ 8 φ(n -1) φ(n) ≥ 8 u(p) u(p ′′ ) .

Constant curvature annuli

Our model for a calibrated triple (M, f, ω) is the hyperbolic plane, f a Busemann function and ω = * df the unit 1-form calibrating a family of concentric horoballs. Denoting w = e -f we have

ω = ι w -1 ∇w vol, |∇w| = w and ∆w = -2w. In coordinates, w = 1 ℑm(z) = 1 y
when the hyperbolic plane is represented as the upper half plane {ℑm(z) > 0} equipped with the metric

dx 2 + dy 2 y 2 = w 2 dx 2 + dw 2 w 2 .
When divided by a translation in the x coordinate, this simply connected model gives rise to complete constant curvature cusps

w 2 dx 2 + dw 2 w 2 , x ∈ R/τ Z, w ∈ (0, +∞).
and then to annuli

A τ,c,c ′ = (R/τ Z × [c, c ′ ], w 2 dx 2 + dw 2 w 2 ). The annulus A τ,c,c ′ has constant curvature -1. Its boundary components have lengths cτ and c ′ τ . Note that A τ,c,c ′ is isometric to A cτ,1,c ′ /c
. The second parameter is only used to adjust the interval of variation of the w coordinate. Note that area(A τ,c,c ′ ) = τ (c ′c), and that the height of A τ,c,c ′ , i.e. the distance between the boundary components, is equal to log(c ′ /c).

Definition 9

We shall call A 1,0,δ (resp. A 1,δ,+∞ ) a constant curvature -1 cusp (resp. anticusp) with boundary a horocycle of length δ.

A(e) P P(e) The height of A(e) is log u(ω(e))log u(α(e)).

In particular, the height is always ≥ log 2.

If α(e) (resp. ω(e)) has two incoming (resp. outgoing) edges, one chooses two points on the left hand (resp. right hand) boundary component of A(e) which separate it into two intervals of lengths proportional to the weights of incoming (resp. outgoing) edges. Then the two marked points are identified. The resulting surface P (e) is a singular pair of pants whose boundary consists of three circles, see figure 6.

By construction, if e, e ′ are incoming and e ′′ is the outgoing edge of some vertex v, the lengths of the right hand boundary components of P (e) and P (e ′ ) fit with the lengths of the left hand boundary components of P (e ′′ ). Therefore one can choose orientation preserving isometries between these circles and glue them together. This applies as well to vertices with one incoming and two outgoing edges.

Morse theory shows that the resulting space

P = e P (e)/ ∼
is homeomorphic to M , in such a way that u is mapped to w, the projection onto the second factor, on each annulus A(e).

The nonsingular part of P

The space P constructed above can be viewed as M equipped with a Riemannian metric g which, away from critical points of u, is locally isometric to hyperbolic plane with curvature -1. More precisely, for a point p ∈ P , let ℓ(p) be the smallest length of a cycle in {u = u(p)} containing p, let r(p) be the min of log 2 and of the injectivity radius of the bi-infinite constant curvature cusp along the horocycle of length equal to ℓ(p). Then the ball B(p, r(p))

• either is isometric to a ball of radius r(p) in the hyperbolic plane,

• or contains a critical point.

Proposition 10 Let M ′ denote the smooth part of P . Then M ′ is an incomplete surface of constant curvature -1, diffeomorphic to M with the critical points of u removed. Its isoperimetric profile is

I M ′ (v) = v. Proof. Let B t = {u ≤ t}. Then area(B t \ B s ) = t -s.
In particular area(B t ) = t is finite.

Indeed, if the interval [s, t] does not contain critical values, B t \ B s is the union of disjoint annuli isometric to {s ≤ w ≤ t} ⊂ A weight(e),value(α(e),value(ω(e) .

Each annulus has area weight(e)(ts), the weights of edges crossing a level set of u sum up to 1, therefore the total area is ts. The general case follows by additivity.

If D is a disk of radius r (small enough) centered at a critical point, then D is an isometric double covering of a hyperbolic disk of radius r (see next paragraph). Thus its area and boundary length tend to 0 as r tends to 0. Each B t can be approximated by domains B t,ǫ by removing B ǫ and tiny disks around critical points, whose total area and boundary length is arbitrarily small.

Since u is smooth, the 1-form ω = ι u -1 ∇u vol is smooth. Since u coincides with coordinate w on each P (e), ω has unit norm, satisfies dω = vol and calibrates the sublevel sets of u. Therefore proposition 3 applies, and the isoperimetric profile of M ′ is linear.

Description of singularities

We claim that at each critical point p of u, P has a conical singularity with angle 4π. More precisely, for r ≤ r(p), the ball B(p, r) in M is isometric to the disk D r of radius tanh(r) in C equipped with the metric

g 0 = 16|z| 2 (1 -|z| 4 ) -2 |dz| 2 in such a way that u = u(p) |z 2 -i| 2 1-|z| 4 .
Indeed, the map z → z ′ = z 2 is a 2-sheeted Riemannian covering of D r \ {0} onto the punctured disk of radius tanh(r) equipped with the hyperbolic metric 4|dz ′ | 2 /(1 -|z| 2 ) 2 . We use the isometry

z ′ → z ′′ = 1 -iz ′ z ′ -i
between the Poincaré disk and the upper half plane equipped with |dz ′′ | 2 /(ℑm(z)) 2 , and pull back the function w ′′ = 1/ℑm(z ′′ ). Taking a constant multiple of it, we get a function

w 0 = u(p) |z 2 -i| 2 1 -|z| 4
on D 0 . The level set {w 0 = u(p)} splits D 0 into four regions, isometric in pairs, and isometric to the intersection of a horoball (resp. the complement of a horoball) and a hyperbolic ball of radius 1/4 centered on the boundary of the horoball. In each region, log(w 0 /u(p)) is the signed distance to the boundary horocycle. Therefore these regions fit together exactly into a ball of radius r centered at a singular point of M . And w 0 is exactly mapped to u. Then we change g 1 into g while keeping the same volume element. At points where ω does not vanish, one can write

Smoothing of singularities

g 1 = ( * 1 ω |ω| 1 ) 2 + g ⊥ 1
where g ⊥ 1 is non negative with kernel the vectorfield ξ dual to ω.

Let g = ( * 1 ω χ(|ω| 1 ) -1 |ω| 1 ) 2 + χ(|ω| 1 ) -2/(n-1) g ⊥ 1 ,
where χ : R + → (0, +∞) is a smooth function such that

• χ(t) ≥ t ; • χ(t) = t for t > 2/3 ; • χ(t) = 1 for t < 1/3.
With respect to the metric g, the 1-form

* 1 ω χ(|ω| 1 ) -1 |ω| 1 has unit norm, so that |ω| = | * ω| = |ω| 1 χ(|ω| 1 ) ≤ 1.
In a neighborhood of the boundary, g 1 = g 0 and ω = ω 0 , thus |ω| 1 = 1 and g = g 1 . At points where |ω| 1 < 1/3, g = g 1 , thus g extends smoothly to all of D, and the inequality |ω| ≤ 1 trivially extends to points where ω vanishes.

Since vol g = vol 1 , dω = vol g and area(D, g) = area(D, g 1 )

= D dω 0 = ∂D ω 0 ≤ vol 0 (∂D).
Proposition 12 Let E be the union of the intervals (c/2, 2c) where c is a critical value of u. There exists a smooth Riemannian metric and a smooth 1-form ω on M with the following properties.

1. |ω| ≤ 1.

2. dω = vol.

3. ω calibrates the sublevel sets {u ≤ t} for all t / ∈ E.

Furthermore, one can arrange that each superlevel set {u ≥ s} have bounded geometry.

Proof. This local construction can be performed in the model (D r , g 0 ). In order to apply Lemma 11 to smooth versions of g 0 and ω 0 , we must check that they satisfy

I = Dr dω 0 > 0. Since D 0 dω 0 = ∂D 0 ω 0 ,
I only depends on the boundary data, and we can compute it from the singular data. Using the double covering z → z 2 of D r onto a ball of hyperbolic radius tanh(r), we see that I is twice the flux of a Busemann vectorfield along the boundary of a hyperbolic ball, i.e. twice the area of the hyperbolic ball. In particular, I > 0.

Therefore, Lemma 11 Remark 13 If c is a critical value of u, then D = {u ≤ c} is not a solution of the isoperimetric problem.

Indeed, solutions have a smooth boundary. Compare [START_REF] Calabi | Simple closed geodesics on convex surfaces[END_REF] page 541 or [START_REF] Pitts | Existence and regularity of minimal surfaces on Riemannian manifolds[END_REF] page 37.

Construction of special Morse functions

Proposition 14 Let M be a connected orientable 2-manifold. Assume that one of the following properties holds.

• M has at least 4 ends.

• M has 3 ends and at least one of the ends has infinite genus.

• M has 2 ends and both have infinite genus.

Then there exists on M a function f such that If furthermore both ∂ + M and ∂ -M are disconnected, then one can arrange that all levels of f be disconnected.

1. For all a ≤ b ∈ R, f -1 [a, b] is compact.
Proof. Compact orientable surfaces with marked boundary are classified by their genus and the number of boundary components of each sign. Indeed, such surfaces are obtained by removing disks from closed orientable surfaces. Closed orientable surfaces are classified by their genus and the diffeomorphism group of a surface is transitive on finite collections of disks. Therefore is suffices to give an example for each value of the triple (genus, number of + boundary components, number ofboundary components). Figure 7 shows such a surface together with the critical levels of a suitable Morse function. If ∂ + M or ∂ -M is connected, the Z-shaped core must be replaced with a > or --shaped surface, in which case the disconnectedness of levels is lost.

Lemma 16 Let M be a connected orientable non compact 2-manifold with non empty compact boundary. There exists a proper Morse function on M which is constant along the boundary and satisfies 1 to 4.

Proof. Start with an arbitrary proper Morse function f which is constant on the boundary. Choose an increasing sequence t j of noncritical values. On each compact submanifold {t j ≤ f ≤ t j+1 }, apply Lemma 15 to improve f . The resulting function has all the required properties. Case k = 4. For t ∈ [-1, 1], the level {f = t} is disconnected as shown on the picture. For t > 1 or t < 1, this follows from the fact that {f = t} intersects at least two connected components of M \ K.

Case k = 3. We can assume that the end where u ≥ 1 has infinite genus. A noncompact end of infinite genus is, up to a compact change, diffeomorphic to the surface with boundary shown on figure 8 ( [START_REF] Richards | On the classification of noncompact surfaces[END_REF], [START_REF] Kerekjarto | Vorlesungen über Topologie I[END_REF]). Therefore, up to enlarging K, we can assume that the connected component of M \ K where f ≥ 1 has a boundary consisting of exactly 2 curves. Figure 8 also shows the critical levels of a Morse function on the end satisfying 1 to 5. We use it to modify f . Thus if t ≥ 1, the level {f = t} is disconnected. For t ∈ [-1, 1], this follows from Lemma 15. For t ≤ 1, this follows from the fact that {f = t} intersects two connected components of M \ K.

Case k = 2. An orientable surface with two ends of infinite genus is the bi-infinite version of figure 8 ( [START_REF] Richards | On the classification of noncompact surfaces[END_REF], [START_REF] Kerekjarto | Vorlesungen über Topologie I[END_REF]) and therefore has an obvious Morse function satisfying 1 to 5.

Proof of Theorem 2

Apply Proposition 14 in order to obtain a Morse function on M having properties 1 to 5. Apply Proposition 12 to produce a smooth metric, a smooth function u and a smooth calibration ω on M which calibrates the boundary of most sublevel sets For each t ∈ (c -, c + ), we must find a submanifold with boundary which is calibrated by ω and has area t.

B t = {u ≤ t} of u,
\ B c ′ + )) -area(B c + \ B c -) = weight(e)(c ′′ --c ′ + ) -(c + -c -) ≥ weight(e)( 1 2 - 1 8 )c ′′ -(2 - 1 2 )c ≥ 1 4
weight(e)c ′′ -2c ≥ 0, Therefore, as s varies, area(C e,s ) and area(D e,s ) together achieve all values in (c -, c + ) (this is the pipe clearing trick). Then Proposition 3 applies, and I M (a) = a for all a > 0.

Surfaces of revolution

This is a preparation for the next section, where caps of revolution with explicit isoperimetric profiles will be needed.

In a surface of revolution, the isoperimetric profile is not always achieved by disks of revolution. It is the case provided the curvature is a nonincreasing function of the distance from the pole. This is not sufficient for our purposes. Indeed, a constant curvature cusp cannot be replaced with a surface of revolution with nonincreasing curvature of the same area. We need just a little more flexibility. Fortunately, the property that disks of revolution are extremal is often stable under small perturbations.

Lemma 17 Let A = ([a, b] × R/2πZ, dr 2 + f (r) 2 dθ 2
), f > 0, be a smooth annulus of revolution such that for all r ∈ [a, b], f ′ (r) 2f ′′ (r)f (r) is not the square of a nonzero integer. There exist constants β > 0 and γ > 0 such that if g is a C 2 function on [a, b] such that gf C 2 < β, D 0 = {a ≤ r < r 0 } and D = {a ≤ r < ρ(θ)} are domains such that area(D) = area(D 0 ) and ρr 0 C 2 < γ whose boundary has constant geodesic curvature in the metric dr 2 + g(r) 2 dθ 2 , then D = D 0 .

Proof.

Given g ∈ C 2 ([a, b]) and ρ ∈ C 2 (R/2πZ), let κ ∈ C 0 (R/2πZ) be the geodesic curvature of the curve {r = ρ(θ)}, κ = -g(ρ)ρ ′′ + g ′ (ρ)g(ρ) 2 + 2g ′ (ρ)ρ ′2 g(ρ) 2 + ρ ′2 3 . ( 1
)
The area of D = {a ≤ r < ρ(θ)} is equal to area(D) = where κ = κ(g, ρ) is given by formula (1),

g = G ′ and ρ = G -1 • (R + v).
Constant geodesic curvature annular domains of area v correspond to solutions of Φ(v, G, R) = 0. For every (v, G) such that 0 < v < G(b), R = 0 is a solution. Let us show that, for (v, G) close to (v, F ) where F (r) = r 0 f (s) ds, this is the only solution. Let us compute the derivative of Φ with respect to R at (v, F, 0). Let r 0 = F -1 (v) be the solution of F (r 0 ) = v. Let ρ t be a smooth family of functions starting from the constant function ρ 0 = r 0 , and ρ = dρ t dt |t=0 . Then

dΦ dt |t=0 = -3f -4 f ′ ρ(f ′ f 2 ) + f -3 (-f ρ′′ + f ′′ f 2 ρ + 2f ′2 f ρ)
where f = f (r 0 ), f ′ = f ′ (r 0 ), and f ′′ = f ′′ (r 0 ). In other words,

∂Φ ∂R (v, F, 0) : C 2 0 (R/2πZ) → C 0 0 (R/2πZ
) is equal to the operator

P : u → -f -2 u ′′ + (f ′′ f -1 -f ′2 f -2 )u.
Since the eigenvalues of u → -u ′′ on functions on R/2πZ with vanishing average are squares of nonzero integers, the assumption made implies that P is invertible on Sobolev spaces H 2 0 → H 0 0 of functions with vanishing average which are in L 2 (resp. have 2 derivatives in L 2 ). Since H 2 0 ⊂ C 0 0 , if P u is continuous, so is u ′′ , thus P -1 maps C 0 0 to C 2 0 . P is a continuous bijection C 2 0 (R/2πZ) → C 0 0 (R/2πZ), and thus an isomorphism. The implicit function theorem applies : there are neighborhoods

I v of v in R, V of F in C 3 0 ([a, b]) and W of 0 in C 2 0 (R/2πZ) such that for (v ′ , G) ∈ I v × V v , the equation Φ(v ′ , G, R) = 0 admits a unique solution in W v , this is R = 0.
Covering the interval [0, F (b)] with a finite number of intervals I v , this translates into constants β > 0 and γ > 0 such that if gf C 2 < β and ρr 0 C 2 < γ, then the curve {r = ρ(θ)} has constant geodesic curvature if and only if ρ is constant.

Optimality of rotationally symmetric disks is often stable

Although we shall not need the following proposition, we state it for its independant interest.

Proposition 18 Let S = (R + ×R/2πZ, dr 2 +f (r) 2 dθ 2 ) be a complete smooth surface of revolution such that • f is nondecreasing and tends to +∞;

• the curvature K(r) = - f ′′ (r) f (r) tends to -∞;
• the disks of revolution {r < t} are the only extremal domains for the isoperimetric problem in S, and for all t ∈ [a, b], they are strictly stable, i.e.

f ′ (t) 2 f (t) 2 - f ′′ (t) f (t) < 1 f (t) 2 . Let g be a C 2 function on R + such that g = f outside [a, b]. If g -f C 2 is
small enough, then in the surface of revolution S ′ = (R 2 , dr 2 + g(r) 2 dθ 2 ), the disks of revolution {r < t} are the only extremal domains for the isoperimetric problem. In particular, for all t > 0,

I S ′ (2π t 0 g(s) ds) = 2πg(t).
Proof. The assumptions on f imply that all surfaces under consideration are ultrahyperbolic. Therefore extremal domains exist and form compact sets.

The proof goes by contradiction. Assume there exists a sequence g ℓ of functions C 2 -converging to f and a sequence of non rotationally symmetric extremal domains D ℓ . One can assume that D ℓ converges in flat norm to some domain D. By lower semi-continuity of the boundary length and continuity of the area, D is extremal in S, thus D is rotationally symmetric, i.e. D = {r ≤ r 0 }.

Let χ : R + → R + be a smooth function such that χ(r 0 ) = 0. By flat convergence,

lim ℓ→+∞ ∂D ℓ χ(r)f (r) dθ = ∂D χ(r)f (r) dθ = 0.
This shows that, for every ǫ > 0, outside the ǫ neighborhood U ǫ of ∂D,

lim ℓ→+∞ ∂D ℓ \Uǫ f (r) dθ = 0.
In particular,

lim ℓ→+∞ length(∂D ℓ ) = length(∂D) = ∂D f (r) dθ = lim ℓ→+∞ ∂D ℓ ∩Uǫ f (r) dθ ≤ lim ℓ→+∞ length(∂D ℓ ∩ U ǫ ), which shows that lim ℓ→+∞ length(∂D ℓ \ U ǫ ) = 0.
Since the projection (r, θ) → r is length decreasing, there exist two sequences r ℓ < r 0 < r ′ ℓ , tending to r 0 such that ∂D ℓ does not intersect the parallels {r = r ℓ } and {r = r ′ ℓ }. Since

lim ℓ→+∞ ∂D ℓ ∩{r ℓ <r<r ′ ℓ } dθ = ∂D dθ = 2π, exactly one component c ℓ of ∂D ℓ ∩ {r ℓ < r < r ′ ℓ } is homotopic to ∂D in the annulus {r ℓ < r < r ′ ℓ }.
Using an arclength parametrization of c ℓ and applying Ascoli's theorem, one can assume that c ℓ converges uniformly, and the limit must be ∂D parametrized by arclength.

∂D ℓ has constant geodesic curvature κ ℓ . At points where the function r restricted to c ℓ achieves its maximum r max or its minimum r min , a comparison principle yields

f ′ (r min ) f (r min ) ≤ κ ℓ ≤ f ′ (r max ) f (r max ) ,
showing that κ ℓ tends to f ′ (r 0 ) f (r 0 ) .

Since short closed curves must have large geodesic curvature somewhere, ∂D ℓ has no short components, thus ∂D ℓ = c ℓ .

Along a long curve with bounded geodesic curvature which makes a large angle with parallels at some point, the function r varies a lot. This shows that the angle of the tangent of c ℓ with parallels tends uniformly to 0. In other words, c ℓ can be viewed as a graph {r = ρ ℓ (θ)} where ρ ℓ converges to r 0 in C 1 . Since the geodesic curvature converges, formula (1) for the geodesic curvature shows that ρ ′′ ℓ converge, i.e. ρ ℓ converges to r 0 in C 2 . Lemma 17 implies that, for ℓ large enough, ρ ℓ is constant, a contradiction.

Finite approximations of a cusp-like surface

What we need is a variant of Proposition 18 where the model surface is non compact.

Proposition 19 Let S = (R×R/2πZ, dr 2 +f (r) 2 dθ 2 ) denote a complete surface of revolution. Let S ℓ = ([-R ℓ , +∞)× R/2πZ, dr 2 + g ℓ (r) 2 dθ 2 ) be a sequence of complete smooth surfaces of revolution converging to S in the following sense : g ℓ = f on R + , and g ℓ converges to f , C 2 -uniformly on compact subsets of R -. Assume that

• f (r) = e r for r ≤ 0; • f ′ /f is nondecreasing;
• f is nondecreasing and tends to +∞;

• the curvature K(r) = - f ′′ (r) f (r) tends to -∞;
• g ℓ is nondecreasing.

Then for ℓ large enough, the disks of revolution {r ≤ t} are extremal domains for the isoperimetric problem in S ℓ . In particular, for all t > -R ℓ ,

I S ℓ (2π t -R ℓ g ℓ (s) ds) = 2πg ℓ (t).
Proof. Since the candidate isoperimetric profile is nondecreasing, one can restrict to domains which are disjoint unions of disks (otherwise, replace a domain with the largest disk spanned by one of its boundary components). Since S ℓ is ultrahyperbolic, there exist extremal domains, which again are disjoint unions of disks. Let D ℓ be extremal domains with area v ℓ in S ℓ which converge in flat norm to a possibly non compact submanifold D ⊂ S. Ultrahyperbolicity holds for S on {r ≥ 0}. Therefore D ∩ {r ≥ 0} is compact. D can be approximated by domains of the form D ∩ {e r ≤ ǫ} with ǫ tending to 0. Let ω = f (r) dθ on S. Then dω = u vol where u = f ′ /f is nondecreasing. Proposition 24 applies, showing that D has calibrated boundary, i.e. D = {r ≤ r 0 } is a rotationally symmetric non compact annulus.

As in the proof of Proposition 18, most of the length of ∂D ℓ concentrates in thin neighborhoods of ∂D. Then one component c ℓ of ∂D ℓ is contained in a thinner and thinner neighborhood of ∂D (being homotopic to ∂D in this neighborhood). One can assume that c ℓ converges uniformly, its constant geodesic curvature converges, thus c ℓ is a graph which C 2 -converges to ∂D. Since f ′ /f is increasing, f ′2f ′′ f ≤ 0 < 1. Lemma 17 implies that c ℓ = {r = r ℓ } is a parallel for ℓ large enough.

For ℓ large, D ℓ contains {r ≤ r ℓ } with r ℓ close to r 0 . The other components of D ℓ are thus contained in {r ≥ r ℓ }. By uniform ultrahyperbolicity on the {r ≥ 0} side, they are contained in fact in a fixed compact set on with the metrics converge C 2 -uniformly. Short closed curves contained in this set cannot have bounded geodesic curvature. We conclude that D ℓ = {r ≤ r ℓ }.

Caps with prescribed isoperimetric profile

Corollary 20 Let δ > 0 be small enough and k, α be large enough. There exists a smooth complete surface of revolution S = S δ,k,α such that 1. S is ultrahyperbolic;

2. S contains a rotationally symmetric annulus of constant curvature -1 and area α + δ, whose inner boundary has length δ and inner disk has area δ;

3. the curvature at the origin is equal to k;

4. all rotationally symmetric disks in S are extremal;

5. the square I 2 S of the isoperimetric profile of S extends to a smooth function on a neighborhood of R + ; 6. I S is nondecreasing and v → I S (v)/v is nonincreasing;

7. for δ ≤ v ≤ α, I S (v) = v; 8. for all v, I S (v) ≥ √ 4πv -k v 2 .
Proof. Let S = (R + × R/2πZ, g = dr 2 + f (r) 2 dθ 2 ) be a surface of revolution. Let us first collect necessary conditions on the function f . Assume that all disks of revolution are extremal domains in S. In other words, the isoperimetric profile I S is determined by

V ′ (r) = I S (V (r)). (2)
where V (r) = 2π r 0 f (s) ds is the area of the disk of revolution of radius r. For f to give rise to a smooth metric on a plane, it is necessary and sufficient that f extends to a smooth odd function on R such that f ′ (0) = 1. Therefore V extends to a smooth even function, i.e. V (r) = G(r 2 ) where G is smooth on a neighborhood of 0. Since V ′′ (0) = 2πf ′ (0) = 2π, V (r) ∼ πr 2 and G ′ (0) = π, so the inverse map G -1 is smooth on a neighborhood of 0. Equation (2) implies that

I S (G(r 2 )) 2 = 4r 2 G ′ (r 2 ) 2 , i.e. for v ≥ 0, I S (v) 2 = 4G -1 (v)G ′ (G -1 (v)) 2
is the restriction of a smooth function.

Differentiating equation (2) yields

V ′′ = V ′ I ′ S (V ) = I S (V )I ′ S (V ) = 1 2 dI 2 S dv (V ),
The first derivative of I S is given by

dI S dv (V ) = V ′′ V ′ = f ′ f , thus I is convex if and only if f ′ /f is nondecreasing. Also dI 2 S dv (0) = 2V ′′ (0) = 4π.
Differentiating once more yields

V ′′′ = 1 2 V ′ d 2 I 2 S dv 2 (V ).
The curvature, as a function of the distance r to the origin, is given by

K(r) = - f ′′ f = - V ′′′ V ′ = - 1 2
d 2 I 2 S dv 2 (V (r)) = -(I S d 2 I S dv 2 + ( dI S dv ) 2 )(V (r)), which yields d 2 I 2 S dv 2 (0) = -2K(0).
Conversely, let α be large. Choose first a smooth convex function I 0 on R + such that I 0 (v) = v for v ≤ α, and I ′ 0 tends to +∞. The corresponding surface of revolution satisfies f (r) = e r on (-∞, log(α/2π)], f is nondecreasing, f ′ /f is nondecreasing and tends to +∞, and the curvature

-f ′′ f tends to -∞.
Given δ < α and k large enough, there exists a function I = I δ,k,α on R + such that

• I 2 extends to a smooth function on a neighborhood of R + ;

• I(0) = 0, dI 2 dv (0) = 4π and d 2 I 2 dv 2 (0) = -2k; • I(v) = I 0 (v) for v ≥ δ; • d 2 I 2 dv 2 ≥ -2k everywhere;
• I is nondecreasing and v → I(v)/v is nondecreasing;

• as δ tends to 0 (and k tends to +∞), I = I δ,k,α converges to I 0 , C ∞ -uniformly on compact subsets of R -. The corresponding surface of revolution S δ,k,α is smooth, it satisfies g δ = f on [log(δ/2π), +∞), g δ is nondecreasing. Proposition 19 implies that for δ small enough, disks of revolution are extremal in S δ,k,α , and the isoperimetric profile of S δ,k,α is equal to I δ,k,α .

Cusp filling

Let M be an orientable surface with at least two ends. Remove two points from M and equip M with a metric g and calibration in such a way that neighborhoods of the deleted points be isometric to constant curvature -1 cusps. According to preceding sections, the isoperimetric profile of (M, g) is linear. Furthermore, away from the cusp neighborhoods, (M, g) has bounded curvature and injectivity radius. Let us cut cusp neighborhoods along horocycles of equal length and fill them with smooth caps of revolution where the isoperimetric profile is achieved by disks of revolution. We show that the isoperimetric profile of the obtained metric is smooth.

The gluing procedure

The following elementary properties will be needed.

Lemma 21 Let f and g be nonnegative functions on R + such that v → f (v)/v and v → g(v)/v are nonincreasing. Then min{f, g} has the same property. Assume furthermore that f (δ) = g(δ) for some δ > 0. Then the function h such that h = f on [0, δ] and h = g on (δ, +∞) also shares the same property. Any such function is subadditive, i.e.

f (v + v ′ ) ≤ f (v) + f (v ′ ) for all v, v ′ ∈ R + .
Lemma 22 Let M denote a 2-dimensional complete Riemannian manifold with curvature bounded from above, K ≤ k. Assume that there exists ǫ > 0 such that M ǫ = {x ∈ M ; inj(x) < ǫ} consists of a disjoint union of annuli.

Let I d denote the modified isoperimetric profile in which the competing domains are restricted to disjoint unions of domains of one of the following two types, 1. disks;

2. domains having at least one boundary component which is homotopic neither to zero nor to a component of ∂M ǫ .

Then there exists η > 0, η < 2π/k, such that

I d M (v) ≥ √ 4πv -kv 2 for v < η.
Proof. First, I d tends to zero, since small disks have small boundary length. A simple closed curve c of length < ǫ which is not homotopic to zero is contained in M ǫ . Then c is homotopic to a boundary component of M ǫ with one of the two possible orientations. Therefore domains of the second type do not contribute to I d (v) for v small. For disks, the Bol-Fiala isoperimetric inequality applies, [START_REF] Bol | Isoperimetrische Ungleichungen für Bereiche auf Flächen[END_REF], [START_REF] Fiala | Sur le problème des isopérimètres sur les surfaces ouvertes à courbure positive[END_REF]. For unions of disks, use the subadditivity of v → √ 4πvkv 2 , which follows from Lemma 21.

Lemma 23 Let m be an integer and 0 < δ < α/m. Let S j δ,α , j = 1, . . . , m, denote m smooth surfaces with boundary, of revolution, with area α. Denote by C j δ,α the rotationaly symmetric disk of revolution in S j δ,α with area δ. Assume that • all S j δ,α \ C j δ,α are constant curvature -1 annuli with inner boundary of length δ;

• for all v ≤ mδ, I S j δ,α is achieved by a rotationally symmetric disk;

• I S j δ,α
is nondecreasing, and v → I S j δ,α (v)/v is nonincreasing;

• for every k, there exists δ(k) > 0 such that for v < δ < δ(k), I S j δ,α

(v) ≤ √ 4πv -kv 2 .
Let M denote an orientable 2-dimensional complete Riemannian manifold. Assume that

• there exists ǫ ≫ δ such that M ǫ = {x ∈ M ; inj(x) < ǫ} consists of m disjoint constant curvature -1 cusps;

• let δ ′ be the number such that area(M δ ′ ) = mδ, then for every v > mδ, I M (v) is achieved by a domain which contains M δ ′ ;

• M has curvature bounded from above, K ≤ k;

• I M is nondecreasing;

• for all j = 1, . . . , m, I M = I S j δ,α on [mδ, mα] where α ≤ area(M ǫ )/m does not depend on α.

Remove each component of M δ ′ and replace it with a disk C j δ,α , to produce a smooth surface N . If δ is small enough, then I N is achieved,

I N (v) = min j I S j δ,α (v) for v ≤ mδ, I N (v) = I M (v) for v ≥ mδ.
Proof. Since the candidate isoperimetric profile is nondecreasing, when establishing the isoperimetric inequality for a domain D in N , we can replace D with any other domain with larger area and shorter boundary. For instance, if some boundary component c of D is null homotopic, it bounds a disk D ′ . Replace D with D ∪ D ′ . This increases the area of D and decreases its boundary length. Therefore, we do not loose generality in assuming that D is a disjoint union of domains which, unless they are disks, have no null homotopic boundary component.

Denote by N ǫ the complement of M \ M ǫ in N . Let w : N ǫ → R be the function, extending the (exponential of minus) horofunction on the constant curvature -1 annuli of N , such that area({w < t}) = t. Then {w < mδ} is the union of the caps C j δ,α , {w < mα} is isometric to j S j δ,α . If length(∂D ∩ {w < mα}) ≥ length({w = mα}), then replacing D with D ∪ {w < mα} increases area and decreases boundary length. Thus we can assume that length(∂D ∩ {w < mα}) < length({w = mα}).

The distance in M (and therefore in N ) between the level sets {w = m 2 δ} and {w = mα} tends to +∞ as δ tends to 0. If δ is small enough, there exists t ∈ (m 2 δ, mα) such that ∂D ∩ {w = t} = ∅. If the j-th component of {w = t} is contained in D, then the whole j-th component of {w < t} is contained in D. Otherwise, let us replace the part of D contained in the j-th component of {w < t} with the disk of the form {w < t j } in the same component which has the same area. Due to the isoperimetric inequality in S j δ,α , this decreases boundary length. Assume that area(D) < mδ. Then no component of D can contain a component of {w < m 2 δ}. Let v j denote the area of the part of D contained in the j-th component of {w < t} and v ′ = area(D)j v j = area(D ∩ {w > t}). Note that Lemma 22 applies to D ∩ {w > t}. We can assume that mδ < η, the constant in this lemma, and that δ < δ(k). Then length(∂D) , the boundary length gets even smaller in M than in N . This

≥ j I S j δ,α (v j ) + I d M (v ′ ) ≥ j min{I S j δ,α }(v j ) + min{I S j δ,α }(v ′ ) ≥ min{I S j δ,α }( j v j + v ′ ) ≥ min{I S j δ,α }(area(D)), since I d M (v) ≥ √ 4πv -k v 2 ≥ min{I S j δ,α } for v ∈ [0,
shows that length(∂D) ≥ I M (area(D)).

We have shown that

I N ≥ min{I S j δ,α
} on [0, mδ] and I N ≥ I M elsewhere. Conversely, let v ≤ mδ.

Let j be the index for which I S j δ,α (v) is minimum. Then there exists in S j δ,α a rotationally symmetric domain D of area v and minimal boundary length. This domains naturally embeds in N . Let v > mδ. By assumption, I M (v) is achieved by a domain which contains M δ ′ . Again, this domain has an obvious counterpart in N with the same area and the same boundary length. We conclude that I N is achieved and is equal to min{I S j δ,α } on [0, mδ] and to I M elsewhere.

Smooth profile for 2 ended surfaces

Let M be an orientable 2-manifold with at least two ends or one end of infinite genus. Remove two points from M to get M . Then M satisfies the assumptions of Theorem 2. One can equip M with a Morse function w : M → (0, +∞) with compact level sets and a metric modelled on constant curvature -1 annuli, such that, for some t > 0, {w < t} consists of two cusps. Note that on {w ≥ t}, the injectivity radius in bounded below and the curvature is bounded above. Theorem 2 yields a calibration ω which garantees that the isoperimetric profile is linear

I M (v) = v.
Choose δ very small, according to Lemma 23. Apply Corollary 20 in order to choose appropriate caps of revolution. Arrange so that one of the caps has an isoperimetric profile which is everywhere smaller than the other. Lemma 23 garantees that the resulting surface N has a smooth isoperimetric profile. This completes the proof of the first part of Theorem 3.

Conformal changes of metrics

Proposition 25 Let (M, g) be an oriented Riemannian manifold, w a smooth positive function on M , ω a calibration on M , i.e. |ω| = 1 and dω = vol. Assume that there exist t 1 < t 0 such that • all critical values of w belong to (-∞, t 0 );

• |∇w| = w on {w > t 0 };

• on {w > t 0 }, ω = ι w -1 |∇w| vol;

• for all t ≥ t 0 or t ≤ t 1 , vol({w < t}) = t = vol({w = t});

• for all v < t 0 , I M (v) = v is asymptotically achieved by a sequence of domains which are contained in {w ≤ t 0 }.

Let I be a smooth convex function on R + such that I(v) = v for v ≤ t 0 . Then there exists a conformal metric g ′ = f 2 g on some sublevel set

M ′ = {w < τ } ⊂ M diffeomorphic to M such that g ′ ≡ g on {w ≤ t 0 }, vol(M ′ ) = +∞ and I M ′ = I.
Proof. Let τ be some positive number, and f :

[0, τ [→ R + a smooth function such that f ≡ 1 on [0, t 0 ]. Let g ′ denote the conformal metric f (w) 2 g on M ′ = {w < τ } ⊂ M . Let V (t) = vol ′ ({w < t})
denote the volumes of sublevel sets of w in the new metric. According to the coarea formula, for t ≥ t 0 ,

V (t) = V (t 0 ) + {t 0 <w<t} f (w) n = t 0 + t t 0 f (s) n ( {w=s} 1 |∇w| ) ds = t 0 + t t 0 f (s) n ds = t 0 f (s) n ds, since vol({w = s}) = s = w = |∇w| on {w = s} when s ≥ t 0 . In particular, f (t) = V ′ (t) 1/n . Similarly, if t ≥ t 0 , vol ′ ({w = t}) = f (t) n-1 vol({w = t}) = f (t) n-1 t.
Assume that for all v ≥ t 0 , the isoperimetric profile I = I (M ′ ,g ′ ) is achieved by sublevel sets of w.

Then, for all t ≥ t 0 , vol ′ ({w = t}) = I(vol ′ ({w < t})).

This implies that

f (t) n-1 t = I( t 0 f (s) n ds), i.e. V ′ (t) n-1 n t = I(V (t)). (3)
This is a differential equation which, together with the initial condition V (t 0 ) = t 0 , uniquely determines V , and therefore f = V ′1/n .

Conversely, given a smooth function

I : R + → R + such that I(v) = v for v ≤ t 0 , let V : [0, τ [→ R + be
= f (w) n vol, ω ′ = f (w) n-1 ω, thus dω ′ = (n -1)f (w) n-2 f ′ (w) dw ∧ ω + f (w) n-1 dω = ((n -1)f (w) n-2 f ′ (w)|∇w| + f (w) n-1 ) vol = u vol ′ ,
where u = h(w) and

h(t) = f (t) -n ((n -1)f (t) n-2 f ′ (t)t + f (t) n-1 ) = (f n ) -1 d dt (f n-1 t) = ( dV dt ) -1 dI(V (t)) dt = dI dv (V (t)).
If I is convex, h is nondecreasing, therefore large sublevel sets of w satisfy

t ≥ t 0 ⇒ {u < h(t)} ⊂ {w < t} ⊂ {u ≤ h(t)}.
Using, as compact approximations to {w < t}, the domains {ǫ < w < t}, Lemma 24 applies, and for v ≥ t 0 , I M ′ (v) is achieved by the sublevel set of w of volume v. For all v, I M ′ (v) ≥ v.

By assumption, there exists for each v < t 0 a sequence of approximate extremal domains of area v contained in {w ≤ t 0 }. These domains are unaffected by the conformal change. Therefore I M ′ (v) = v for v < t 0 . This proves that I M ′ = I.

Curvature and injectivity radius

Let M be a compact orientable 2-dimensional Riemannian manifold. Remove finitely many (but at least two) points from M to get M . Choose two more points a and b. Apply Theorem 2 to M \ {a, b}. Get a Morse function w : M \ {a, b} → (0, +∞) with compact level sets, a metric g modelled on constant curvature -1 annuli except near critical points of w. Let t 0 (resp. t 1 ) be slightly larger (resp. smaller) than the largest (resp. smallest) critical value. Then {w < t 1 } consists of two constant curvature -1 cusps and {w > t 0 } of finitely many constant curvature -1 anticusps. Choose some smooth convex function I on R + such that I(v) = v for v ≤ t 0 and I(v) = v log(v) for large v. Apply Proposition 25 to get a conformal metric on M ′ = {w < τ } with isoperimetric profile equal to I.

As observed in [START_REF] Bavard | Sur le volume minimal de R 2[END_REF], the second variation formula relates the first and second derivatives of the isoperimetric profile at v to the Ricci curvature in the normal direction along the boundary of an extremal domain of volume v. Here, since the Gauss curvature K is constant along the boundary of the extremal domain of volume v, one gets

d 2 I 2 M ′ dv 2 (v) = -2K.
Therefore, K tends to -∞ as w tends to τ .

Recall that the conformal change of metric is g ′ = f (w) 2 g where f = V ′1/2 and V is the solution of V ′ (t)

1 2 t = I(V (t)) with initial condition V (t 0 ) = t 0 . The formulae 1 t - 1 τ = +∞ V (t) I(s) -2 ds and f (t) = V ′ (t) 1/2 = I(V (t)) t
show that the asymptotic behaviour of f (t) as t tends to τ depends only on the asymptotic behaviour of I(v) as v tends to +∞. An explicit calculation with

I(v) = v log(v) shows that 1 t -1 τ ∼ V (t) -1 (log(V (t))) -2 , log(V (t)) ∼ log(τ t/τ -t) and f (t) ∼ τ t τ -t (log( τ t τ -t )) -1 .
In particular,

τ t 0 f (t)
t dt = +∞. This implies that the metric

g ′′ = g + f (w) 2 w 2 dw 2 on {w < τ } is complete. Since g ′ ≥ g + f (w) 2 -1 w 2 dw 2 ,
the identity map ({f (w) ≥ 2}, g ′ ) → ({f (w) ≥ 2}, g ′′ ) is Lipschitz. This proves that (M ′ , g ′ ) is complete.

Let x j be a sequence of points in M ′ such that w(x j ) tends to τ . Assume that inj(x j ) is bounded by L. Since M ′ is complete and negatively curved away from some sublevel set of w, for j large, inj(x j ) is equal to half the length of a geodesic loop ℓ j which not null homotopic. Since M has injectivity radius bounded below away from {w < t 1 }, the g-length of ℓ j is bounded below. Since ℓ j ⊂ B ′ (x j , L) ⊂ {w > w(x j )-ǫ j } where ǫ j tends to 0, f is large at each point of ℓ j , a contradiction. We conclude that inj(x) tends to +∞ as w(x) tends to τ .

To complete the proof of Theorem 3, there remains to fill the cusps as described in paragraph 9.2.

Remark 26

In the examples produced in Theorem 3, the isoperimetric profiles passes through three different regimes.

1. At small areas, the profile is subadditive. One can arrange that the profile be achieved by a unique extremal domain. For this, it suffices to arrange that one of the caps be more positively curved than the other. There is some flexibility in the choice of the prescribed profile.

2. At medium areas, the profile is exactly linear. There is no flexibility. The profile is achieved by large families of extremal domains.

3. At large areas, the profile is superadditive. There is again flexibility in the choice of the profile.

It is achieved by a unique extremal domain.

11 Appendix : ultrahyperbolicity

Proposition 27 Let M be a non compact complete 2-dimensional Riemannian manifold M which is ultrahyperbolic, i.e.

• curvature tends to -∞ at infinity;

• injectivity radius tends to +∞ at infinity.

On such a surface, the isoperimetric profile is achieved. Therefore, it is continuous on R + . Furthermore, for every V < area(M ), the set of domains D such that area(D) ∈ [0, V ] and length(∂D) = I M (area(D)), is compact.

Proof. Given ρ < 0 and L > 0, let C(ρ, L) denote the tubular neighborhood of width 2L of the set of points x ∈ M such that either inj(x) ≤ L or K(x) ≥ ρ. By assumption, this is a compact set. Denote by J the function defined on R + by J ρ (v) = 4πvρv 2 .

We first show that for every a > 0, if ρ < -( 4 L cosh -1 (1+ a 2π )) 2 , then every connected domain D with area ≤ a and boundary length ≤ L which is not contained in C(ρ, L) satisfies the isoperimetric inequality length(∂D) ≥ J ρ (area(D)). Let x ∈ D \ C(ρ, L). Then on the ball B(x, 2L), the injectivity radius is everywhere ≥ L and the curvature everywhere ≤ ρ. Let y ∈ B(x, L). Since area(B(y, L/4)) ≥ 2π(cosh( √ -ρL/4) -1) > a, the ball B(y, L/4) overlaps the boundary of D.

This implies that d(x, y) < L 4 + L 4 + 1 2 length(∂D) ≤ L. This show that D is contained in B(x, L). The Bol-Fiala inequality for disks ( [START_REF] Bol | Isoperimetrische Ungleichungen für Bereiche auf Flächen[END_REF], [START_REF] Fiala | Sur le problème des isopérimètres sur les surfaces ouvertes à courbure positive[END_REF]) implies that length(∂D) ≥ J ρ (area(D)). Indeed, D is homeomorphic to a disk with holes, filling the holes yields a disk D ′ , to which the Bol-Fiala isoperimetric inequality applies : length(∂D) ≥ length(∂D ′ ) ≥ J ρ (area(D ′ ))

≥ J ρ (area(D)), since J ρ is nondecreasing. Note that this implies that √ -ρarea(D) ≤ length(∂D).

Let D ℓ be a sequence of domains such that area(D ℓ ) converges to v, staying less than a, and length(∂D ℓ ) converges to I M (v), staying less than L. According to the compactness theorem for integral currents [START_REF] Federer | Geometric measure theory[END_REF], one can diagonally extract a subsequence such that for every ρ < 0, D ℓ ∩C(ρ, L) converges in flat norm. Since area(∂D ℓ \ C(ρ, L)) ≤ L/ √ -ρ, the limiting current with unbounded support D has area v. By semi-continuity, length(∂D) ≤ I M (v). Since D minimizes boundary length for compactly supported area preserving perturbations, D is a locally finite union of smooth domains, [START_REF] Fleming | Flat chains over a finite coefficient group[END_REF].

Fix a point x 0 ∈ M \ D ∞ . Choose ρ ′ < K(x 0 ). When r tends to 0, length(∂B(x 0 , r)) and area(B(x 0 , r)) behave asymptotically like in constant curvature K(x 0 ). It follows that, for w small enough (say w ≤ v 0 ), the function I such that length(∂B(x 0 , r)) = I(area(B(x 0 , r)) satisfies I(w) ≤ 4πwρ ′ w 2 . Let v 1 = min{v 0 , area(B(x 0 , dist(x 0 , D)))}. Given a and L, choose ρ < ρ ′ such that ρ < -( 4 L cosh -1 (1 + a 2π )) 2 and L √ -ρ < v 1 . Let B = B(x 0 , r 0 ) denote the ball centered at x 0 with area v 1 . By construction, B does not intersect D.

Assume that D is not contained in C(ρ, L). Let D 2 be the union of components of D which are not contained in C(ρ, L), and D 1 = D\D 2 . For r > 0, let D r = D 1 ∪B(x 0 , r). Since every component ∆ of D which is not contained in C(ρ, L) satisfies √ -ρarea(∆) ≤ length(∂∆), area(D 2 ) ≤ L √ -ρ < v 0 . Therefore there exists r ∈ (0, r 0 ) such that area(D r ) = area(D). Then length(∂D r )length(∂D 1 ) = length(∂B(x 0 , r))

≤ I(area(B(x 0 , r)))

≤ 4πα -kα 2 ,
where α = area(D 2 ). Since v → J ρ (v)/v is nonincreasing, J ρ is subadditive. Therefore the inequality length(∂∆) ≥ J ρ (area(∆)), valid for each component ∆ of D 2 , implies length(∂D 2 ) ≥ J ρ (area(D 2 )). It follows that length(∂D r ) < length(∂D). This contradicts the extremality of D.

We conclude that D is compact.

This proves the existence of compact extremal domains. According to [START_REF] Bavard | Sur le volume minimal de R 2[END_REF], this implies that the isoperimetric profile is continuous. The compactness of the set of extremal domains follows.

Remark 28 Here is a relative version of Proposition 27.

Figure 1 :

 1 Figure 1: Calibrated regions in a singular surface

Figure 2 :Figure 3 :

 23 Figure 2: The pipe clearing trick

Proposition 3

 3 Let M be an oriented Riemannian manifold with volume element vol M , let ω be an n -1-form on M such that |ω| ≤ 1, and dω = c vol M for some constant c. Let V ⊂ M be a submanifold with compact boundary and finite volume v. Assume that • ω calibrates ∂V , i.e. ω |∂V = vol ∂V .

Proof.

  Let ξ denote the Busemann vectorfield attached to ζ (i.e. ξ(x) is the initial speed of the unit speed geodesic starting at x and converging to ζ). Then ξ descends to a well defined unit vectorfield on M . Consider the n -1-form ω = ι ξ vol M . Then |ω| = |ξ| = 1, and dω = (n -1)vol. Horospheres centered at ζ descend to hypersurfaces orthogonal to ξ in M . Each horoball centered at ζ descends to an open set B v of finite volume v, and v takes every value in (0, +∞). With the orientation induced from horoballs, horospheres are calibrated by ω. Proposition 3 applies to B v,ǫ = B v \ B ǫ .

Figure 4 :

 4 Figure 4: The level graph of a function

Figure 6 :

 6 Figure 6: Gluing together singular pairs of pants

Lemma 11

 11 Let D be a n-dimensional disk, g 0 a smooth Riemannian metric and ω 0 a smooth n -1-form on D such that |ω 0 | 0 = 1 and dω 0 = vol 0 in a neighborhood of the boundary. Assume that D dω 0 > 0. Then there exists a smooth Riemannian metric g and a smooth n -1-form ω on D such that 1. g = g 0 and ω = ω 0 in a neighborhood of ∂D, 2. |ω| ≤ 1 and dω = vol in D, 3. vol(D, g) = D dω 0 ≤ vol 0 (∂D). Proof. First change g 0 into g 1 away from the boundary, in order that vol(D, g 1 ) = D dω 0 . This is possible since D dω 0 > 0 by assumption. Let η be a compactly supported n-form on D. According to the Poincaré Lemma, there exists a compactly supported n -1-form β on D such that dβ = η if and only if D η = 0. Therefore there exists a compactly supported n -1-form β such that ω = ω 0 + β satisfies dω = vol 1 .

Figure 7 :

 7 Figure 7: A Morse function with disconnected levels on a compact surface with boundary

Figure 8 :

 8 Figure 8: Infinite genus end

  but unfortunately not all of them. This works for levels which are not in an open neighborhood E of the set of critical values. E is a discrete union of intervals of the form (c -, c + ), each of them containing exactly one critical value c, c -= c/2, c + = 2c.

Figure 9 :

 9 Figure 9: C e,c ′′ -

  s) ds. Therefore C 2 metrics and C 2 annular domains of a given area v are conveniently described by pairs of functions G : [a, b] → R, R : R/2πZ → R such that G(0) = 0, G ′ > 0 and 2π 0 R(θ) dθ = 0. g and ρ are recovered via g = G ′ and ρ = G -1 • (R + v). Let C 3 0 ([a, b]) denote the space of C 3 functions G on [a, b] such that G(a) = 0. Let C 2 0 (R/2πZ) denote the space of C 2 functions R such that 2π 0 R(θ) dθ = 0. Let U ⊂ R × C 3 0 ([a, b]) × C 2 v (R/2πZ) denote the open subset of triples (v, G, R) such that v > 0, G ′ > 0 and 0 < R + v < G(b). Let C 0 0 (R/2πZ) denote the space of continuous functions with vanishing average. Let Φ : U → C 0 0 (R/2πZ) denote the C 1 map defined by Φ(v, G, R) = κ -

Figure 10 :

 10 Figure 10: Prescribed isoperimetric profile

  mδ] by Lemma 22 and min{I S j δ,α } is subadditive, by Lemma 21. If area(D) > mδ, let us construct a domain D ′ in M with the same area as D. The components of D which do not intersect {w < t} naturally live in M . The components of D which contain components of {w < t} also have obvious counterparts in M . To those components of D which are contained in {w < t}, we associate a domain of the form {w < t j } in the corresponding end of M with equal area. Since I M ≤ I S j δ,α

10 Construction of ultrahyperbolic surfaces 10 . 1

 101 The rearrangement argumentWe need a generalization of Lemma 3.Proposition 24 Let M be an oriented Riemannian manifold with volume element vol M , let ω be an n -1-form on M such that |ω| ≤ 1, and dω = u vol M for some nonnegative function u. Let V ⊂ M be a submanifold with compact boundary and finite volume v. Assume that• ω calibrates ∂V , i.e. ω |∂V = vol ∂V ;

  can be applied near each critical point p. Let ℓ(p) denote the minimum length of a boundary component of an annulus that contains p. Then ℓ(p) increases with u(p). Indeed, when moving from a critical value to the next, the total length of the critical level is multiplied at least by 16, and weights are at most divided by 2. The injectivity radius of the bi-infinite curvature

-1 cusp at a point on the horocycle of length ℓ(p) also increases. Therefore, the radius r(p) of the ball centered at a critical point p to which Lemma 11 can be applied increases with u(p). On the ball B(p, r(p)), u takes values in (e -r(p) u(p), e r(p) u(p)). Since E contains the interval (u(p)/2, 2u(p)), one can perform exactly the same smoothing on B(p, log 2) for all p, provided u(p) is large enough. This gives bounded geometry.

  the maximal solution of Equation 3 with initial condition V (t 0 ) = t 0 . Then V (t) = t for t ≤ t 0 and lim t→τ V (t) = +∞. One sets f = V ′1/n . Let ξ = ∇w |∇w| . By assumption, the given calibration ω is equal to ι ξ vol on {w ≥ t 0 }. Let ξ ′ = ∇w |∇w| ′ and ω ′ = ι ξ ′ vol ′ . By definition, |ω ′ | ′ ≤ 1 and ω ′ calibrates level sets {w = t} for t ≥ t 0 . Since |∇w| ′ = f (w) -1 |∇w|, and vol ′

• there exists t such that {u < t} ⊂ V ⊂ {u ≤ t};

• there exist compact domains

Since ω calibrates ∂V , ∂V ω = vol(∂V ). We have shown that I M (v) ≥ vol(∂V ), and that equality holds asymptotically for the domains V ǫ . We conclude that I M (v) = vol(∂V ).


S A 2013 Barannikov

François Laudenbach

Keywords:

Given a Morse function f on a closed manifold M with distinct critical values, and given a field F, there is a canonical complex, called the Morse-Barannikov complex, which is equivalent to any Morse complex associated with f and whose form is simple. In particular, the homology of M with coefficients in F is immediately readable on this complex. The bifurcation theory of this complex in a generic one-parameter family of functions will be investigated. Applications to the boundary manifolds will be given.

Here is an expanded version of the lectures given in the Winter school organized in La Llagonne by the University Paul Sabatier in Toulouse (January 2013) 1 . There are three parts:

I) The Morse-Barannikov complex (after C. Viterbo), II) Bifurcations, III) The non-empty boundary case.

The Morse-Barannikov complex

We adopt a presentation which is a mix of the presentation given by S. Barannikov in [START_REF] Barannikov | The framed Morse complex and its invariants[END_REF] and a more abstract one given by C. Viterbo in his joint work with D. Le Peutrec and F. Nier [START_REF] Le Peutrec | Precise Arrhenius law for p-forms, The Witten Laplacian and Morse-Barannikov complex[END_REF] which we slightly simplify.

We are given a closed manifold M , a Morse function f : M → R whose critical values are distinct, and a field F. For each integer k the critical points of index k are numbered in the increasing order of the critical values: f (p 1 ) < f (p 2 ) < . . . (the function is just generic and it is not assumed to be ordered). We shall often identify the set of critical points and the set of critical values.

For defining the Morse complex it is necessary to have two extra data:

-A (decreasing) pseudo-gradient, that is, a vector field on M which satifies X • f < 0 out of the critical points, and some non-degeneracy condition for the vanishing of X at each critical point; therefore, the zeroes of X are hyperbolic. As a consequence, each critical point p has a stable manifold W s (p) and an unstable manifold W u (p). This pseudogradient is chosen Morse-Smale, a generic property meaning that the stable manifolds are transverse to the unstable manifolds. -An orientation of the unstable manifolds.

The Morse complex C * (f, X) is made as follows. In degree k the module C k (f, X) is the free Z-module generated by the critical points of index k and the differential ∂ k : C k (f, X) → C k-1 (f, X) counts the signed number of connecting orbits. Observe that, for (p, q), a pair of critical points of respective index k and k -1, W u (p) ∩ W s (q) is made of a finite number of connecting orbits. Since W s (q) is co-oriented by the orientation of W u (q), each orbit descending from p to q gets a sign. Define a q p to be the signed number of the connecting orbits and define the Morse differential C k (f, X) by ∂ k (p) = a q p q .

Theorem 1.1. (Milnor [6], Th. 7.2). The Morse complex is a chain complex:

∂ • ∂ = 0.
Moreover, H (C * (f, X)) ∼ = H * (M ; Z). A fortiori, H (C * (f, X); F) ∼ = H * (M ; F).

Definition 1.2. A chain complex C * with coeffcients in F is said to be F-equivalent to C * (f, X) if it has the same generators and if its differential δ is made from ∂ by conjugating in each degree by an invertible upper triangular matrix T with coefficients in F:

C k+1 (f, X) ⊗ F T ∂ / / C k (f, X) ⊗ F ∂ / / T C k-1 (f, X) ⊗ F T C k+1 δ / / C k δ / / C k-1
Here, C k (f, X) ⊗ F is a vector space equipped with its canonical ordered basis.

Remark 1.3. When changing the pseudo-gradient or the orientations of the unstable manifolds the Morse complex is changed by Z-equivalence. Conversely, if dim M > 1 and if the level sets of f are connected (or f has one local minimum and one local maximum only, i.e. f is polar in Morse's terminology) every Z-equivalence is realizable by such changes. When the coefficients are in a field an F-equivalence has no longer such a geometrical meaning in general. But, an F-equivalence keeps the memory of the filtration by the sub-level sets of the function f . This fact will be used in the last step of the proof of Barannikov's theorem.

Theorem 1.4. (Barannikov [1]) The Morse complex C * (f, X) is F-equivalent to a simple complex (C * , ∂ B ), that is, for every generator p, ∂ B (p) is 0 or a generator and ∂ B (p) = ∂ B (p ) if p = p and ∂ B (p) = 0. Moreover, (C * , ∂ B ) is unique and depends only on C * (f, X) ⊗ F for any pseudo-gradient X.
This complex is called the Morse-Barannikov complex associated with the Morse function f ; it depends on the field F. One important point in the statement is the coupling of some critical points, the unpaired generators being "isolated" in the complex. This fact plays a deep rôle in the work by Y. Chekanov & P. Pushkar [START_REF] Chekanov | Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures[END_REF]. When the Morse complex is concentrated in two degrees, the statement amounts to the fact that the double coset GL(n, Z)/T (n) × T (n) is isomophic to the symmetric group S n , a fact which was important in Cerf's work on pseudo-isotopy (here, T (n) denotes the sub-group of invertible upper triangular matrices) [START_REF] Cerf | La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudoisotopie[END_REF]. In Barannikov's paper the proof of existence follows more or less from Gauss' algorithm. The proof of uniqueness remains mysterious. It is clarified by C. Viterbo in [START_REF] Le Peutrec | Precise Arrhenius law for p-forms, The Witten Laplacian and Morse-Barannikov complex[END_REF].

Viterbo's important remark is that the critical points of the given Morse function f are divided in three types: upper, lower and homological, depending of the place of a zero map in the diagram below of F-vector spaces and F-linear maps, in which c denotes a critical value of index k + 1 and, for brevity, c + ε stands for the sub-level set

f c+ε := f -1 (-∞, c + ε] : F 0 / / H k+1 (c -ε) / / H k+1 (c + ε) J / / H k+1 (c + ε, c -ε) ∼ = O O ∆ / / I H k (c -ε) / / H k (c + ε) / / 0 H k+1 (+∞, c -ε) ∆ 6 6
The horizontal line is an exact sequence. The critical point p such that f (p) = c is said to be of upper type when J = 0, implying ∆ injective. It is said to be of lower type when J is surjective and I = 0. It is said of homological type when I is injective and ∆ = 0. Clearly, since F is a field, all possibilities are covered, making a partition of the critical points.

The type of a critical point p of f is readable on the Morse complex with coefficients in F, that is C * (f, X) ⊗ F. For instance, p is of lower type if there is an F-linear combination of critical points higher than p whose boundary is p ; this property does not depend on the chosen pseudo-gradient X.

The coupling of critical points. If p is a critical point of index k + 1, the local unstable manifold W u loc (p) is unique up to isotopy and orientation. Set where σ runs among the k-cycles of the sub-level set f c-ε representing ∆([p]); it is a critical value λ(p) = f (q) where q is a critical point of index k. By identifying critical point and critical value, we set q := λ(p).

Lemma 1.6. The critical point q := λ(p) is of lower type.

Proof. Denote ∆ q ([p]) the class of ∆([p]) modulo the sub-level set f (q) -ε in H k f (p) -ε, f (q) -ε .
By definition of the minimax, this class is not zero and we have

∆ q ([p]) = αI p q [q] , α ∈ F, α = 0, where I p q : H k (f (q) + ε, f (q) -ε) → H k (f (p) -ε, f (q) -ε
) is induced by the inclusion. Thus, W u loc (q) is the boundary of 1 α W u (p) in the pair (+∞, f (q) -ε). Hence I([q]) = 0. Lemma 1.7. The map λ defines a bijection from the set of critical points of upper type onto the set of critical points of lower type.

The

Proof. 1)

Injectivity. Let q = ∂ B (p) = ∂ B (p ) with f (p) > f (p ).
We are using the same notation as in Lemma 1.6. We have

∆ q ([p]) = αI p q ([q]) = 0 in H k f (p) -ε, f (q) -ε and ∆ q ([p ]) = α I p q ([q]) = 0 in H k f (p ) -ε, f (
q) -ε . By construction I p q ∆ q ([p ]) = 0. Thus, we have:

∆ q ([p]) = ∆ q ([p]) - α α I p q ∆ q ([p ]) = αI p q ([q]) - α α α I p q ([q]) = 0 .
Therefore, f (q) is not the minimax value associated with p.

2) Surjectivity. Let q be a critical point of lower type and index k. Set

µ(q) = inf σ max(f |σ)
where σ runs among the relative chains of (+∞, f (q) -ε) whose boundary is a relative cycle representing the class [q]. This µ(q) is a critical value of index k + 1 with µ(q) = f (p) for some critical point p. A chain σ approximating the infimum has a non vanishing class in

H k+1 f (p) + ε, f (p) -ε ; hence, [σ] = β[p] in the pair (f (p) + ε, f (p) -ε) with β ∈ F, β = 0. We have ∆ q [p] = I p q ( 1 β [q]
). If this element is zero, this means that there is another relative chain bounded by W u (q) under the level of p, contradicting the definition of µ(q). Then, ∆ q ([p]) = 0. A fortiori, ∆([p]) = 0 and p is of upper type.

We have also to show that λ(p) = q. By the above σ, ∆([p]) is homologous to [q] up a non zero scalar. If it is homologous to [q ] with f (q ) < f (q), then I p q ([q]) = 0, and this is not the case.

At this point we have the uniqueness part in Barannikov's theorem.

Lemma 1.8. The Morse-Barannikov complex is F-equivalent to the Morse complex. In particular, its homology is isomorphic to H * (M, F).

Proof. Suppose we have a chain complex (C * , ∂), F-equivalent to the Morse complex and which is simple until the degree k. Then, ∂ (C k+1 ) is orthogonal to the critical points of upper type (with respect to the canonical scalar product of a based vector space). If not, ∂ • ∂ = 0.

Let p 1 , . . . , p m be the critical points of index k + 1, with f (p 1 ) < . . . < f (p m ). Let q 1 , . . . , q r be the critical points of index k whose type is lower or homological; f (q 1 ) < . . . < f (q r ). We assume that, for some j ≤ r, we have ∂p i = 0 or ∂p i = q k(i) for every i < j, the map i → k(i) being injective. We have

∂p j = i<j,∂p i =0 α i q k(i) +              0 or β k 0 q k 0 + k<k 0 ,k =k(i) β k q k , with β k 0 = 0 .
In the first case we use the following upper triangular matrix in degree k + 1:

T (p j ) = p j - i<j,∂p i =0 α i p i T (p ) = p if = j
and we set ∂ = ∂ • T . We get ∂(p j ) = 0 and ∂(p i ) = ∂(p i ) for every i < j; so we have improved the simpleness of the differential.

In the second case, we use an upper triangular matrix in both degree k + 1 and k:

T (p j ) = 1 β k 0 p j - i<j, ∂p i =0 α i p i , T (p ) = p if = j, T (q k 0 ) = q k 0 + k<k 0 ,k =k(i) β k β k 0 q k T (q k ) = q k if k = k 0 . We set ∂k+1 = T -1 • ∂ k+1 • T and ∂k = ∂ k • T .
We observe that ∂k = 0 on the k-cycles as T keeps this set invariant. We have

∂(p j ) = q k 0 and ∂(p i ) = ∂(p i ) if i < j.
Thus, ∂ has the simple form for 1 ≤ i ≤ j. Arguing this way recursively, we get a simple complex which is F-equivalent to the Morse complex.

Since the equivalence relation involves upper triangular matrices only, ∂(p) remains the class of ∆([p]) in H * f (p) -ε; F as it is in the Morse complex. Therefore, when the F-equivalent complex is simple, the type of each critical point can be easily derived and this complex is the Morse-Barannikov complex. The proof of Theorem 1.4 is completed.

Bifurcations

Bifurcations occur in a path of functions. It follows from Thom's transversality theorems, as it is explained by J. Cerf in the beginning of [START_REF] Cerf | La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudoisotopie[END_REF], that the space F of real smooth functions on M has a natural stratification whose strata of codimension ≤ 1 are the following:

0) The stratum F 0 , an open dense set in F, is formed by Morse functions whose critical values are all simple. The next two strata are of codimension one. 1) The stratum F 1 is formed by the functions whose critical values are simple and whose critical points are all non-degenerate but one where the Hessian has a kernel of dimension one.

2) The stratum F 2 is formed by the Morse functions whose critical values are all simple but one which has multiplicity 2.

3) The complement R in F of the preceding strata. A generic path has its end points in F 0 , avoids R which in turn is said to be of codimension greater than 1, and crosses F 1 ∪ F 2 transversely in a finite number of points.

Since the Morse-Barannikov complex is well defined for functions in F 0 only, it is necessary to study the bifurcation when crossing F 1 and F 2 . In that aim, it is convenient to introduce the Barannikov The critical values of f of index k are marked on D k . When the pair of critical points (p, q) are coupled in the Barannikov complex a segment is drawn from f (p) to f (q); clearly, the slope of this segment is negative (Fig. 1). The critical values, of homological type, in particular max f and min f , remain non-connected to another one.

= (f t ) t∈[0,1] . Its Cerf diagram is the union in [0, 1] × R of {t} × f t (critf t ).
It is made of finitely many smooth arcs transverse to the verticals {t} × R, ending at cusp points or in {0, 1} × R, crossing one another transversely.

2.1. Bifurcation at birth times. This event is the crossing of the stratum F 1 , which is co-oriented: the crossing in the positive direction corresponds to the birth of a pair of critical points of index k and k + 1 respectively. The crossing in the opposite direction corresponds to the cancellation of a pair of critical points.

The birth is modeled by the following formula:

f (x, y) = c + Q(y) + x 3 -(t -t 0 )x,
where t 0 is the birth time, c is the critical value of the birth point, (x, y) ∈ R × R n-1 are local coordinates at the birth point p 0 and Q is a non-degenerate quadratic form of index k on R n-1 .

In the Cerf diagram, there is a cusp of coordinates (t 0 , c). For 0 < t 0 -t small, there are no critical points in the vicinity of p 0 . For 0 < t -t 0 small, there are two critical points p t , q t of index k + 1 and k close to p 0 (Fig. 2).

Figure 2 It follows from the model that p t is of upper type and q t is of lower type and this pair is coupled in the Barannikov diagram. The other critical points keep their type and coupling when crossing the birth time.

Bifurcation at double critical value time.

This event is the crossing of the stratum F 2 , say at time t 0 . Since the functions f t are Morse for t close to t 0 , the critical points can be followed continuously on (t 0 -η, t 0 +η). Generically, the pseudo-gradient is Morse-Smale at the time t 0 . Therefore, the Morse complex remains the same on a small interval. But, as the order of the critical values is modified the Barannikov complex could change.

Denote by (p 1 t , p 2 t ) the pair of critical points whose values cross at time t 0 ; say f t (p 1 t ) < f t (p 2 t ) when t < t 0 ; hence, f t (p 1 t ) > f t (p 2 t ) when t > t 0 . The question is how the types and coupling of critical points are changing when t crosses the time t 0 .

With Barannikov we limit ourselves to the case when M is the n-sphere S n and the crossing does not involve the extremal values, this latter question being left as an exercise to the reader. Since on a sphere the only critical points of homological type are the extrema, the crossing deals with critical values of upper/lower type.

We shall say that there is no bifurcation if the crossing keeps all critical points with their initial types and coupling (remember that, near a crossing time all the functions of the considered path are Morse and the critical points can be followed smoothly in time due to the implicit function theorem).

One checks by hand that there is no bifurcation if p 1 t and p 2 t have distinct indices. Now, we are reduced to the case where the two crossing critical values have the same index, say k.

We are going to prove in the next three propositions that, in our restrictive setting, there are only three types of bifurcations which are shown with their Barannikov diagrams before and after crossing (Fig. 3).

2.3.

Notation. Before stating the bifurcation propositions, it is useful to introduce some notation. Denote by c 0 the double critical value:

c 0 = f t 0 (p 1 t 0 ) = f t 0 (p 2 t 0
). We have two (k -1)-spheres Σ 1 and Σ 2 traced in the level set f t 0 = c 0 -ε by the respective unstable manifolds of p 1 t 0 and p 2 t 0 . When t runs in a small interval around t 0 , these objects move by a small isotopy: Σ 1 t , Σ 2 t ⊂ (f t = c 0 -ε). Finally, if α is a non-zero (k -1)-homology class is the sub-level set c 0 -ε, we denote by λ(α) the critical value which is the infimum of c such that α vanishes in the relative homology H k-1 (c 0 -ε, c; F) of the pair of the denoted sub-level sets.

Proposition 2.4. We assume that, for t < t 0 , p 1 t and p 2 t have not the same type upper/lower. Then, the following holds true. 1) A bifurcation can occur only in case (A left), that is: p 1 t is of upper type and below p 2 t which is of lower type.

2) Assuming the above necessary condition, a bifurcation occurs if and only if

[Σ 1 ] = [Σ 2
] in H k-1 (c 0 ; F) up to a scalar. In that case, for t > t 0 , p 1 t becomes of lower type and p 2 t becomes of upper type; the new coupling is shown on (A right).

Proof. 1) Assume p 1 t and p 2 t are respectively of lower and upper type for t < t 0 . When t is close to t 0 , t < t 0 , by assumption, Σ 1 t is null-homologous in its sub-level set while Σ 2 t is nonhomologous to 0. Since these spheres change with t by an isotopy, this property persists up to t = t 0 and still a little further, up to some t > t 0 . Therefore the types are unchanged after crossing and it is easy to check that the coupling is also unchanged. Hence, no bifurcation.

2) Then, we assume (A left). Again, for t < t 0 and close to t 0 , certainly

[Σ 1 t ] is not homologous to 0 in H k-1 ((f t = c 0 -ε); F). The unstable manifold W u (p 2 t ) traces in f t = f t (p 2 t ) -ε , ε > 0, a sphere which is homologous to 0 in its sub-level set if f t (p 2 t ) -ε > f t (p 1 t ). Thus, for Σ 2 t in f t = c 0 -ε, there are two possibilities: (i) [Σ 2 t ]=0 or (ii) [Σ 2 t ] is a non-zero multiple of [Σ 1
t ] since it should vanish when passing in a sub-level set containing p 1 t . In both cases, this property persists up to t 0 and up to some t > t 0 . One checks easily that in case (i) there is no bifurcation.

Consider case (ii). The point p 2 t is of upper type, by the very definition. Now, with ε > 0 small enough so that f t (p 2 t ) < f t (p 1 t ) -ε , we see that the trace of W u (p 1 t ) in the level set

f t = f t (p 1 t )
-ε is homologous to 0 in its sub-level set. Therefore, p 1 t is of lower type and the types of the two critical points exchange. One checks that the coupling is as shown on (A right).

Assume now p 1 t and p 2 t are both of upper type. By the same homological argument as before, the type cannot change. But the coupling could change. Denote q 1 t and q 2 t the points associated with p 1 t and p 2 t respectively before t 0 . Proposition 2.5. In this situation, there is a bifurcation (exchange of coupling) if and only if

λ([Σ 1 ]) = λ([Σ 2 ]). In that case, when t < t 0 , f t (q 1 t ) > f t (q 2 t ) as shown on (B left). Proof. Assume λ([Σ 1 ]) < λ([Σ 2 ]
) (the opposite inequality is treated similarly). This implies that [Σ 1 ] is 0 in the homology of the pair of sub-level sets c 0 -ε, λ([Σ 2 ]) . But this vanishing holds true for every t close t 0 . This proves that the coupling of p 1 t remains unchanged; hence, no bifurcation.

What happens in case of equality? First, we look at t < t 0 . Since the function f t , restricted to the sub-level set c 0 -ε, moves by isotopy when t runs in (t 0 -η, t 0 ], one derives that λ([Σ 1 t ]) varies continuously on this interval. But we know λ([

Σ 1 t ]) = f t (q 1 t ); as a consequence we have λ([Σ 1 ]) = λ([Σ 2 ]) = f t 0 (q 1 t 0 ) . For a small ε so that f t (p 2 t ) -ε > f t (p 1 t ), the trace of W u (p 2 t ) in f t = f t (p 2 t ) -ε is homologous to 0 in the pair of sub-level sets (f t (p 2 t ) -ε , f t (q 1 t ) -ε); indeed, the class of W u (q 1 t ) is null in this pair which contains p 1 t . Therefore, λ([Σ 2 t ]) < λ([Σ 1 t ]
) and, hence, f t (q 2 t ) < f t (q 1 t ) as shown on (B left).

Second, we look at t > t 0 . Now, λ([Σ 2 t ]) varies continuously on [t 0 , t 0 + η); thus, this value is f t (q 1 t ) due to the above equality at time t 0 . Thus p 2 t is coupled with q 1 t . Therefore, p 1 t must be coupled with q 1 t ; there is, indeed, no other free place!

The last case to consider is when p 1 t and p 2 t are both of lower type. For homological reasons, there is no change of types. Let q 1 t and q 2 t be the points of index k + 1 with which they are coupled respectively when t < t 0 .

At time t 0 , denotes by e i , i = 1, 2, the k-cell traced by the unstable manifold of p i t 0 in the level set f = c 0 -ε. The map µ that we introduced in Lemma 1.7 is still defined: µ([e i ]) is the infimum of c such that the class of e i is 0 in the pair (c, c 0 -ε) with coefficients in F. Arguing similarly as in the previous proposition, one proves the following. Proposition 2.6. In this situation, there is a bifurcation (exchange of coupling) if and only if µ([e 1 ]) = µ([e 2 ]). In that case f t (q 1 t ) > f t (q 2 t ) when t < t 0 as shown on (C left).

The non-empty boundary case

Following S. Barannikov, we discuss in this section the problem of extending without critical points a germ of function given along the boundary M of a compact (n+1)-dimensional manifold W , M = ∂W . This setting was already considered in 1934 by Morse-van Schaack [START_REF] Morse | The critical point theory under general boundary conditions[END_REF] where the Morse inequalities have been formulated and proved for manifolds with non-empty boundary. Notice that generically a function F : W → R is a Morse function whose critical points lie in the interior of W and whose restriction to the boundary is Morse.

Actually, the problem considered by Barannikov in [START_REF] Barannikov | The framed Morse complex and its invariants[END_REF] was more ambitious, that is, given a generic germ f along the boundary M , to give a bound from below of the number of critical points of any generic extension F : W → R of f , as acute as possible. We focus on M = S n , the n-sphere, and W = D n+1 . Moreover, we limit ourselves to answer the question of knowing when this bound is positive. This problem was completely solved by S. Blank & F. Laudenbach [START_REF] Blank | Extension à une variété de dimension 2 d'un germe de fonction donné au bord[END_REF] for n = 1 and by C. Curley [START_REF] Curley | Non-singular extension of Morse functions[END_REF] for n = 2. This yields some information about the non-existence of an extension without critical points F : W → R of the germ f . For instance, if the framing points up at the maximum, the maximum principle tells us that any extension must have at least one critical point in the interior. Barannikov's discussion will be of course more subtle. The framing may be attached to a Morse function on M , an F-equivalence class of Morse complexes or the associated Morse-Barannikov complex C B (f ) as well. We will speak of the framed Barannikov diagram. We are going to look at the framed Barannikov complex and get some information in relation to the extension problem.

From now on, we restrict ourselves to M = S n and W = D n+1 . A framed function f : S n → R is said to be standard if f has two critical points only, the framing pointing down at the maximum and pointing up at the minimum. In this case, there is a standard extension to the (n + 1)-ball without critical points. Now, start with a generic germ f along S n and assume there is an extension without critical points F : D n+1 → R. Then, there is a one-parameter family of spheres S t ⊂ D n+1 , t ∈ [0, 1], such that S 0 = M , S t lies inside S t when t > t and the germ of F along S 1 is standard. Set f t := F |S t ; it is a function thought of as defined on M = S n , equipped with a framing due to the knowledge of S t for t = t + ε, ε > 0. The framing says the direction of moving of the spheres near a critical point p of f t (up to isotopy, F may be locally thought of as the height function in R 3 ).

Generically, for such a family of spheres, the map t → f t is a generic path of functions in the sense of Section 2. Therefore, there is a sequence of bifurcations starting from a given germ and ending at the standard germ.

There is no bifurcation of framing. At a birth time the pair (p t , q t ) which has born with f t (p t ) > f t (q t ) has the standard framing: the arrows are both up or both down; such a pair will be said to be a standard pair. At a crossing time t 0 involving the pair of critical values f t (p 1 t ) < f t (p 2 t ), with the same indices, each critical point keeps its framing during the crossing. But, the rule of moving added to the rule of numbering implies that, for t < t 0 , the arrow of p 1 t is up and the arrow of p 2 t is down (Compare Fig. 4). At each generic time, we have a well-defined Morse-Barannikov framed complex and the bifurcations are those allowed by Propositions 2.4 to 2.6. Of course, in each case the bifurcation occurs only when the required homological condition is satisfied. If it is not, the crossing yields no bifurcation.

3.2. Barannikov idea's. Not any sequence of allowed bifurcations of the framed Barannikov's complex is realizable by a sequence of bifurcations of framed Morse complex; at the level of framed Barannikov diagram there is no longer homological condition. Therefore, there is a formal problem which is the following: given a framed Barannikov complex, does there exist a sequence of allowed bifurcations connecting it to the standard Barannikov complex (i.e. one maximum down and one minimum up)? If there is no solution to the formal problem, a fortiori there is no solution to the extension problem without critical point. Now, the question is whether it is possible to answer the formal problem in finite time. This question is solved by the last theorem in Barannikov's article.

Theorem 3.3. Given a framed Barannikov complex C 0 , if it is connected the standard Barannikov complex C st , then C 0 is connected to C st without any birth.

In particular, the formal problem reduces to a finite combinatorics.

Definition 3.4. Let f be a framed Morse function. A coupled pair of critical points is said to be inverted when one of its two arrows points up and the other points down.

There are two types of such inverted pairs: in type I (resp. type II) the upper point is equipped with an arrow pointing up (resp. down).

The index of a coupled pair (inverted or standard) will be the index of the upper point.

One checks on the list of bifurcations that such a pair could not disappear alone. At best, it is possible to shift the indices of the involved critical points (use the bifurcation of Fig. 4 one pair being inverted and the other being standard in the sense of the birth bifurcation). But, two inverted pairs involving of a bifurcation as shown on Fig. 4 become both standard and then, each of them can be cancelled.

As a consequence, an obvious obstruction to extending without critical points is the parity of the number of inverted pairs in the initial framed Morse function. The obstruction which follows from Theorem 3.3 is more subtle as shown in the next example. The proof of Theorem 3.3 is based on the fact that there is a very short list of bifurcations involving at least one inverted pair; moreover, the role of the two types, I and II, are completely different. Here is the list of these bifurcations: In particular, it is impossible to change the index of a pair of type II. For proving Theorem 3.3, without losing generality we may assume that the initial complex C 0 has no standard pairs. All the coupled pairs are inverted and we are facing the problem of canceling them, maybe with the help of introducing standard pairs (births) whose bifurcations could create new inverted pairs in good positions for a total cancellation. One checks that this event cannot happen.

According to the previous list, for canceling one pair A of type II it is required to have one pair B of type I and of the same index is the right position allowing the bifurcation [START_REF] Barannikov | The framed Morse complex and its invariants[END_REF]. If B comes from births followed by the bifurcation (3), then B is born with another pair of type II still with the same index. So the price to pay the cancellation of A with B is the appearance of C which is almost in the same position as A in the Barannikov diagram, up to a shift of the heights in the direction of the arrows of the framing. So, nothing is gained. The complete proof follows the same line. We refer to Barannikov's article for the details.

Corollary 1 . 5 .

 15 The homology H * (M ; F) is graded isomorphic to the sub-space generated by the critical points having the homological type, in the sense given below: ∂ B (p) = 0 and p / ∈ Im ∂ B .

  [p] := [W u loc (p)] ∈ H k+1 (c+ε, c-ε) where c = f (p). If p is of upper type, one defines λ(p) := inf σ max(f |σ)

  Barannikov differential. Set ∂ B (p) = λ(p) if p is of upper type and ∂ B (p) = 0 in the two other cases. According to the previous lemma, ∂ B • ∂ B = 0.

  diagram and the Cerf diagram, which are defined as follows. The Barannikov diagram deals with a generic Morse function f ∈ F 0 . Let n = dim M . The vertical lines D k , k = 0, 1, . . . , n, are drawn in the plane, the x-coordinate of D k being n -k.

Figure 1 The

 1 Figure 1 The Cerf diagram deals with a generic path γ= (f t ) t∈[0,1] . Its Cerf diagram is the union in [0, 1] × R of {t} × f t (critf t ).It is made of finitely many smooth arcs transverse to the verticals {t} × R, ending at cusp points or in {0, 1} × R, crossing one another transversely.

3. 1 .

 1 The framing. Here we use Barannikov's terminology. Given a Morse function f : M → R a framing of f is the data of one vertical arrow at each critical value of f . A generic germ f : M × [0, ε) → R along the boundary determines a framing according the following rule: for the critical p of f the arrow at f (p) points up (resp. down) if < d f (p), n(p) > is positive (resp. negative), where n(p) is a tangent vector at p pointing inwards. Conversely, the framing classifies the germ f up to isotopy fixing the boundary.

Figure 5

 5 Figure 5

( 1 )

 1 a pair of type I and a pair of type II of the same indices yield two standard pairs; (2) two pairs of type I whose indices differ by 1 yield two standard pairs whose indices differ by 1; (3) two standard pairs having the same index k yield a pair of type I and a pair of type II both having the index k; (4) two standard pairs whose indices are k and k + 1 yield two pairs of type II whose indices are still k and k + 1; (5) a pair of type I and index k and a standard pair of index k ± 1 yield a pair of type II and index k ± 1 and a standard pair of index k.