Razvan Gabriel Iagar
email: razvan.iagar@imar.ro.

Philippe Laurençot
email: philippe.laurencot@math.univ-toulouse.fr

Existence and uniqueness of very singular solutions for a fast diffusion equation with gradient absorption *

Keywords: AMS Subject Classification: 35K67, 35K92, 34B40, 34C11, 35B33 Very singular solution, singular diffusion, gradient absorption, self-similar solutions, p-Laplacian, uniqueness

Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption

where 2N/(N + 1) < p < 2 and p/2 < q < p -N/(N + 1), thereby extending previous results restricted to q > 1.

Introduction

The singular diffusion equation with gradient absorption

∂ t u -∆ p u + |∇u| q = 0, (t, x) ∈ Q ∞ := (0, ∞) × R N , (1.1)
with p ∈ (1, 2) and q > 0 features a singular diffusion term and an absorption term depending solely on the gradient. Taken apart, these two terms lead to two completely different behaviors for large times so that, putting them together, a competition between them is expected. In fact, we study recently in [START_REF] Iagar | Positivity, decay and extinction for a singular diffusion equation with gradient absorption[END_REF] qualitative properties and decay estimates for nonnegative and bounded solutions to (1.1) and identify ranges of exponents p ∈ (1, 2) and q > 0 with different behaviors. In particular, given p ∈ (2N/(N + 1), 2), if q > q * := p -(N/(N + 1)), the diffusion term dominates for large times while it is the absorption term that dominates when q ∈ (0, p -1) leading to finite time extinction. Finite time extinction also occurs for q ∈ (p -1, p/2) but is expected to be of a different nature, some influence of the diffusion term persisting near the extinction time. Finally, when q ∈ (p/2, q * ), the solutions to (1.1) with initial data decaying sufficiently rapidly at infinity decay to zero at a faster algebraic rate than the one that would result from the diffusion alone, a feature which reveals an interplay between diffusion and absorption. By analogy with the existing literature on related problems (see, e.g., [START_REF] Benachour | Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations[END_REF][START_REF] Escobedo | Asymptotic behaviour of positive solutions of a nonlinear heat equation[END_REF][START_REF] Vázquez | Asymptotic behaviour of nonlinear parabolic equations. Anomalous exponents[END_REF] and the references therein), the large time behavior in that case is expected to be described by a particular self-similar solution to (1.1) which is called a very singular solution. Recall that a very singular solution to a partial differential equation is a solution U in Q ∞ (in a weak or classical sense) such that lim

t→0 sup |x|>ε {U (t, x)} = 0, (1.2)
and lim

t→0 |x|<ε U (t, x) dx = ∞, (1.3)
for any ε > 0 [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF]. In other words, the initial condition for U is zero in R N \ {0} and it has a stronger singularity at the origin x = 0 than the Dirac mass. Recall that a solution to a partial differential equation is usually referred to as a singular solution if it satisfies (1.2), the main examples being the so-called fundamental solutions, that is, solutions having a Dirac mass as initial condition, and the very singular solutions. The existence and non-existence of singular solutions have been thoroughly studied for diffusion equations with a zero-order absorption term and we refer to, e.g., [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF][START_REF] Kamin | Classification of singular solutions of a nonlinear heat equation[END_REF][START_REF] Kamin | Existence and uniqueness of the very singular solution of the porous media equation with absorption[END_REF][START_REF] Leoni | A very singular solution for the porous media equation u t = ∆ (u m ) -u p when 0 < m < 1[END_REF][START_REF] Leoni | On very singular self-similar solutions for the porous media equation with absorption[END_REF][START_REF] Peletier | A very singular solution of the porous media equation with absorption[END_REF] for ∂ t u -∆u m + u q = 0, m > 0, and [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF][START_REF] Kamin | Singular solutions of some nonlinear parabolic equations[END_REF][START_REF] Peletier | A very singular solution of a quasilinear degenerate diffusion equation with absorption[END_REF] for ∂ t u -∆ p u + u q = 0, p > 1. The case of diffusion equations with an absorption term depending solely on the gradient has been investigated more recently, see, e.g., [START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. I. Existence[END_REF][START_REF] Benachour | Very singular solutions to a nonlinear parabolic equation with absorption. II. Uniqueness[END_REF][START_REF] Qi | The self-similar profiles of generalized KPZ equation[END_REF] for ∂ t u -∆u m + |∇u| q = 0, m ≥ 1, and [START_REF] Shi | Self-similar singular solution of a p-Laplacian evolution equation with gradient absorption term[END_REF] for ∂ t u -∆ p u + |∇u| q = 0, p > 2.

In this paper we focus on the singular diffusion equation with gradient absorption (1.1) for the particular range of exponents p and q for which very singular solutions are likely to exist, namely:

p c := 2N N + 1 < p < 2, p 2 < q < q * = p - N N + 1 , (1.4)
as already mentioned. Owing to the homogeneity of (1.1), we actually look for a (forward) self-similar and radially symmetric very singular solution u to (1.1) of the form

u(t, x) = t -α f (|x|t -β ), (t, x) ∈ Q ∞ ,
for some profile f and exponents α and β to be determined. Inserting this ansatz in (1.1) gives the values of α and β α = p -q 2q -p , β = q -p + 1 2q -p , (1.5) and implies that the profile f = f (r), r = |x|t -β , is a solution of the ordinary differential equation

(|f ′ | p-2 f ′ ) ′ (r) + N -1 r (|f ′ | p-2 f ′ )(r) + αf (r) + βrf ′ (r) -|f ′ (r)| q = 0 , r > 0, (1.6)
with f ′ (0) = 0. The previous conditions (1.2) and (1.3) become lim r→∞ r (p-q)/(q-p+1) f (r) = 0, lim

t→0 t N β-α r<εt -β f (r)r N -1 dr = ∞, (1.7)
the second one being satisfied if f ∈ L 1 (0, ∞; r N -1 dr) and α -N β > 0, that is q < q * .

We then prove the following result:

Theorem 1.1. Assume that p and q satisfy (1.4). There is a unique nonnegative solution f to (1.6) which satisfies f ′ (0) = 0 and

lim r→∞ r (p-q)/(q-p+1) f (r) = 0 . (1.8)
In fact, there is a positive constant w * defined in (2. [START_REF] Peletier | A very singular solution of a quasilinear degenerate diffusion equation with absorption[END_REF]) below such that

lim r→∞ r p/(2-p) f (r) = w * .
Let us first mention that Theorem 1.1 is given in [START_REF] Shi | Self-similar very singular solution of a p-Laplacian equation with gradient absorption: existence and uniqueness[END_REF] under the additional restriction that q > 1 besides the constraints (1.4) on p and q. However, the proof given there does not seem to apply to the case q ∈ (p/2, 1] and some new ideas have to be introduced which actually work for the whole range defined in (1.4). The proof given below is thus done for p and q satisfying (1.4). As usual, Theorem 1.1 is a consequence of a detailed study of the initial value problem associated to (1.6), that is, we consider the solution f (.; a) to (1.6) with initial data f (0; a) = a > 0 and f ′ (0; a) = 0. We first establish the well-posedness of this problem at the beginning of Section 2 together with some basic properties of its solutions in Section 2.1. In particular, we show that, if a > 0 is such that f (.; a) is positive in (0, ∞), then |f ′ (.; a)| is controlled by f (.; a) 2/p as a consequence of sharp gradient estimates established in [START_REF] Iagar | Positivity, decay and extinction for a singular diffusion equation with gradient absorption[END_REF]. This turns out to be a cornerstone of the proof as it allows us to transfer some asymptotic properties of f (.; a) to f ′ (.; a). We next prove in Section 2.2 that a monotonicity property with respect to the initial condition a is enjoyed by the solutions f (.; a) under suitable conditions. Such a monotonicity property is also true for the singular diffusion equation with zero-order absorption studied in [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF]. We then split the range of initial conditions a ∈ (0, ∞) into three disjoint sets according to the behavior of the derivative of r → r p/(2-p) f (r; a) and characterize these three sets in Sections 2.3, 2.4, and 2.5. This study then guarantees the existence of at least one profile f satisfying the properties listed in Theorem 1.1. We finally borrow an argument from [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF] to prove the uniqueness of this solution.

2 Self-similar and radially symmetric very singular solutions

In order to construct a solution to (1.6) satisfying (1.8), we use a shooting method which leads us to the following initial value problem:

     (|f ′ | p-2 f ′ ) ′ (r) + N -1 r (|f ′ | p-2 f ′ )(r) + αf (r) + βrf ′ (r) -|f ′ (r)| q = 0, f (0) = a, f ′ (0) = 0, (2.1)
where a > 0 is an arbitrary positive real number and the condition f ′ (0) = 0 follows from the radial symmetry and expected smoothness of very singular solutions. Introducing

F := -|f ′ | p-2 f ′ , we observe that (2.1) also reads              f ′ (r) = -|F (r)| (2-p)/(p-1) F (r), F ′ (r) + N -1 r F (r) = αf (r) -βr|F (r)| (2-p)/(p-1) F (r) -|F (r)| q/(p-1) , f (0) = a, g(0) = 0. (2.2)
Since q > p -1 and (2 -p)/(p -1) > 0, the right-hand side of (2.2) is locally Lipschitz continuous. The term involving (N -1)/r being handled as usual, there is a unique maximal C 1 -smooth solution (f (.; a), F (.; a)) to (2.2). Owing to the continuity of f (.; a) and the positivity of a, f (.; a) is clearly positive in a right-neighborhood of r = 0 and we define R(a) := inf {r ≥ 0 : f (r; a) = 0} > 0.

(

In the sequel, where there is no risk of confusion, we will omit a from the notation and let f = f (.; a).

Basic properties of f (.; a)

We first prove some basic properties of the profile f = f (.; a).

Lemma 2.1. Let a > 0. We have f ′ (r; a) < 0 for any r ∈ (0, R(a)) and |f ′ (r; a)| ≤ (αa

) 1/q in (0, R(a)). Moreover, if R(a) = ∞, then lim r→∞ f (r; a) = lim r→∞ f ′ (r; a) = 0.
Proof. It readily follows from (2.2) that F ′ (0) = αa/N > 0 so that there is δ > 0 such that f ′ (r) < 0 for r ∈ (0, δ). Introducing r 0 := inf {r ∈ (0, R(a)) : f ′ (r) = 0}, we assume for contradiction that r 0 < R(a). Then, on the one hand, F (r 0 ) = f ′ (r 0 ) = 0 and we deduce from (2.2) that F ′ (r 0 ) = αf (r 0 ) > 0. On the other hand, F (r) ≥ 0 = F (r 0 ) for r ∈ (0, r 0 ) which implies that F ′ (r 0 ) ≤ 0, whence a contradiction. Consequently, r 0 ≥ R(a) and f ′ < 0 in (0, R(a)). Consider now R ∈ (0, R(a)) and let r m be a point of minimum of f ′ in [0, R]. Clearly, r m > 0 and either r m ∈ (0, R) and f ′′ (r m ) = 0 or r m = R and f ′′ (r m ) ≤ 0.

In both cases, it follows from (2.1) that |f ′ (r m )| q -αf (r m ) ≤ 0, whence

|f ′ (r)| q ≤ |f ′ (r m )| q ≤ αf (r m ) ≤ αf (0) = αa , r ∈ [0, R] .
Assume now that R(a) = ∞. Since f is decreasing, there exists l := lim r→∞ f (r) ≥ 0. Defining the following energy:

E(r) := p -1 p |f ′ (r)| p + α 2 f (r) 2 , r ≥ 0, it follows from (2.1) that E ′ (r) = (p -1)(|f ′ | p-2 f ′ f ′′ )(r) + α(f f ′ )(r) = - N -1 r |f ′ (r)| p -βr|f ′ (r)| 2 -|f ′ (r)| q+1 < 0. (2.4)
Hence E is decreasing and positive in (0, ∞), and has thus a limit as r → ∞. This property ensures that f ′ has a limit as r → ∞ while (2.4) implies that f ′ ∈ L q+1 (0, ∞). Combining these two facts entails that f ′ (r) -→ 0 as r → ∞ and so does F . It then follows from (2.3) that F ′ (r) -→ αl as r → ∞. Assume for contradiction that l > 0. Then F ′ (r) ≥ αl/2 for r large enough whence F (r) ≥ αlr/4 for r large enough. Therefore F (r) -→ ∞ as r → ∞ and a contradiction. We have thus shown that l = 0.

Another useful result is the following expansion of f (.; a) near the origin:

Lemma 2.2. Let a > 0. There exist three positive constants C 1 , C 2 , and C 3 such that, as r → 0,

f (r; a) = a -C 1 aα N 1/(p-1)
r p/(p-1) + C 2 aα N (q-p+2)/(p-1)
r (p+q)/(p-1)

+ C 3 aα N (3-p)/(p-1)
r 2p/(p-1) + o(r 2p/(p-1) ), (2.5)
and

∂ a f (r; a) = 1 - C 1 (p -1)a aα N 1/(p-1)
r p/(p-1)

+ (2 + q -p)C 2 (p -1)a aα N (q-p+2)/(p-1)
r (p+q)/(p-1) + o(r (p+q)/(p-1) ).

(

Proof. We start with the fact that (|f ′ | p-2 f ′ ) ′ (0) = -(aα/N ), so that, as r → 0,

f ′ (r) = - aα N 1/(p-1)
r 1/(p-1) + o(r 1/(p-1) ).

(2.7)

After integration, we obtain

f (r) = a - p -1 p aα N 1/(p-1)
r p/(p-1) + o(r p/(p-1) ).

(2.8) Since (2.1) also reads

d dr r N -1 |f ′ (r)| p-2 f ′ (r) = r N -1 |f ′ (r)| q -βrf ′ (r) -αf (r) , (2.9)
we infer from (2.7) and (2.8) that, as r → 0,

1 r N -1 d dr r N -1 |f ′ (r)| p-2 f ′ (r) = -aα + aα N q/(p-1)
r q/(p-1) + o(r q/(p-1) ).

Integrating once, we obtain the following expansion for f ′ :

f ′ (r) = - aα N 1/(p-1)
r 1/(p-1)

+ 1 q + N (p -1) aα N (q-p+2)/(p-1)
r (q+1)/(p-1) + o(r (q+1)/(p-1) ).

(2.10) Another integration gives that, as r → 0,

f (r) = a - p -1 p aα N 1/(p-1)
r p/(p-1)

+ p -1 (p + q)(q + N (p -1)) aα N (q-p+2)/(p-1)
r (p+q)/(p-1) + o(r (p+q)/(p-1) ).

(2.11) Inserting (2.10) and (2.11) in (2.9) then leads to (2.5) after lengthy but simple computations, with the constants

C 1 = p -1 p , C 2 = p -1 (p + q)(q + N (p -1)) , C 3 = (p -1)q 2p 2 (p + N (p -1))(2q -p)
.

Concerning ∂ a f , we first observe that differentiating (2.9) with respect to a gives (p -1)

d dr r N -1 |f ′ (r)| p-2 ∂ a f ′ (r) = qr N -1 |f ′ (r)| q-2 f ′ (r)∂ a f ′ (r) -βr N ∂ a f ′ (r) -αr N -1 ∂ a f (r) .
(2.12)

Setting Y ′ (r) := q|f ′ (r)| q-p f ′ (r)/(p -1) for r ∈ [0, R(a)) with Y (0) = 0, we have d dr r N -1 |f ′ (r)| p-2 ∂ a f ′ (r)e -Y (r) = -βr N ∂ a f ′ (r) + αr N -1 ∂ a f (r) e -Y (r) p -1 .
We next use the properties ∂ a f (0; a) = 1, ∂ a f ′ (0; a) = 0, and Y ′ (r) → 0 as r → 0 (the latter being true since q > p -1) to conclude that

d dr r N -1 |f ′ (r)| p-2 ∂ a f ′ (r)e -Y (r) = - α p -1 r N -1 + o(r N -1 )
as r → 0. Integrating this identity, we infer from (2.5) that

∂ a f ′ (r) = - α N (p -1) aα N (2-p)/(p-1) r 1/(p-1) + o(r 1/(p-1) ).
Integrating once more with respect to r gives an expansion of ∂ a f (r) as r → 0. Inserting these expansions in (2.12) and arguing as in the proof of (2.5) give (2.6) after some computations.

We now prove that, if a > 0 is such that R(a) = ∞, then |f ′ (.; a)| is controlled by f (.; a) 2/p . To this end, we first check that f (.; a) can be associated to a solution to (1.1) which turns out to be a viscosity solution in the sense of Definition A.1 below. This then allows us to apply the optimal gradient estimates obtained in [START_REF] Iagar | Positivity, decay and extinction for a singular diffusion equation with gradient absorption[END_REF].

Lemma 2.3. Let a > 0 be such that R(a) = ∞ and ε > 0. Setting U ε (t, x) := (t + ε) -α f (|x|(t + ε) -β ; a) , (t, x) ∈ [0, ∞) × R N , the function U ε is a viscosity solution to (1.1) with initial condition U ε (0) defined by U ε (0, x) := ε -α f (|x|ε -β ; a) for x ∈ R N .
Proof. Let us first observe that, owing to (2.7)-(2.9), we have as r → 0

(p -1)f ′′ (r) = - aα N 1/(p-1)
r (2-p)/(p-1) + o(r (2-p)/(p-1) ).

(2.13) Since p < 2 and f clearly belongs to C 2 ((0, ∞)), we deduce from (2.13

) that f ∈ C 2 ([0, ∞)) with f ′′ (0) = 0. Consequently, U ε belongs to C 2 ([0, ∞) × R N ) and is clearly a classical solution to (1.1) in (0, ∞) × R N \ {0}
where ∇U ε does not vanish. Then U ε is a viscosity solution to (1.1) in (0, ∞)×R N \{0} as it obviously satisfies Definition A.1 for all (t 0 , x 0 ) ∈ (0, ∞) × R N \ {0}. Consider now t 0 > 0 and x 0 = 0. We first prove that U ε is a viscosity subsolution. To this end, let ψ ∈ A be such that U ε (t 0 , 0) = ψ(t 0 , 0) and U ε (t, x) < ψ(t, x) for any (t, x) ∈ Q ∞ \{(t 0 , 0)}, the set A of admissible comparison functions being defined in the Appendix. Since U ε and ψ are both C 1 -smooth, this property implies that ∇ψ(t

0 , 0) = ∇U ε (t 0 , 0) = 0 and ∂ t ψ(t 0 , 0) = ∂ t U ε (t 0 , 0) = -αa(t 0 + ε) α+β < 0, so that Definition A.1 is satisfied. We next prove that U ε is a viscosity supersolution, that is, -U ε is a viscosity subsolution. Let ψ ∈ A be such that -U ε (t 0 , 0) = ψ(t 0 , 0) and -U ε (t, x) < ψ(t, x) for any (t, x) ∈ Q ∞ \ {(t 0 , 0)}. Since U ε and ψ are both C 1 -smooth, we have ∇ψ(t 0 , 0) = ∇U ε (t 0 , 0) = 0 and ∂ t ψ(t 0 , 0) = -∂ t U ε (t 0 , 0). Therefore, for x ∈ R N , ψ(t 0 , x) -ψ(t 0 , 0) ≥ -U ε (t 0 , x) + U ε (t 0 , 0) = -(t 0 + ε) -α f (|x|(t 0 + ε) -β ; a) -f (0; a) ,
and it follows from (2.5) that, as x → 0,

ψ(t 0 , x) -ψ(t 0 , 0) ≥ C 1 (t 0 + ε) -α aα N 1/(p-1) |x|(t 0 + ε) -β p/(p-1)
+ o(|x| p/(p-1) ) .

However, since ψ ∈ A, there is ξ ∈ Ξ such that, as x → 0, ξ(|x|) ≥ ψ(t 0 , x) -ψ(t 0 , 0). Combining the above two inequalities leads us to lim inf r→0 ξ(r) r p/(p-1) > 0 , which contradicts (A.2). This situation thus cannot occur and the proof is complete.

As a consequence of Lemma 2.3 and since p > p c , we may apply the gradient estimates proved for solutions to (1.1) in [10, Theorems 1.2 and 1

.3]. Proposition 2.4. Let a > 0 be such that R(a) = ∞. Then f (.; a) satisfies f ′ (r; a) ≤ C 4 f (r; a) 2/p , r ≥ 0 , (2.14)
for some constant C 4 depending only on N , p, q and a.

Proof. Fix ε > 0. According to Lemma 2.3, the function

U ε defined by U ε (t, x) = (t + ε) -α f (|x|(t + ε) -β ; a) for (t, x) ∈ [0, ∞) × R N is a viscosity solution to (1.1) with a nonnegative initial condition which belongs to W 1,∞ (R N ) by Lemma 2.1.
If q ≥ 1, we infer from [10, Theorem 1.2] that there is a positive constant C depending only on N and p such that

∇U -(2-p)/p ε (t, x) ≤ C t -1/p , (t, x) ∈ Q ∞ .
In terms of f (.; a), we obtain

f ′ (|x|(t + ε) -β ) ≤ C t + ε t 1/p f (|x|(t + ε) -β ) 2/p , (t, x) ∈ Q ∞ .
Letting ε → 0 and choosing t = 1 give (2.14). If q ∈ (p/2, 1), we infer from [10, Theorem 1.3] that there is a positive constant C depending only on N , p, and q such that

∇U -(2-p)/p ε (t, x) ≤ C U ε (t/2) 1/αp ∞ + t -1/p , (t, x) ∈ Q ∞ .
In terms of f (.; a), we obtain

f ′ (|x|(t + ε) -β ) ≤ C (t + ε) 1/p a 1/αp (t + 2ε) -1/p + t -1/p f (|x|(t + ε) -β ) 2/p for (t, x) ∈ Q ∞ .
Letting ε → 0 and choosing t = 1 give (2.14).

Decay rates and monotonicity

We now study the possible decay rates as r → ∞ of the profile f (.; a) (when R(a) = ∞).

Since we expect an algebraic decay, we make the following ansatz:

f (r; a) ∼ Cr 1-γ , f ′ (r; a) ∼ C(1 -γ)r -γ , as r → ∞,
for some γ > 1. Inserting this ansatz in (2.1), we easily find that there are only two possibilities: γ = 2/(2 -p) and γ = 1/(q -p + 1). In the former case, f (r; a) ∼ Cr -p/(2-p) as r → ∞, while f (r; a) ∼ Cr -(p-q)/(q-p+1) as r → ∞ in the latter. Observing that (p -q)/(q -p + 1) < p/(2 -p) as q > p/2, the solutions to (2.1) decaying with the first rate are called fast orbits while those decaying with the second one are called slow orbits [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF]. Complying with (1.2) requires f to be a fast orbit, and we now proceed to show the existence of such solutions to (2.1).

To this end, following [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF][START_REF] Shi | Self-similar very singular solution of a p-Laplacian equation with gradient absorption: existence and uniqueness[END_REF], we introduce the new unknown function

w(r) = w(r; a) := r p/(2-p) f (r; a), (r, a) ∈ [0, R(a)) × (0, ∞),
which is a solution of the following differential equation:

(p -1)r 2 w ′′ (r) + [N -1 -2µ(p -1)]rw ′ (r) + µ(µ -N )w(r) + |rw ′ (r) -µw(r)| 2-p (α -βµ)w(r) + βrw ′ (r) -r η |rw ′ (r) -µw(r)| q = 0 (2.15) for r ∈ [0, R(a))
, where

µ := p 2 -p > N, η := - 2q -p 2 -p < 0, α -βµ = - 1 2 -p < 0.
We end the preliminary results with the following monotonicity property with respect to the initial value a which will be very useful in the uniqueness proof later on. We define a linear differential operator L a by:

L a (ϕ)(r) := (p -1)r 2 ϕ ′′ (r) + [N -1 -2µ(p -1)]rϕ ′ (r) + µ(µ -N )ϕ(r) + (2 -p)(|W | -p W )(r)(rϕ ′ (r) -µϕ(r)) (α -βµ)w(r) + βrw ′ (r) -r η |W (r)| q + |W (r)| 2-p (α -βµ)ϕ(r) + βrϕ ′ (r) -qr η (|W | q-2 W )(r)(rϕ ′ (r) -µϕ(r)) , (2.16)
where W (r) := rw ′ (r)-µw(r) and w(r) = w(r; a) for r ∈ [0, R(a)) in (2.16). Before stating the monotonicity lemma, we gather some properties of L a in the following two results.

Lemma 2.5. Given a > 0, we have L a (∂ a w(.; a)) = 0 and L a (rw ′ (.; a)) < 0 in (0, R(a)) .

(2.17)

Moreover, for l ∈ (0, µa], there exists a small interval (0, s l (a)) such that µaw a (r; a) > rw ′ (r; a) for r ∈ (0, s µa (a)) , lw a (r; a) < rw ′ (r; a) for l ∈ (0, µa) and r ∈ (0, s l (a)).

Proof. To simplify notations, we set w a := ∂ a w(.; a) and f a := ∂ a f (.; a). Differentiating the ordinary differential equation (2.15) with respect to the parameter a, we readily obtain that L a (w a ) = 0 in (0, R(a)). Next, differentiating (2.15) with respect to r and multiplying the resulting identity by r give, by a straightforward computation

L a (rw ′ ) = ηr η |rw ′ (r) -µw(r)| q-p+2 < 0 in (0, R(a)).
We now prove the last two assertions and consider l ∈ (0, µa]. Since w a (r) = r µ f a (r) and rw ′ (r) = r µ (rf ′ (r) + µf (r)), we have lw a (r) -rw ′ (r) = r µ (lf a (r) -rf ′ (r) -µf (r)) and use Lemma 2.2 to identify the behavior of this function in a right-neighborhood of r = 0. Indeed, we infer from Lemma 2.2, (2.10), and the definitions of C 1 and C 2 that, as r → 0,

lf a (r) -rf ′ (r) -µf (r) = l -µa -C 1 aα N 1/(p-1) l (p -1)a -µ - p p -1 r p/(p-1) + C 2 aα N (q-p+2)/(p-1) (2 + q -p)l (p -1)a - (p + q) p -1 -µ r (p+q)/(p-1)
+ o(r (p+q)/(p-1) ) .

On the one hand, if l < µa, we have lf a (r) -rf ′ (r) -µf (r) ∼ l -µa < 0 as r → 0, which implies that this function is negative in (0, s l (a)) for some sufficiently small s l (a) > 0. On the other hand, if l = µa, we have

l -µa = 0 , l (p -1)a -µ - p p -1 = µ (p -1) -µ - p p -1 = 0 and (2 + q -p)l (p -1)a - (p + q) p -1 -µ = µ(2 + q -p) p -1 - p + q p -1 -µ = (2q -p)(p -1) 2 -p > 0,
hence µaf a (r) -rf ′ (r) -µf (r) > 0 for r > 0 sufficiently small. We are now in a position to state and prove the monotonicity result with respect to a.

Lemma 2.7 (Monotonicity lemma). Let a > 0 and suppose that w ′ (.; a) > 0 in a finite interval (0, r 0 ) with r 0 < R(a). Then we have µa∂ a w(r; a) > rw ′ (r; a) for r ∈ (0, r 0 ) and ∂ a w(.; a) > 0 on [0, r 0 ].

Proof. We adapt the proof of [6, Lemma 2.2]. To simplify notations, we again set w a := ∂ a w(.; a) and f a := ∂ a f (.; a) and first show that µaw a (r) > rw ′ (r) for r ∈ (0, r 0 ). Assume for contradiction that there exists some point r 2 ∈ (0, r 0 ) such that µaw a (r) -rw ′ (r) > 0 for r ∈ (0, r 2 ) and µaw a (r 2 ) -r 2 w ′ (r 2 ) = 0 at r = r 2 . Then, given l ∈ (0, µa), we have lw a (r 2 ) -r 2 w ′ (r 2 ) < 0 and, since lw a (r) -rw ′ (r) < 0 for r ∈ (0, s l (a)) by Lemma 2.5, there is l ∈ (0, µa) sufficiently close to µa such that we have lw a (r 1 ) -r 1 w ′ (r 1 ) = lw a (r 3 )r 3 w ′ (r 3 ) = 0 and lw a (r) -rw ′ (r) > 0 for r ∈ (r 1 , r 3 ) for some 0 < r 1 < r 3 < r 2 . We are then in a position to apply Lemma 2.6 to the function h(r

) := lw a (r) -rw ′ (r) in the interval [r 1 , r 3 ] ⊂ (0, r 2 ), recalling that L a (h) = lL a (w a ) -L a (rw ′ ) = -L a (rw ′ ) > 0 in (r 1 , r 3
) by Lemma 2.5. This implies that lw a (r) -rw ′ (r) ≤ 0 for r ∈ (r 1 , r 3 ) and a contradiction. Therefore

µaw a (r) > rw ′ (r) for r ∈ (0, r 0 ), (2.18)
and as a direct consequence, w a > 0 in (0, r 0 ). It remains to check that w a (r 0 ) > 0. Fix s 0 ∈ (0, r 0 ) and define l 0 := s 0 w ′ (s 0 )/w a (s 0 ) and k 0 := (l 0 w a -rw ′ ) ′ (s 0 ). We first observe that, since s 0 ∈ (0, r 0 ), (2.18) guarantees that l 0 ∈ (0, µa). We next show that k 0 > 0. Indeed, if k 0 < 0, we have l 0 w a (r) -rw ′ (r) > 0 in a left-neighborhood of s 0 as l 0 w a (s 0 ) -s 0 w ′ (s 0 ) = 0. Defining

s 1 := sup {r ∈ (0, s 0 ) : l 0 w a (r) -rw ′ (r) = 0} ,
we have just established that s 1 < s 0 while Lemma 2.5 guarantees that s 1 > 0 as l 0 < µa. Therefore, setting h(r) := l 0 w a (r) -rw ′ (r) for r ∈ [s 1 , s 0 ], we have h(s 1 ) = h(s 0 ) = 0 and L a (l 0 w a -rw ′ ) > 0 in (s 1 , s 0 ). This implies that h ≤ 0 in (s 1 , s 0 ) by Lemma 2.6 and a contradiction. Next, if k 0 = 0, it follows from Lemma 2.5 that (p -1)s 2 0 (l 0 w arw ′ ) ′′ (s 0 ) = L a (l 0 w a -rw ′ )(s 0 ) > 0 from which we deduce that l 0 w a (r) -rw ′ (r) > 0 in a left-neighborhood of s 0 . Arguing as in the previous case, we are again led to a contradiction and conclude that k 0 > 0. Consider now the solution ψ of L a (ψ) = 0 in (0, R(a)) with initial conditions ψ(s 0 ) = 0 and ψ ′ (s 0 ) = 1. We claim that ψ cannot vanish in (s 0 , r 0 ]. Indeed, since L a (w a ) = 0 with w a (s 0 ) > s 0 w ′ (s 0 )/µa > 0 by Lemma 2.5 and (2.18), w a and ψ are linearly independent solutions to the same second-order ordinary differential equation and the oscillation theorem guarantees that there is at least one zero of w a between two zeros of ψ. Thus, if ψ(s) = ψ(s 0 ) = 0 for some s ∈ (s 0 , r 0 ], the function w a has to vanish at least once in (s 0 , s) which contradicts (2.18). Therefore, ψ(r) > 0 for r ∈ (s 0 , r 0 ]. Define now the function ϕ := l 0 w a -k 0 ψ. Then, by construction and Lemma 2.5,

L a (ϕ -rw ′ ) > 0 in (0, R(a)), ϕ(s 0 ) -s 0 w ′ (s 0 ) = (ϕ -rw ′ ) ′ (s 0 ) = 0, so that (p -1)s 2 0 (ϕ -rw ′ ) ′′ (s 0 ) = L a (ϕ -rw ′ )(s 0 ) > 0.
Consequently, ϕ(r) -rw ′ (r) > 0 in a right-neighborhood of s 0 . Assume for contradiction that there is s 2 ∈ (s 0 , r 0 ] such that ϕ(r) -rw ′ (r) > 0 for r ∈ (s 0 , s 2 ) and ϕ(s 2 ) -s 2 w ′ (s 2 ) = 0. As L a (ϕ -rw ′ ) > 0 in (s 0 , s 2 ), we deduce from Lemma 2.6 that ϕ(r) -rw ′ (r) ≤ 0 for r ∈ (s 0 , s 2 ) and a contradiction. Consequently, ϕ(r) -rw ′ (r) > 0 for r ∈ (s 0 , r 0 ]. In particular, l 0 w a (r 0 ) = ϕ(r 0 ) + k 0 ψ(r 0 ) > r 0 w ′ (r 0 ) + k 0 ψ(r 0 ) > 0, which ends the proof.

As in [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF][START_REF] Shi | Self-similar very singular solution of a p-Laplacian equation with gradient absorption: existence and uniqueness[END_REF], we split the range of a in three disjoint sets:

A := a > 0 : there exists R 1 (a) ∈ (0, R(a)), w ′ (R 1 (a); a) = 0 , B := a > 0 : w ′ (.; a) > 0 in (0, ∞), lim r→∞ w(r; a) < ∞ , C := a > 0 : w ′ (.; a) > 0 in (0, ∞), lim r→∞ w(r; a) = ∞ .
Since w ′ (r; a) ∼ µar µ-1 as r → 0, we have w ′ (.; a) > 0 in a right-neighborhood of r = 0 from which we deduce that the three sets are disjoint and A ∪ B ∪ C = (0, ∞). According to the discussion at the beginning of Section 2.2 and the definition of w, a fast orbit corresponds to a ∈ B while a slow orbit starts from a ∈ C. Thus, the first step is to show that B is non-empty which will follow from the fact that A and C are both open intervals. In a second step, we shall prove that B reduces to a single point, thereby showing the existence and uniqueness of a radially symmetric self-similar very singular solution a stated in Theorem 1.1.

Characterization of A

We show that, for all a ∈ A, the profile f (.; a) crosses the r-axis, that is, R(a) < ∞. To this end, we introduce

w * := µ p-1 (µ -N ) µβ -α 1/(2-p) , (2.19)
which is the constant solution of (2.15) without the term r η |rw ′ (r) -µw(r)| 2+q-p which is expected to be negligible in the limit r → ∞ as η < 0. Proof. We adapt the technique in [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF]. Let a ∈ A and denote the smallest positive zero of w ′ in (0, R(a)) by R 1 (a), its existence being guaranteed as a ∈ A. Then, w ′ > 0 in (0, R 1 (a)) and w ′′ (R 1 (a)) ≤ 0. Assume for contradiction that w ′′ (R 1 (a)) = 0. Differentiating (2.15) with respect to r and taking r = R 1 (a) in the resulting identity give

(p -1)R 1 (a) 2 w ′′′ (R 1 (a)) = ηR 1 (a) η-1 (µw(R 1 (a)) 2+q-p < 0.
Consequently, there exists some δ > 0 such that w ′′ > 0 in (R 1 (a) -δ, R 1 (a)), which entails that w ′ (r) < w ′ (R 1 (a)) = 0 for r ∈ (R 1 (a) -δ, R 1 (a)) and a contradiction with the definition of R 1 (a). Hence, w ′′ (R 1 (a)) < 0. We show next that w ′ < 0 in (R 1 (a), R(a)). Since w ′ (R 1 (a)) = 0 and w ′′ (R 1 (a)) < 0, we clearly have w ′ < 0 in a right-neighborhood of R 1 (a). Assume for contradiction that there is R 2 (a) ∈ (R 1 (a), R(a)) such that w ′ < 0 in (R 1 (a), R 2 (a)) and w ′ (R 2 (a)) = 0. Then w ′′ (R 2 (a)) ≥ 0. We evaluate (2.15) at R i (a), i = 1, 2, and find

µ(µ -N )w(R i (a)) + (α -βµ)(µw(R i (a))) 2-p w(R i (a)) -R i (a) η (µw(R i (a))) q-p+2 = -(p -1)R i (a) 2 w ′′ (R i (a)), for i = 1, 2. Since w ′′ (R 1 (a)) < 0 ≤ w ′′ (R 2 (a)) we deduce (µw(R 1 (a))) 2-p βµ -α + µR 1 (a) η (µw(R 1 (a))) q-1 < µ(µ -N ) ≤ (µw(R 2 (a))) 2-p βµ -α + µR 2 (a) η (µw(R 2 (a))) q-1 . (2.21) But this is a contradiction, since w(R 1 (a)) > w(R 2 (a)), 2-p > 0, 2-p+q-1 = q-p+1 > 0, βµ -α > 0 and R 1 (a) η > R 2 (a) η . Thus, w ′ < 0 in (R 1 (a), R(a)).
The converse statement being obviously true, it remains to show (2.20). If a ∈ A, w(.) has a single critical point R 1 (a) in (0, R(a)) and w attains its maximum at this point. The inequality (2.20) then readily follows from the first inequality in (2.21).

The next step is to show that any profile f (.; a) starting from a ∈ A crosses the r-axis.

The following lemma which is implicitly contained in [6, Lemma 3.1] will be useful.

Lemma 2.9. Let h be a nonnegative function in

C 1 ([0, ∞)) such that there is a sequence (r k ) k≥1 , r k → ∞ as k → ∞, for which h(r k ) -→ 0 as k → ∞. Then, there is a sequence (ρ k ) k≥1 , ρ k → ∞ as k → ∞, such that h(ρ k ) -→ 0 and ρ k h ′ (ρ k ) -→ 0 as k → ∞.
Proof. If h oscillates infinitely many times, then one can select (ρ k ) k≥1 to be a sequence of local minimum points of h. Otherwise, if h does not oscillate infinitely many times, it eventually decreases monotonically to zero and h ′ belongs to L 1 (0, ∞). Then, one can select

ρ k > k such that ρ k |h ′ (ρ k )| ≤ 1/k.
The next result provides a complete description of A in terms of R(a).

Lemma 2.10. Let a > 0. Then a ∈ A if and only if R(a) < ∞.

Proof. The proof is divided into two technical steps. Let a ∈ A and suppose for contradiction that R(a) = ∞.

Step 1. Interpretation in terms of decay. By Lemma 2.8, we deduce that w ′ < 0 in (R 1 (a), ∞) and the non-negativity of w guarantees that w has a limit ℓ ≥ 0 as r → ∞. In addition, w ′ belongs to L 1 (R 1 (a), ∞) from which we deduce that there exists a sequence r k → ∞ such that r k w ′ (r k ) -→ 0 as k → ∞. Since w ′ < 0, it follows from Lemma 2.9 that we may also assume that

r k (r k w ′′ (r k ) + w ′ (r k )) -→ 0 as k → ∞, whence r 2 k w ′′ (r k ) -→ 0 as k → ∞.
We evaluate (2.15) at r = r k and pass to the limit as k → ∞ to deduce that ℓ ∈ {0, w * } (recall that η < 0). Owing to (2.20), the possibility ℓ = w * is excluded and we conclude that lim r→∞ w(r) = 0.

(2.22)

Step 2. The contradiction. Since p < 2, we infer from (2.14) and (2.22) that

r|f ′ (r)| f (r) ≤ C rf (r) (2-p)/p = C w(r) (2-p)/p -→ 0.
Consequently, there exists r * > R 1 (a) such that -µ ≤ rf ′ (r) f (r) ≤ 0 for any r > r * , from which we deduce that

w ′ (r) = r µ f ′ (r) + µf (r) ≥ 0 , r ∈ (r * , ∞) ,
and a contradiction with (2.22). Therefore R(a) < ∞.

Conversely, if R(a) < ∞, then w(R(a)) = 0 = w(0), which implies that w has a maximum point in (0, R(a)), hence a ∈ A.

Proposition 2.11. The set A is an open interval of the form (0, a * ) for some a * > 0.

Proof. We first show that A is non-empty and that (0, a) ⊂ A for a sufficiently small. The proof is the same as that of [START_REF] Shi | Self-similar very singular solution of a p-Laplacian equation with gradient absorption: existence and uniqueness[END_REF]Theorem 2] and we only sketch it here for the sake of completeness. We perform a rescaling in the r-variable and, for a > 0, we define g(.; a) by f (r; a) = ag ra (2-p)/p ; a , r ∈ [0, R(a)) .

Then g = g(.; a) solves

     (|g ′ | p-2 g ′ ) ′ (s) + N -1 s (|g ′ | p-2 g ′ )(s) + βsg ′ (s) + αg(s) -a (2q-p)/p |g ′ (s)| q = 0, g(0) = 1, g ′ (0) = 0.
(2.23)

Owing to Lemma 2.1, we have g(s) > 0 and -α 1/q a -(2q-p)/pq ≤ g ′ (s) < 0 for s ∈ 0, R(a)a (2-p)/p . (2.24)

We next study the limit of (2.23) as a → 0 which reads, since q > p/2,

     (|h ′ | p-2 h ′ ) ′ (s) + N -1 s (|h ′ | p-2 h ′ )(s) + βsh ′ (s) + αh(s) = 0, h(0) = 1, h ′ (0) = 0.
(2.25)

Arguing as in Lemma 2.1, there is S 0 > 0 such that the solution h to (2.25) satisfies h(S 0 ) = 0, h(s) > 0 and h ′ (s) < 0 for s ∈ (0, S 0 ). Moreover, proceeding as in the proof of [22, Theorem 2], we show that S 0 < ∞ with h ′ (S 0 ) < 0. It is then easy to show, using the continuous dependence of solutions for (2.23), that for a > 0 sufficiently small, g(.; a) also vanishes and thus R(a) < ∞. Consequently, thanks to Lemma 2.10, (0, a) ⊂ A for a small enough. Since A is open, we have a 1 < a 2 , a 1 ∈ A, and (a 1 , a 2 ) ⊂ A. Setting

It remains to show that

ρ := lim inf aցa 1 R 1 (a) ∈ [0, ∞] ,
there are three possibilities: If ρ = ∞, we actually have R 1 (a) -→ ∞ as a ց a 1 and it follows by continuous dependence that, for any r > 0, we have r < R 1 (a) for a > a 1 close enough to a 1 and thus w ′ (r; a 1 ) = lim

aցa 1 w ′ (r; a) ≥ 0 .
In addition, by Lemma 2.10 and (2.26), w(r; a 1 ) = lim

aցa 1 w(r; a) ≤ m((a 1 + a 2 )/2) < w * , (2.27)
so that w(.; a 1 ) is a non-decreasing and bounded function. It thus has a limit ℓ ∈ [0, m((a 1 + a 2 )/2)] as r → ∞. If ℓ > 0, we argue as in Step 1 of the proof of Lemma 2.10 to obtain that ℓ = w * and a contradiction. Consequently, ℓ = 0 so that w(.; a 1 ) ≡ 0 and thus a 1 = 0. If ρ ∈ (0, ∞), there is a sequence (a j ) ∈ (a 1 , a 2 ) such that a j -→ a 1 and R 1 (a j ) -→ ρ as j → ∞. By continuous dependence we have

w ′ (ρ; a 1 ) = lim j→∞ w ′ (R 1 (a j ); a j ) = 0 ,
whence a 1 ∈ A and a contradiction. If ρ = 0, there is a sequence (a j ) ∈ (a 1 , a 2 ) such that a j -→ a 1 and R 1 (a j ) -→ 0 as j → ∞. Let us assume for contradiction that a 1 > 0. Then there is r 1 > 0 such that w ′ (r; a 1 ) > 0 for r ∈ (0, r 1 ) by Lemma 2.2. Given r ∈ (0, r 1 ), we have R 1 (a j ) < r 1 for j large enough whence w ′ (r; a 1 ) = lim j→∞ w ′ (r; a j ) ≤ 0 , by continuous dependence, and a contradiction. Consequently, a 1 = 0 in this case as well.

We have thus shown that, given a 2 ∈ A, the interval (0, a 2 ) is included in A from which Proposition 2.11 follows.

Characterization of C

We begin with the following useful result.

Lemma 2.12. Let a > 0. Then a ∈ C if and only if

sup r∈(0,R(a))
w(r; a) > w * .

(2.28)

Proof. The direct implication is obvious. Conversely, if a > 0 is such that (2.28) is satisfied, it follows from Lemma 2.8 that a ∈ A. Then a ∈ B ∪C and w(.; a) is an increasing function in (0, ∞). If it is bounded, it has a finite limit ℓ as r → ∞ and we argue as in the first step of the proof of Lemma 2.10 to deduce that ℓ ∈ {0, w * }, clearly contradicting (2.28). Consequently, w(.; a) is unbounded and a ∈ C. Proof. Let us first show that C is non-empty. Given a > 0, by Lemma 2.1 we have f ′ (r) ≥ -(αa) 1/q for r ∈ (0, R(a)), whence f (r) ≥ a -(αa) 1/q r for r ∈ (0, R(a)).

(2.29)

In particular, R(a) ≥ a(αa) -1/q = α -1/q a (q-1)/q . Consequently, α -1/q a (q-1)/q /2 ∈ (0, R(a)) and it follows from (2.29) that w α -1/q a (q-1)/q /2; a ≥ a 2 (α -1/q a (q-1)/q /2) µ = α -p/q(2-p) 2 -2/(2-p) a (2q-p)/q(2-p) , which can be chosen greater that 2w * for a large enough since q > p/2, and guarantees that a ∈ C for a large enough, according to Lemma 2.12. This shows that C is non-empty and that a ∈ C for a large enough. In addition, C is clearly an open set thanks to Lemma 2.12. Now, let a 0 ∈ C. Then w ′ (.; a 0 ) > 0 in (0, ∞) and also ∂ a w(.; a 0 ) > 0 in (0, ∞) by Lemma 2.7. Therefore, C is an open interval.

We now show that the set C is composed in fact of the slow orbits.

Proposition 2.14. For every a ∈ C, there exists k(a) ∈ (0, ∞) such that lim r→∞ r (p-q)/(q-p+1) f (r; a) = k(a).

While the outcome of Proposition 2.14 is similar to that of [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF]Theorem 4.1], only the first step of the proof of Proposition 2.14 borrows some ideas of the proof of [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF]Theorem 4.1], that is, the introduction of a new variable and a new unknown function. The other two steps are different owing to the different nature of the absorption on the one hand (Step 2) and the fact that q takes values below 1 on the other hand (Step 3). Concerning the latter, we prove a general lemma.

Lemma 2.15. Let X ∈ C 1 ([0, ∞)) satisfy the differential inequality

X ′ (t) + C 5 e γt X(t) ≤ C 6 (1 + e δt ), t > 0 , (2.30)
for some 0 < δ < γ and C 5 > 0, C 6 > 0. Given θ ∈ (0, γ -δ), there exist t θ > 0 and K θ > 0 depending only on γ, δ, C 5 , C 6 , X(0), and θ such that X(t) ≤ K θ e -θt for all t ≥ t θ .

Proof. Make the change of variable X(t) = Y (e γt ) = Y (τ ) for t ≥ 0, where τ := e γt ≥ 1.

Then the inequality (2.30) becomes

γY ′ (τ ) + C 5 Y (τ ) ≤ C 6 τ 1 + τ δ/γ , τ ≥ 1,
or, equivalently, after a straightforward transformation

d dτ Y (τ )e C 7 τ ≤ C 6 1 τ + 1 τ (γ-δ)/γ e C 7 τ , τ ≥ 1,
with C 7 := C 5 /γ. After integration over (1, τ ), τ ≥ 1, we obtain

Y (τ ) ≤ Y (1)e -C 7 (τ -1) + C 6 τ 1 1 σ + 1 σ (γ-δ)/γ e C 7 (σ-τ ) dσ, τ ≥ 1.
(2.31)

We take τ ≥ 2 sufficiently large such that τ / log τ ≥ 2/C 7 . In order to estimate the integral in the right-hand side of (2.31), we split it and handle differently the contributions to the integral of the intervals (1, τ /2), (τ /2, τ -(log τ )/C 7 ) and (τ -(log τ )/C 7 , τ ). First,

τ /2 1 1 σ + 1 σ (γ-δ)/γ e C 7 (σ-τ ) dσ ≤ 2 τ /2 1 e -C 7 τ /2 dσ ≤ τ e -C 7 τ /2 . Next, σ -τ ≤ -(log τ )/C 7 for σ ∈ (τ /2, τ -(log τ )/C 7 ) so that τ -(log τ )/C 7 τ /2 1 σ + 1 σ (γ-δ)/γ e C 7 (σ-τ ) dσ ≤ 1 τ τ -(log τ )/C 7 τ /2 1 σ + 1 σ (γ-δ)/γ dσ ≤ 1 τ log σ + γ δ σ γ/δ σ=τ σ=τ /2 ≤ γ δ 1 τ + 1 τ (γ-δ)/γ .
Finally, if σ ∈ (τ -(log τ )/C 7 , τ ), we have σ ≥ τ /2 and σ -τ < 0 and

τ τ -(log τ )/C 7 1 σ + 1 σ (γ-δ)/γ e C 7 (σ-τ ) dσ ≤ τ τ -(log τ )/C 7 2 τ + 2 τ (γ-δ)/γ dσ ≤ 2 log τ C 7 1 τ + 1 τ (γ-δ)/γ .
Plugging these estimates into (2.31), we obtain that, given ̺ ∈ (0, (γ -δ)/δ), there are K ̺ > 0 and τ ̺ ≥ 2 such that Y (τ ) ≤ K ̺ τ -̺ for all τ ≥ τ ̺ . Coming back to X gives the claim.

The proof of Proposition 2.14 is divided into several steps.

) + Λ(τ ) 2 -Λ ′ (τ )],
and it follows from (2.15) that Λ solves the following differential equation

(p -1)Λ ′ (τ ) = F (τ, Λ(τ )) , τ ∈ R , (2.33)
with F (τ, Λ) := (p -1)Λ 2 + (p -N )Λ + Λ 2-p w(e τ ; a) 2-p α -βΛ -Λ q e -qτ f (e τ ; a) q-1 .

Since a ∈ C, we have R(a) = ∞ and it follows from Lemma 2.1 that f ′ (.; a) < 0, whence Λ > 0. In addition, for τ ∈ R, w ′ (e τ ; a) = e (µ-1)τ e τ f ′ (e τ ; a) + µf (e τ ; a) = e (µ-1)τ (µ -Λ)f (e τ ; a) > 0 since a ∈ C, from which we deduce that Λ(τ ) < µ for all τ ∈ R.

Step 2. Limit of Λ as τ → ∞. We claim that lim τ →∞ e -qτ f (e τ ; a) q-1 = 0.

(2.34) Indeed, either q ≥ 1 and (2.34) is obvious thanks to the boundedness of f which guarantees that e -qτ f (e τ ; a) q-1 ≤ a q-1 e -qτ , τ ∈ R , or q ∈ (p/2, 1) and e -qτ f (e τ ; a) q-1 = e -qτ e -µ(q-1)τ w(e τ ; a) q-1 = e -(2q-p)τ /(2-p) w(e τ ; a) q-1 , and both terms converge to zero as τ → ∞ since a ∈ C and q ∈ (p/2, 1). Hence (2.34) holds true. Consequently, since a ∈ C, we have

lim τ →∞ F (τ, γ) = +∞ if γ < α β , lim τ →∞ F (τ, γ) = -∞ if γ > α β , which ensures that lim τ →∞ Λ(τ ) = α β . ( 2

.35)

Step 3. Exponential decay of Λ -α/β. We study the rate of convergence of Λ(τ ) to α/β as τ → ∞. This last step of the proof is done in [START_REF] Chen | Self-similar singular solutions of a p-Laplacian evolution equation with absorption[END_REF] by construction of suitable suband supersolutions. We give here an alternative and direct approach based on Lemma 2.15 which allows us to handle also the case q < 1. Since Λ(τ ) → α/β as τ → ∞, taking into account the definition of Λ, we find

lim r→∞ rf ′ (r) f (r) = - α β ,
hence, for any ε ∈ (0, α/β), there exists r ε ≥ 1 such that

- α β -ε ≤ rf ′ (r) f (r) = -Λ(log r) ≤ - α β + ε, r > r ε . (2.36)
By integrating, we obtain that there exists a constant K ε > 0 such that

K ε r -(α/β)-ε ≤ f (r) ≤ K ε r -(α/β)+ε , r > r ε . (2.37)
We then come back to the differential equation (2.33) and check that the conditions required to apply Lemma 2.15 with either X = (Λ -α/β) + or X = (α/β -Λ) + are fulfilled. Indeed, we can write (2.33) in the form

(p -1)Λ ′ (τ ) + βΛ(τ ) 2-p w(e τ ; a) 2-p Λ(τ ) - α β = S(τ ) , τ ∈ R , (2.38)
with S(τ ) := (p -1)Λ(τ ) 2 + (p -N )Λ(τ ) -Λ(τ ) q-p+2 w(e τ ; a) 2-p e -qτ f (e τ ; a) q-1 .

On the one hand, owing to the boundedness of Λ, the non-negativity of f (.; a), w(.; a), and Λ, and (2.37), we have

S(τ ) ≤ (p -1)µ 2 + pµ , τ ∈ R , (2.39)
and S(τ ) ≥ -N µ -µ q-p+2 e (p-q)τ f (e τ ; a) q-p+1 ≥ -N µ -µ q-p+2 e (p-q)τ K q-p+1 ε e (ε(q-p+1)+q-p)τ S(τ ) ≥ -N µ -µ q-p+2 K q-p+1 ε e ε(q-p+1)τ , τ ≥ log r ε .

(2.40)

On the other hand, it follows from (2.35), (2.36), and (2.37) that

βΛ(τ ) 2-p w(e τ ; a) 2-p ≥ β α β -ε 2-p K 2-p ε e (2-p)(µ-(α/β)-ε)τ , τ ≥ log r ε . (2.41) We now fix ε ∈ (0, α/β) such that ε ≤ α 2β and 2ε(q -p + 1) < 1 β -ε(2 -p) = (2 -p) µ - α β -ε ,
(p -1)X ′ 1 (τ ) + β αK ε 2β 2-p e (2-p)(µ-(α/β)-ε)τ X 1 (τ ) ≤ S(τ ) + ≤ (p -1)µ 2 + pµ
for τ ≥ log r ε . Consequently, X 1 satisfies a differential inequality of the form of (2.30) with γ = (µ -(α/β) -ε)(2 -p) > 0 and δ = 0. We then apply Lemma 2.15 and conclude that Λ(τ ) -α β + ≤ C e -ε(q-p+1)τ for τ large enough. Similarly, setting X 2 := (α/β -Λ) + , it follows from (2.38) and (2.40) that (p -1)X ′ 2 (τ ) + βΛ(τ ) 2-p w(e τ ; a) 2-p X 2 (τ ) ≤ (-S(τ )) + ≤ N µ + µ q-p+2 K q-p+1 ε e ε(q-p+1)τ , and from (2.41), the non-negativity of X 1 , and the choice of ε that

(p -1)X ′ 2 (τ ) + β αK ε 2β 2-p e (2-p)(µ-(α/β)-ε)τ X 2 (τ ) ≤ N µ + µ q-p+2 K q-p+1 ε e ε(q-p+1)τ
for τ ≥ log r ε . Consequently, X 2 satisfies a differential inequality of the form of (2.30) with γ = (µ -(α/β) -ε)(2 -p) and δ = ε(q -p + 1), and the choice of ε guarantees that γ -δ ≥ δ > 0. We then apply Lemma 2.15 and conclude that

α β -Λ(τ ) + ≤ C e -ε(q-p+1)τ
for τ large enough. We have thus established that |Λ(τ ) -α/β| ≤ C e -ε(q-p+1)τ for τ large enough. In particular, Λ -α/β belongs to L 1 (1, ∞) and, recalling (2.32), we realize that r α/β f (r; a) = f (1; a) exp -

Characterization of B. Uniqueness

Since A = (0, a * ) and C = (a * , ∞), we already know that B is non-empty, that is, there exists at least one radially symmetric self-similar very singular solution. We complete the proof of Theorem 1.1 by the uniqueness result. (2.42)

As in [START_REF] Shi | Self-similar very singular solution of a p-Laplacian equation with gradient absorption: existence and uniqueness[END_REF], the proof is divided into two steps, with a different argument to prove the first step.

Step 1. We show that rw ′ (r; a) -→ 0 as r → ∞ for a ∈ B. Indeed, we notice that the differential equation (2.1) also reads

Lemma 2 . 6 (

 26 Comparison principle). Let a > 0, r 1 ∈ (0, R(a)), and r 2 ∈ (r 1 , R(a)) and assume that w ′ (.;a) > 0 in [r 1 , r 2 ]. Then, for any function h ∈ C 2 (r 1 , r 2 ) satisfying h(r 1 ) = h(r 2 ) = 0 and L a (h) ≥ 0 in (r 1 , r 2 ), we have h ≤ 0 in (r 1 , r 2 ).Proof. Lemma 2.6 is a variant of the classical maximum principle [9, Corollary 3.2] and is proved in[4, p. 48]. The proof relies on the existence of a function ϕ > 0 on [r 1 , r 2 ] such that L a (ϕ) ≤ 0. Here this is satisfied by ϕ(r) = rw ′ (r; a) in view of Lemma 2.5 and the assumption on w ′ (.; a).

Lemma 2 . 8 .

 28 Let a > 0. Then, a ∈ A if and only if there exists R 1 (a) ∈ (0, R(a)) such that w ′ (.; a) > 0 in (0, R 1 (a)), w ′ (.; a) < 0 in (R 1 (a), R(a)) and w ′′ (R 1 (a); a) < 0. Moreover, if a ∈ A, then sup r∈(0,R(a))w(r; a) < w * .(2.20)

  A is an open interval. By Lemma 2.8, if a ∈ A, we have w ′ (R 1 (a); a) = 0 with w ′′ (R 1 (a); a) < 0. By continuous dependence, this property readily implies that A is open. In addition, we can apply the implicit function theorem and conclude that a → R 1 (a) belongs to C 1 (A). Define m(a) := w(R 1 (a); a) for a ∈ A. Then m ∈ C 1 (A) and we infer from Lemma 2.7 (with r 0 = R 1 (a)) and Lemma 2.8 that dm da (a) = w ′ (R 1 (a); a) dR 1 da (a) + ∂ a w(R 1 (a); a) = ∂ a w(R 1 (a); a) > 0 , a ∈ A . (2.26) Consider now a 2 ∈ A and define a 1 ≥ 0 by a 1 := sup {a ∈ (0, a 2 ) : a ∈ A} .

Proposition 2 . 13 .

 213 The set C is an open interval of the form (a * , ∞).

Proof of Proposition 2 . 14 .

 214 Step 1. Introducing a new function. Fix a ∈ C. As in [6, Theorem 4.1], we define a new function Λ by f (e τ ; a) = f (1; a) exp -τ 0 Λ(s) ds , τ ∈ R. (2.32) Then f ′ (e τ ; a) = -e -τ Λ(τ )f (e τ ; a), f ′′ (e τ ; a) = e -2τ f (e τ ; a)[Λ(τ

  k(a) ∈ (0, ∞) , as stated in Proposition 2.14.

Proposition 2 . 16 .

 216 We have a * = a * , thus B contains only one element. Proof. Let a ∈ B. Then, R(a) = ∞, w(.; a) is increasing and has a finite limit as r → ∞. Arguing as in the first step of the proof of Lemma 2.10, we deduce that lim r→∞ r µ f (r; a) = lim r→∞ w(r; a) = w * .

Step 2 .

 2 -1 |f ′ (r)| p-2 f ′ (r) + βr N f (r)) = r N -1 (|f ′ (r)| q -(α -βN )f (r)) , r ≥ 0 .Since q > p/2, it follows from Lemma 2.1 and Proposition 2.4 that, as r → ∞, |f ′ (r; a)| q ≤ C 4 f (r; a) 2q/p = o(f (r; a)), and we deduce from (2.42) that, as r → ∞,d dr (r N -1 |f ′ (r)| p-2 f ′ (r) + βr N f (r)) ∼ -(α -βN )w * r N -1-µ .Since µ > N , we obtain after integration-r N -1 (-f ′ (r; a)) p-1 + βr N f (r; a) ∼ (α -βN )w * r -(µ-N ) µ -N , hence, using again (2.42), -(-f ′ (r; a)) p-1 ∼ -βrf (r; a) + α -βN µ -N w * r -(µ-1) ∼ α -βN µ -N -β w * r -(µ-1)as r → ∞. Noticing thatα -βN µ -N -β = -1 p(N + 1) -2N < 0, we conclude that, as r → ∞, f ′ (r; a) ∼ -w * p(N + 1) -2N 1/(p-1) r -2/(2-p) . Consequently, lim r→∞ rw ′ (r; a) = lim r→∞ r 2/(2-p) f ′ (r; a) + µr p/(2-p) f (r; a) = -w * p(N + 1) -2N 1/(p-1)+ µw * , But from the definition and a simple computation we findw * = µ (p-1)/(2-p) (p(N + 1) -2N ) 1/(2-p) , that is µ = (w * ) (2-p)/(p-1) (p(N + 1) -2N ) -1/(p-1) , hence µw * = (w * ) 1+(2-p)/(p-1) (p(N + 1) -2N ) -1/(p-1) = w * p(N + 1) -2N 1/(p-1).We have thus shown that lim r→∞ rw ′ (r; a) = 0 . (2.43) We now follow the argument in [6, Theorem 5.1]. Let a ∈ B. Since w(r; a) -→ w * and rw ′ (r; a) -→ 0 as r → ∞ by (2.42) and (2.43), the homogeneous linear operator L a defined by (2.16) can be written as L a (ϕ) = r 2 ϕ ′′ (r) + P 1 (r) r ϕ ′ (r) + P 0 (r) ϕ(r) , with lim r→∞ P 1 (r) = b := N -1 p -1 -2µ + p(N + 1) -2N p -1 (1 + βµ) , lim r→∞ P 0 (r) = -c , c := µ p(N + 1) -2N p -1 > 0 .

( 2 .

 2 44)Owing to the negativity of c, we claim that, given two linearly independent solutions of the equation L a (ϕ) = 0, one of them has to be unbounded as r → ∞. Taking this result for granted, we define the function ψ as the solution to L a (ψ) = 0 in (1, ∞) with ψ(1) = 0, ψ ′ (1) = 1. Arguing as in Lemma 2.7 with r 0 = ∞ (recall that R(a) = ∞ as a ∈ B), we conclude that there is a positive real number κ > 0 such that ∂ a w(.; a) > κψ > 0 in (1, ∞). Since L a (∂ a w(.; a)) = 0 with ∂ a w(1; a) > 0 by Lemma 2.5 and Lemma 2.7, ∂ a w(.; a) and ψ are clearly linearly independent solutions of the equation L a (ϕ) = 0. The above claim implies that one of them must be unbounded and thus that ∂ a w(.; a) is unbounded since it dominates ψ. We then deduce that, for all a ∈ B, ∂ a w(r; a) -→ ∞ as r → ∞. Therefore, if a * < a * , we can apply Fatou's lemma to get 0 = lim r→∞ [w(r, a * ) -w(r, a * )] = lim r→∞ a * a * ∂ a w(r; a) da ≥ a * a * lim inf r→∞ ∂ a w(r; a) da = ∞,

which is a contradiction. Hence a * = a * . Let us finally sketch the proof of the above claim. Introducing the new variable τ = log r, the linear operator L a transforms into the linear operator La (ψ)(τ ) := φ′′ (τ ) + P1 (τ ) φ′ (τ ) + P0 (τ ) φ(τ ) with P1 (τ ) := P 1 (e τ ) -1 and P0 (τ ) := P 0 (e τ ). Introducing next the change of function

we obtain the canonical form K a of the operator La which reads

At this point, we note that (2.44) ensures that lim

Then, if ψ is a solution to K a (ψ) = 0, the variation of constants formula implies that there are ψ 0 ∈ R and

Arguing by contradiction, it is now easy to show that, if ψ 1 > 0 and λ ∈ (0, λ 0 ), then the function τ → e -λτ ψ(τ ) cannot be bounded. Coming back to L a gives the expected result after choosing λ appropriately close to λ 0 .

A Viscosity solutions to (1.1)

We recall here the definition of viscosity solutions for singular diffusion equations such as (1.1) as introduced in [START_REF] Ishii | Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor[END_REF][START_REF] Ohnuma | Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation[END_REF]. Owing to the singularity of the diffusion, it differs from the usual definition of vicosity solutions (see, e.g., [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]) by restricting the class of comparison functions. More precisely, let Ξ be the set of functions ξ ∈ C 2 ([0, ∞)) satisfying

For example, ξ(r) = r σ with σ > p/(p -1) > 2 belongs to Ξ. In fact, if ξ ∈ Ξ, it readily follows from (A.1) that lim r→0 ξ(r) r p/(p-1) = 0.

(A.2)

We next introduce the class A of admissible comparison functions ψ ∈ C 2 (Q ∞ ) defined as follows: ψ ∈ A if, for any (t 0 , x 0 ) ∈ Q ∞ where ∇ψ(t 0 , x 0 ) = 0, there exist a constant δ > 0, a function ξ ∈ Ξ, and a nonnegative function ω ∈ C([0, ∞)) satisfying ω(r)/r -→ 0 as r → 0, such that, for all (t, x)

A lower semicontinuous function u : Q ∞ → R is a viscosity supersolution to (1.1) in Q ∞ if -u is a viscosity subsolution to (1.1) in Q ∞ . A continuous function u : Q ∞ → R is a viscosity solution to (1.1) in Q ∞ if it is a viscosity subsolution and supersolution.


J L Cabrerizo

M Fernández

J S Gómez

Isotropy and Marginally Trapped Surfaces in a Spacetime

Keywords: numbers: 04.20.Cv, 02.40.Ky, 02.40.Ma

In this note we shall study the notions of isotropic and marginally trapped surface in a spacetime by using a differential geometric approach. We first consider spacelike isotropic surfaces in a Lorentzian manifold and, in particular, in a four-dimensional spacetime, where the isotropy function appears to be determined by the mean curvature vector field of the surface. Explicit examples of isotropic marginally outer trapped surfaces are given in the standard fourdimensional space forms: Minkowski, De Sitter and anti De Sitter spaces. Then we prove ridigity theorems for complete spacelike 0-isotropic surfaces without flat points in these standard space forms. As a consequence, we also obtain characterizations of complete spacelike isotropic marginally trapped surfaces in these backgrounds.

Introduction

A precise definition of isotropy depends on the subject area. In Mathematics, a Riemannian manifold is isotropic if, roughly speaking, the geometry of the manifold is the same regardless of direction. The concept of isotropic immersion of a Riemannian manifold S in another M was introduced by B. O'Neill in 1965 [1] as an isometric immersion such that all its normal curvature vectors have the same length. These submanifolds can be considered as a generalization of the totally umbilical submanifolds, and constitute a distinguished family in Submanifold Theory. Later, Y. H. Kim [2] introduced the notion of pseudo-isotropic submanifold by extending the O'Neill's notion to the case of pseudo-Riemannian submanifolds of pseudo-Euclidean space. Since both notions are basically identical, from now on we shall omit the prefix pseudo and call them all isotropic submanifolds. However, new proof methods are usually required to achive similar results to those obtained in the Riemannian case [3,4].

On the other hand, in studies of gravitational collapse, an useful local property is the closed trapped surface, introduced by R. Penrose in 1965 [5]. This is a closed spacelike two-dimensional surface S at any instant of time which has the property that the outgoing light rays from it are convergent. In the spherical symmetric case the existence of a trapped surface garantees that the system must collapse to a black hole assuming only that the matter has positive energy density (the week energy condition). The idea that matter concentration causes gravitational collapse is the trapped surface conjecture: a trapped surface forms if a sufficient amount of matter is packed into a small enough volume. In [6] the authors give necessary and sufficient conditions for the existence of trapped surfaces in sphericaly symmetric spacetimes, and in [7] the authors by using the spin coefficient formalism generalize from the Schwarzschild solution and prove the existence of a wide class of solutions possessing trapped surfaces, includding the so-called "asymptotically trapped surfaces". In [8] the non-existence of these closed surfaces is proved for strictly stationary spacetimes. In it is shown that the existence of one trapped surface in a spherical symmetric spacetime is sufficient to ensure the formation of a black hole and the completeness of null infinity. A generalization of the concept of trapped surface is introduced in [9] for codimension-2 submanifolds of a Lorentzian space.

From a geometrical point of view, the light converging condition means that the mean curvature vector H is everywhere timelike on the surface. If the mean curvature vector is future or past-pointing all over the surface, the trapped surface is accordingly called future or past trapped. On the other hand, consider two futurepointing lightlike normal vectors at each point of the surface S and suppose the mean curvature vector H is proportional to some of them by a function Θ. The following cases may occurs: (1) S is marginally trapped if Θ never vanishes; (2) partly marginally trapped if Θ ≥ 0 or Θ ≤ 0; (3) marginally outer trapped (MOTS) if Θ is arbitrary. The study of these family of surfaces has been quite active in recent years (see for instance [9,10,11,12,13], etc).

It is clear that the second fundamental form of S is a suitable tool to study these families of surfaces. Thus, a classification of spacelike marginally trapped surfaces in the Lorentzian space forms R n 1 , S n 1 and H n 1 having the property that the second fundamental form is a quadratic degenerated form (in the sense that the surfaces have positive relative nullity) have been obtained in [14]. The same authors proved the nonexistence of marginally trapped surfaces with positive relative nullity for a Robertson-Walker spacetime which does not contain any open subset of constant curvature [15].

The plan of the paper is as follows. Basic formulas, definitions and facts on n-dimensional Lorentz manifolds M n 1 are introduced in Section II. In Section III we first recall the notion of isotropic surface S in M n 1 and, as a consequence, we see that it is pseudo-umbilical, but for n = 3 the surface is totally umbilical. Section IV deals with the interesting case of a surface embedded into a 4-dimensional spacetime M 4 1 , where the notions of λ-isotropy and pseudo-umbilicity for surfaces with lightlike mean curvature vector are seen to be equivalent. Some properties of the isotropy function λ are obtained, in particular, characterizations of 0-isotropic surfaces. In Section V we first exhibit examples of 0-isotropic MOTS in the 4dimensional spacetimes R 4 1 , S 4 1 and H 4 1 , and then we give sufficient conditions for them to be non-totally umbilical marginally trapped surfaces. Finally, we present our main results that provide ridigity theorems, that is, explicit determination of complete spacelike 0-isotropic surfaces without flat points in these spacetimes. In fact, they are seen to be MOTS congruent to the corresponding immersed surfaces introduced as Examples 5.1, 5.2 and 5.3. As a consequence, we obtain a characterization of complete spacelike isotropic marginally trapped surfaces as well as complete pseudo-umbilical marginally trapped surfaces in these standard space forms. We close the paper with a summary of conclusions.

Preliminaries

Let M n 1 be a n-dimensional Lorentzian manifold with metric tensor g. All the manifolds will be assumed to be connected without boundary. Let φ : S → M n 1 be an isometric immersion of a 2-dimensional spacelike surface S. For all local formulae and computations we may assume φ is an imbedding and thus we shall often identify p ∈ S with φ(p) ∈ M n 1 , and the tangent space T p S with the subspace φ * (T p S) of

T p M n 1 .
The normal bundle will be denoted by T ⊥ S. We shall use letters X, Y, Z (resp. ξ, η, ζ) to denote vectors fields tangent (resp. normal) to S. A tangent vector v ∈ T p M n 1 is said to be spacelike (timelike or lightlike) if g(v, v) ≥ 0 (g(v, v) < 0, g(v, v) = 0 and v = 0, respectively). Let ∇ and ∇ be the Levi-Civita connections of M n 1 and S, respectively. Then, the Gauss-Weingarten formulas are given by

∇X Y = ∇ X Y + h(X, Y ), (2.1) ∇X ξ = -A ξ X + D X ξ, ( 2.2)
where h denotes the second fundamental form of φ, A ξ the shape operator, and D is the normal connection. The shape operator and the second fundamental form are related by g(A ξ X, Y ) = g(h(X, Y ), ξ).

Denote by R and R the curvature tensor of M n 1 and S, respectively. If

M n 1 is of constant sectional curvature c, then R is given by R(X, Y )Z = c{g(Y, Z)X - g(X, Z)Y }. The curvature tensor R is given by R(X, Y )Z = K{g(Y, Z)X-g(X, Z)Y },
where K is the Gauss curvature of S.

If p ∈ S and v ∈ T p M n 1 we shall denote by v and v ⊥ the tangential and normal components of v, respectively. The equations of Gauss and Codazzi are given, respectively, by

( R(X, Y )Z) = R(X, Y )Z + A h(X,Z) Y -A h(Y,Z) X, (2.3) ( R(X, Y )Z) ⊥ = ( ∇X h)(Y, Z) -( ∇Y h)(X, Z), (2.4)
where ∇h is defined by

( ∇X h)(Y, Z) = D X (h(Y, Z)) -h(∇ X Y, Z) -h(Y, ∇ X Z).
If p ∈ S and x, y ∈ T p S are linearly independent tangent vectors, we define the discriminant D p at p as D p = g(h(x, x), h(y, y)) -g(h(x, y), h(x, y)) g(x, x)g(y, y) -g(x, y) 2 .

(2.5)

If K denotes the sectional curvature of M n 1 , from Eq. (2.3) we easily have

K(x, y) = K p -D p .
(2.6)

Note that if D ≡ 0 then S is extrinsically flat: observers in M n 1 see no curving. This does not mean that S is intrinsically flat, but the surface S has the same intrinsic curvature as M n 1 . Recall that a point p ∈ S is called umbilic [START_REF] Neill | Semi-Riemannian Geometry with Applications to Relativity[END_REF] provided there exists a vector ξ p ∈ T ⊥ p S such that h(u, v) = g(u, v)ξ p for any u, v ∈ T p S. The immersion φ is said to be totally umbilical when every point of S is umbilic. In such a case, it is easy to see that h(X, Y ) = g(X, Y )H for any X, Y ∈ X(S), where H = (1/2)trace g (h) is the mean curvature vector of S. We also recall that a point p ∈ S is said to be flat if h = 0 at p, and the immersion φ is totally geodesic provided every point of S is flat.

The immersion φ is called pseudo-umbilical if its second fundamental form h satisfies g(h(X, Y ), H) = ρg(X, Y ) for some function ρ. Necessarily this function ρ is given by ρ = g(H, H). Clearly, any totally umbilical immersion is pseudoumbilical.

The first normal space of φ at p ∈ S is defined as the subspace spanned by the second fundamental form h at p, that is,

Im(h) p = span{h(u, v) : u, v ∈ T p S}. A simple computation shows that {ξ ∈ T ⊥ p S : A ξ = 0} is the orthogonal complement of Im(h) p in T ⊥ p S. Observe that if the dimension of Im(h) is constant along S, then Im(h) is a subbundle of T ⊥ S.

Isotropic immersion

We first recall that an isometric immersion φ :

S → M n 1 is called isotropic at p ∈ S [2] if g(h(u, u), h(u, u)) = λ(p) ∈ R (3.1)
does not depends on the choice of the unit tangent vector u ∈ T p S, and φ is said to be isotropic if φ is isotropic at each point of S. In such a case, the smooth function λ : S → R defined by equation (3.1) is called the isotropy function, and the isometric immersion φ is said to be λ-isotropic. In particular, if λ is a constant function, then φ is said to be a constant isotropic immersion.

Remark 3.1 It is clear that every totally umbilical immersion φ : S → M n 1 is an isotropic immersion with isotropy function λ = g(H, H), but not vice versa. Notice also that if φ is totally umbilical and M n 1 has constant sectional curvature c, then H is a parallel vector field and the function g(H, H) is a constant [17, p.11].

Similar to the Riemannian case [1], it is easy to prove the following.

Lemma 3.2 Let φ : S → M n
1 be a spacelike isometric immersion. Then, the following conditions are equivalent:

(1) φ is isotropic at p.

(2) g(h(x, y), h(z, w))+g(h(y, z), h(x, w))+g(h(z, x), h(y, w

)) = λ(p){g(x, y)g(z, w)+ g(y, z)g(x, w) + g(x, z)g(y, w)} for all x, y, z, w ∈ T p S.
Furthermore, if φ is isotropic at p we have

g(h(u, u), h(u, v)) = 0, (3.2) g(h(u, u), h(v, v)) + 2g(h(u, v), h(u, v)) = λ(p), (3.3)
for any orthonormal tangent vectors u, v ∈ T p S.

Remark 3.3 As a consequence of equation (3.2) we see that an isotropic immersion of a spacelike surface S into a 3-dimensional Lorentzian manifold M 3 1 is totally umbilical, since T ⊥ S is a 1-dimensional negative defined space [4].

Lemma 3.4 Let φ : S → M n 1 be a spacelike λ-isotropic immersion.

Isotropic surfaces in 4-dimensional spacetimes

When the mean curvature vector field H of an immersed spacelike surface in a spacetime M n 1 is a lightlike vector field and n=4, then the notions of isotropy and pseudo-umbilicity are equivalent. In fact, we have the following. Proposition 4.1 Let φ : S → M 4 1 be a spacelike surface with lightlike mean curvature vector. Then, the following assertions are equivalent.

(1) φ is pseudo-umbilical.

(2) φ is 0-isotropic.

(3) φ is isotropic.

PROOF. (1) ⇒ (2). Since g(h(X, X), H) = 0 and the normal space is a Lorentzian plane we have g(h(X, X), h(X, X)) = 0. Now (2) ⇒ (3) is obvious, and (3) ⇒ (1) follows from Lemma 3.4. Theorem 4.2 Let S be a spacelike λ-isotropic surface in a spacetime M 4 1 . Then, we have

(a) If p ∈ S is not an umbilical point, then λ(p) = g(H, H) p = D p = 0. (b) The isotropy function λ is given by λ = g(H, H) = D. (c) For any X, Y, Z, W ∈ X(S) we have g(h(X, Y ), h(Z, W )) = λg(X, Y )g(Z, W ). ( 4
∼ = R 4
1 (e 4 is a unit timelike vector), and δ = [λ(p)] 1/2 > 0, µ = [-g(h(e 1 , e 2 ), h(e 1 , e 2 ))] 1/2 > 0. On the other hand, from Eq. (3.2) we have also that g(h(e 1 , e 2 ), h(e 2 , e 2 )) = 0. Therefore, equations (4.2) and (4. F (x, y, z, w) = g(h(z, x), h(y, w)) -g(h(y, z), h(x, w)) -λ(p) {g(z, x)g(y, w) -g(y, z)g(x, w)} .

(4.6)

Then, F is a curvaturelike function, i.e., F has the following symmetries:

(i) F (x, y, z, w) + F (y, z, x, w) + F (z, x, y, w) = 0, (ii) F (y, x, z, w) = -F (x, y, z, w), (iii) F (x, y, w, z) = -F (x, y, z, w), (iv) F (z, w, x, y) = F (x, y, z, w),
But Eq. (2.5) yields F (x, y, y, x) = 0 for any x, y ∈ T p S such that Q(x, y) = g(x, x)g(y, y) -g(x, y) 2 = 0, and therefore [16] F = 0. In particular,

g(h(z, x), h(y, w)) -λ(p)g(z, x)g(y, w) = g(h(y, z), h(x, w)) -λ(p)g(y, z)g(x, w) = = g(h(x, y), h(z, w)) -λ(p)g(x, y)g(z, w).
Now, using (2) of Lemma 3.2, Eq. (4.1) follows.

Remark 4.3 Observe that an isotropic surface into a Riemannian manifold which satisfies Eq. (4.1) is totally umbilical. The Lorentzian version, however, is not true.

As a immediate consequence we have the following two corollaries.

Corollary 4.4 Let φ : S → M 4 1 be a spacelike isotropic immersion. Then, the following assertions are equivalent.

(1) φ is 0-isotropic.

(2) g(h(X, Y ), h(Z, W )) = 0 for any X, Y, Z, W ∈ X(S).

(3) g(H, H) = 0.

(4) S is extrinsically flat, i.e., the discriminant D ≡ 0.

(5) The first normal space Im(h) is a lightlike line at each non-flat point.

Corollary 4.5 Let φ : S → M 4 1 be a spacelike isotropic immersion without umbilical points. Then, φ is 0-isotropic. Corollary 4.6 Let φ : S → M 4 1 be a non-totally umbilical spacelike constant isotropic immersion. Then, φ is 0-isotropic. PROOF. Set U = {p ∈ S : h(u, v) = 0 for some orthonormal vectors u, v ∈ T p S}. Note that U is a non-empty open set which is the set of not umbilical points of S. From Theorem 4.2 the isotropy function λ satisfies λ| U = g(H, H)| U = 0. Now, let p ∈ U be an umbilical point in the boundary of U and take a sequence {p N } N ∈N in S which converges to p and such that every p N is not umbilical. Let X be a local unitary vector field around p and consider the function λ = g(h(X, X), h(X, X)). We have

λ(p) = lim N g (h(X(p N ), X(p N )), h(X(p N ), X(p N ))) = lim N λ(p N ) = 0.
If U is dense the result follows. If U is not dense, the set S -U is an open submanifold of S which is totally umbilical, and thus constant g(H, H)-isotropic. Now we take a sequence of umbilical points converging to a point in the boundary of U, we obtain λ = g(H, H) = 0.

The following result can also be proved in a similar way as Corollary 4.6.

Corollary 4.7 Let S be a non-totally umbilical spacelike isotropic surface of a spacetime M 4 1 (c) of constant sectional curvature c. Then, φ is 0-isotropic.

Isotropic surfaces in a Lorentzian space form

Let R n s be the n-dimensional pseudo-Euclidean space provided with the usual inner product of signature s defined by

x, y = - s i=1 x i y i + n i=s+1
x i y i , where x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ). For a positive number c, the standard space form S n s (c) is the hypersurface S n s (c) = {x ∈ R n+1 s : x, x = 1/c} endowed with the induced metric of signature s. For a negative number c, the standard space form H n s (c) is the hypersurface H n s (c) = {x ∈ R n+1 s+1 : x, x = 1/c} endowed with the induced metric of signature s. We simply denote S n s (1), and H n s (-1) by S n s , H n s , and the manifolds S n s , H n s ,R n 1 , are known as the Lorentzian spaces form models of constant curvature c = 1, c = -1, and c = 0, respectively. In general relativity, R 4 1 , S 4 1 and H 4 1 are called respectively as the Minkowski, de Sitter, and anti-de Sitter spacetimes.

It is well known that for every point p ∈ S n 1 (resp. H n 1 ), any complete connected totally geodesic hypersurface of S n 1 (resp. H n 1 ) through p is the connected component of the intersection of S n 1 (resp. H n 1 ) with a n-dimensional vector subspace

V of R n+1 1 (resp. R n+1 2
) which contains p, and vice versa [16, p. 105]. These totally geodesic hypersurfaces are degenerates if and only if the n-dimensional vector subspace V is degenerate, or equivalently, if the orthogonal complement of V is a lightlike line. We now show an example of an isotropic and marginally trapped surface in R 4 1 , which is not totally umbilical. is a 0-isotropic MOTS. In fact, it is easy to see that the second fundamental form h and the mean curvature vector H are given respectively by

h(X, Y ) = ∇ 2 f (X, Y ) η, H = 1 2 ∆f η,
where η is the lightlike vector (1, 0, 0, 1) ∈ R 4 1 , ∇ 2 f denotes the Hessian of f in R 2 and ∆f = trace g (∇ 2 f ) is the corresponding Laplacian. Moreover, H is lightlike if and only if ∆f is nowhere zero, and ψ is totally umbilical if and only if f is given by

f (x, y) = a x 2 + y 2 + bx + cy + d, with a, b, c, d ∈ R. Notice that ψ(R 2 ) is contained in a degenerate hyperplane x 1 = x 4
(in the natural coordinates of R 4 1 ). Obviously, there exist functions f ∈ C ∞ (R 2 ) such that H is a non-parallel lightlike vector field. In such cases R 2 is a non-totally umbilical isotropic marginally trapped surface embedded in R 4 1 .

In the next example we show an isotropic marginally trapped immersion of S 2 into S 4 1 which is non-totally umbilical.

Example 5.2 Let f : S 2 → R be a smooth function. The isometric immersion ψ : S 2 → S 4 1 defined by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)),

where (x, y, z) denote the canonical coordinates of R 3 , is a 0-isotropic MOTS in the Lorentzian space S 4 1 . In fact, let ∇ 2 f be the Hessian of the function f in the sphere S 2 . A straightforward computation shows [18] that the second fundamental form h of ψ is given by

h(X, Y ) = (∇ 2 f )(X, Y ) + f g(X, Y ) η,
where η denote the lightlike vector (1, 0, 0, 0, 1) ∈ R 5 1 . Thus, the immersion ψ is 0-isotropic with mean curvature vector

H = 1 2 (∆f + 2f ) η.
Note that g(ψ, η) = 0 means that ψ(S 2 ) is contained in the degenerate totally geodesic hypersurface Π of S 4 1 defined as the intersection of S 4 1 with the degenerate hyperplane x 1 = x 5 of R 5 1 . It is clear that H is lightlike if and only if ∆f +2f is nowhere zero. On the other hand, an isometric immersion of a compact pseudo-umbilical spacelike surface S into the De Sitter spacetime S 4 1 with parallel lightlike mean curvature vector, is totally umbilical [19]. Moreover, if S is a pseudo-umbilical surface in S 4 1 with g(H, H) = 0, then it is easy to see that H is parallel if and only if H is a fixed vector of R 5 1 . Since an eigenfunction of -∆ corresponding to the eigenvalue -2 is the restriction to S 2 of a homogeneous polynomial of degree 1, [START_REF] Berger | Le spectre d'une variété Riemannienne[END_REF], we have that ψ is totally umbilical if and only if f is given by

f (x, y, z) = ax + by + cz + d,
where a, b, c, d ∈ R. Notice that there exist functions f ∈ C ∞ (S 2 ) such that H is a lightlike non-parallel vector field. In fact, it suffices to take f (x, y, z) = xy -2 and then we have H = -2(xy + 1)η. In such cases, S 2 is a compact marginally trapped surface embedded in S 4 1 and ψ is a non-totally umbilical.

Example 5.3 Let f : H 2 → R be a smooth function. Then, the isometric immersion ψ :

H 2 → H 4 1 defined by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)) is a 0-isotropic MOTS in H 4 1 . In fact, the second fundamental form h of ψ is given by h(X, Y ) = (∇ 2 f )(X, Y ) -f g(X, Y ) η,
where ∇ 2 f is the Hessian of f in H 2 and η denotes the lightlike vector (1, 0, 0, 0, 1) ∈ R 5 2 . Thus, the immersion ψ is 0-isotropic with mean curvature vector

H = 1 2 (∆f -2f ) η.
Moreover, ψ(H 2 ) is contained in the degenerate totally geodesic hypersurface Π of H 4 1 , defined as the intersection of H 4 1 with the degenerate hyperplane x 1 = x 5 of R 5 2 . Note that there exist functions f ∈ C ∞ (H 2 ) such that H is a non-parallel lightlike vector field. In fact, it suffices to take f (x, y, z) = x 2 + y 2 and then H = 2(x 2 + y 2 )η. In such cases H 2 is a non-totally umbilical isotropic marginally trapped surface embedded in H 4 1 .

For a 4-dimensional Lorentzian space form M 4 1 (c) of constant sectional curvature c we have the following result. Lemma 5.4 Let φ : S → M 4 1 (c) be a spacelike 0-isotropic immersion without flat points. Then, φ(S) is contained in a degenerate totally geodesic hypersurface Π of M 4 1 (c). In particular, there exits a constant lightlike normal vector field η such that g(φ, η) = 0 and H is proportional to η. PROOF. From Corollary 4.4 we have that the first normal space Im(h) is a normal subbundle spanned by a lightlike vector at each point p ∈ S and

g(h(X, Y ), h(Z, W )) = 0, (5.1)
holds for any X, Y, Z, W ∈ X(S). Now, a simple differentiation of equation (5.1) with respect to any tangent vector field T in S gives

g(( ∇T h)(X, Y ), h(Z, W )) = -g(h(X, Y ), ( ∇T h)(Z, W )). (5.2)
Combining repeatedly Codazzi's equation ( ∇X h)(Y, Z) = ( ∇Y h)(X, Z) and equation (5.2), we find g(( ∇T h)(X, Y ), h(Z, W )) = 0. Since the normal space is Lorentzian and two-dimensional we have ( ∇T h)(X, Y ) ∈ Im(h) for any X, Y, T ∈ X(S). This shows that the first normal space Im(h) is D-parallel. Therefore φ(S) is contained in a degenerate totally geodesic hypersurface Π of M 4 1 (c) [21], and consequently, there exits a lightlike constant normal vector field η such that g(φ, η) = 0 and H is proportional to η.

The flat points hypothesis in Lemma 5.4 is necessary as shown in the following example.

Example 5.5 Let (x, y) be the canonical coordinates of R 2 and consider the open set S = {(x, y) ∈ R 2 : 1 < x 2 + y 2 < 4}. Let f : S → R be a smooth function such that f > 0 on the sets

U = {(x, y) ∈ S : 2 -1 -x 2 < y < 4 -x 2 } and V = {(x, y) ∈ S : -4 -x 2 < y < -2 + 1 -x 2 },
while f = 0 on S -(U ∪ V ). Define φ : S → R 4 1 as the isometric immersion given by φ(x, y) = (f (x, y), x, y, f (x, y)) if y ≥ 0, (f (x, y), x, y, -f (x, y)) if y < 0.

Then φ is a spacelike 0-isotropic immersion with flat points, which is full in R 4 1 , i. e., φ(S) is contained in no hyperplane of R 4 1 . The following result determines complete spacelike 0-isotropic surfaces of R 4 1 without flat points.

Theorem 5.6 Let φ : S → R 4 1 be a complete spacelike 0-isotropic immersion without flat points. Then, (a) S is isometric to R 2 .

(b) φ is congruent to an isometric immersion ψ : R 2 → R 4 1 given by ψ(x, y) = (f (x, y), x, y, f (x, y)), for some smooth function f : R 2 → R. Moreover, S is a MOTS in R 4 1 with mean curvature vector

H = 1 2 ∆f η,
where η is a constant lightlike vector field of R 4 1 .

PROOF. (a) From Lemma 5.4 the codimension can be reduced so that φ(S) is contained in a degenerate hyperplane Π of R 4 1 defined as the orthogonal complement of a constant lightlike vector field η of R 4 1 . Now, consider a rigid motion A of R 4 1 sending η (or a proportional vector to η) into (1, 0, 0, 1), and hence Π onto the hyperplane x 1 = x 4 of R 4 1 . Let π : R 4 1 → R 2 be the projection map onto the coordinates (x 2 , x 3 ). Then π • A • φ : S → R 2 is an isometric immersion and, consequently, a local isometry. Now, by the completeness hypothesis, π • A • φ is an isometry [START_REF] Neill | Semi-Riemannian Geometry with Applications to Relativity[END_REF]Corollaries 7.27,7.29].

(b) As π • • φ is an isometry, there exists a smooth function f : R 2 → R such that A sends φ(S) into the set of points (f (x, y), x, y, f (x, y)). Observe that the mean curvature vector of A • φ is given by A(H). Now, from Example 5.1 we have A(H) = 1 2 ∆f (1, 0, 0, 1) and the result follows.

From Proposition 4.1 and Theorem 5.6 we easily obtain the following ridigity results.

Corollary 5.7 Let S be a complete spacelike surface of R 4 1 . Then, S is an isotropic marginally trapped surface if and only if S is congruent to a surface defined by ψ(x, y) = (f (x, y), x, y, f (x, y)), where f : R 2 → R is a smooth function such that ∆f is nowhere zero. Corollary 5.8 Let S be a complete spacelike surface of R 4 1 . Then, S is a pseudoumbilical marginally trapped surface if and only if S is congruent to a surface defined by ψ(x, y) = (f (x, y), x, y, f (x, y)), where f : R 2 → R is a smooth function such that ∆f is nowhere zero. Remark 5.9 The last result completely determines the complete spacelike pseudoumbilical surfaces with lightlike mean curvature vector in R 4 1 . Compare this result with Theorem 4.1 of [14]. On the other hand, note that the marginally trapped surface S of this Corollary is totally umbilical if and only if f (x, y) = a(x 2 + y 2 ) + bx + cy + d with a, b, c, d ∈ R, a = 0 (see Example 5.1). Now, we prove a result analogous to Theorem 5.6 for complete spacelike 0isotropic surfaces of S 4 1 without flat points.

Theorem 5.10 Let φ : S → S 4 1 be a complete spacelike 0-isotropic immersion without flat points. Then, (a) S is isometric to S 2 . (b) φ is congruent to an isometric immersion ψ : S 2 → S 4 1 given by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)),

for some smooth function f : S 2 → R, and S is a compact MOTS of S 4 1 with mean curvature vector

H = 1 2 (∆f + 2f ) η,
where η is a constant lightlike vector field of R 5 1 .

PROOF. (a) From Lemma 5.4 φ(S) is contained in a degenerate hypersurface Π of S 4 1 defined as the intersection of S 4 1 with the orthogonal complement of a constant lightlike vector field η of R 5 1 . Now we consider a rigid motion A of S 4 1 such that A (viewed as a rigid motion of R 5 1 ) sends η into (1, 0, 0, 0, 1), and hence Π onto the hyperplane x 1 = x 5 . Let π : R 5 1 → R 3 be the projection onto the coordinates (x 2 , x 3 , x 4 ). Then π • A • φ : S → S 2 is an isometric immersion and, by the completeness hypothesis, an isometry.

(b) As π • A • φ is an isometry, there a smooth function f : S 2 → R such that A sends φ(S) into the set of points (f (x, y, z), x, y, z, f (x, y, z)). Observe that the mean curvature vector of A • φ is given by A(H). Now, from Example 5.2, A(H) = 1 2 (∆f + 2f ) (1, 0, 0, 0, 1) and the result follows.

By Proposition 4.1 and Theorem 5.10 we have also the following ridigity results.

Corollary 5.11 Let S be a complete spacelike surface of S 4 1 . Then, S is an isotropic marginally trapped surface if and only if S is congruent to a surface defined by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)), where f : S 2 → R is a smooth function such that ∆f + 2f is nowhere zero. Corollary 5.12 Let S be a complete spacelike surface of S 4 1 . Then, S is a pseudoumbilical marginally trapped surface if and only if S is congruent to a surface defined by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)), where f : S 2 → R is a smooth function such that ∆f + 2f is nowhere zero.

We can also prove the corresponding H 4 1 -versions of Theorem 5.10 and Corollaries 5.11, 5.12 as follows.

Theorem 5.13 Let φ : S → H 4 1 be a complete spacelike 0-isotropic immersion without flat points. Then,

(a) S is isometric to H 2 . (b) φ is congruent to an isometric immersion ψ : H 2 → H 4
1 given by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)),

for some smooth function f : H 2 → R, and S is a MOTS in H 4 1 with mean curvature vector

H = 1 2 (∆f -2f ) η,
where η is a constant lightlike vector field of R 5 2 .

Corollary 5.14 Let S be a complete spacelike surface of H 4 1 . Then, S is an isotropic marginally trapped surface if and only if S is congruent to a surface defined by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)), where f : H 2 → R is a smooth function such that ∆f -2f is nowhere zero. Corollary 5.15 Let S be a complete spacelike surface of H 4 1 . Then, S is a pseudoumbilical marginally trapped surface if and only if S is congruent to a surface defined by ψ(x, y, z) = (f (x, y, z), x, y, z, f (x, y, z)), where f : H 2 → R is a smooth function such that ∆f -2f is nowhere zero.

Summary

In this paper we have considered two families of surfaces in spacetimes: isotropic and marginally trapped surfaces. The notion of λ-isotropic surface means that the geometry of the surface is the same at each point regardless of direction, but may vary from point to point (unless λ is constant). Then we have shown that the isotropy function λ is the squared norm of the mean curvature vector field of the surface and also satisfies a key equation (Eq. (4.1)) involving the extrinsic geometry of the surface. As a consequence, we have seen that λ ≡ 0 is equivalent to the surface being extrinsically flat. On the other hand, we have developed a method to produce examples of isotropic MOTS in the classical models of spacetime: (i) for any differentiable function f (x, y) on R 2 we have given an example of 0-isotropic MOTS in the Minkowski spacetime R 4 1 ; (ii) for any differentiable f (x, y, z) on the two-sphere S 2 we have exhibited a 0-isotropic MOTS in the De Sitter spacetime S 4 1 ; (iii) for any differentiable function f (x, y, z) on the hyperbolic plane H 2 we have shown a 0-isotropic MOTS in the anti De Sitter space H 4 1 . Furthermore, we have seen that it is possible to take an appropriate function f such that the resulting embedded surface has a non-parallel null mean curvature vector field, and hence the surface is marginally trapped. It is worth pointing out that we have also provided ridigity results from these examples, that is, each of these examples serve as a model of 0-isotropic marginally trapped surfaces in the Minkowski, De Sitter and anti De Sitter spacetime, respectively. In fact, we have shown that any isotropic marginally trapped surface in R 4 1 , S 4 1 or H 4 1 is congruent to the surface given in example (i),(ii) or (iii), respectively.

  4) give h(e 2 , e 2 ) = ±δ e 3 . If h(e 2 , e 2 ) = δ e 3 , then from Eq. (3.3) we have g(h(e 1 , e 1 ), h(e 2 , e 2 )) + 2g(h(e 1 , e 2 ), h(e 1 , e 2 )) = λ(p).(4.5) Thus equations (4.3) and (4.4) yield g(h(e 1 , e 2 ), h(e 1 , e 2 )) = 0, which is a contradiction because µ > 0. The case h(e 2 , e 2 ) = -δ e 3 also gives a contradiction with µ 2 = -λ(p). Case (ii). Assume λ(p) < 0. As in Case (i), if we write h(e 1 , e 1 ) = δ e 4 , h(e 1 , e 2 ) = µ e 3 , with δ = [-λ(p)] 1/2 and µ = [g(h(e 1 , e 2 ), h(e 1 , e 2 ))] 1/2 we get a contradiction. Finally, using equations (2.5) and (3.4), we have λ(p) = g(H, H) p = D p = 0. (b) From Remark 3.1, at any umbilical point we have λ = g(H, H), and then Eq. (3.4) gives λ = D = g(H, H). Therefore, in a non-umbilical point the result follows from assertion (a).

( c )

 c For any p ∈ S and x, y, z, w ∈ T p S define the multilinear function F on T p S as follows.

Example 5 . 1

 51 Let f : R 2 → R be a smooth function. The isometric immersion ψ : R 2 → R4 1 defined by ψ(x, y) = (f (x, y), x, y, f (x, y))

Author to whom the correspondence should be directed

ACKNOWLEDGMENTS

Research partially supported by OGICYT Grant No. MTM 2007-61284 and a PAY FQM-327 of the Junta de Andalucía(Spain)