The r-stability of Hypersurfaces with Zero Gauss-Kronecker Curvature

Marcos P. A. Cavalcante ^{∗}

Abstract In this paper we give sufficient conditions for a bounded domain in a r-minimal hypersuface of the Euclidean space to be r-stable. The Gauss-Kronecker curvature of this hypersurfaces may be zero on a set of capacity zero.

1 Introduction

Let M be an oriented hypersurface of the (n + 1)-dimensional Euclidean space and let g : M → S^{n} denote its Gauss map. The shape operator of M is the self-adjoint map given by B := −dg, that is, for each p ∈ M,

B_{p} : T_{p}M → T_{p}M, B_{p}(X) = −dg_{p}(X).

The eigenvalues of B_{p} are called the principal curvatures of M at p. We denote them by k_{1}(p), …, k_{n}(p) and we define the r-mean curvature of M as the normalized r-elementary symmetric function of the principal curvatures of M, namely,

H_{0} = 1, H_{r} = \binom{n}{r}^{−1}S_{r}, r = 1, …, n,

where S_{r} = ∑_{i_{1}<...<i_{r}} k_{i_{1}}...k_{i_{r}}. Notice that H_{1}, H_{2}, H_{n} are the mean, scalar

and Gauss-Kronecker curvatures of M, respectively.

We say that M is r-minimal when H_{r+1} = 0. It is well known that the r-minimal hypersurfaces of the Euclidean space are critical points of the r-area functional A_{r} = ∫_{M} S_{r}dM for compactly supported variations of M.

12000 Mathematics Subject Classification. Primary 53C42, 53A07; Secondary 35P15.

2Key words and phrases: r-minimal immersions, r-stability, capacity.

∗The author was fully supported by CNPq-Brazil

In order to state our results we need more notations. Let P_{r} be the Newton transformations of B, which can be defined inductively by

P_{0} = I, P_{r} = S_{r}I − BP_{r−1}, r = 1, 2, ...n.

Let C_{0}^{∞}(M) denote the set of smooth functions with compact support on M. Using the Newton transformations we define the linear operator

L_{r}(f) = div(P_{r}∇f), for f ∈ C_{0}^{∞}(M).

We denote by T_{r} the Jacobi operator,

T_{r}f = L_{r}f − (r + 2)S_{r+2}f,

and by I_{r}(f_{1}, f_{2}) = − ∫_{M} f_{1}T_{r}(f_{2})dM the associated bilinear symmetric form.

Let D be a regular domain on M, that is, D is bounded and has piecewise smooth boundary. Following [1], we say that D is r-stable if I_{r}(f, f) ≥ 0 for all f ∈ C_{0}^{∞}(D) or if I_{r}(f, f) ≤ 0 for all f ∈ C_{0}^{∞}(D). Otherwise we say that D is r-unstable.

In the study of r-stability we need to suppose that L_{r} is elliptic. This is equivalent to P_{r} being positive definite or negative definite everywhere. On the other hand, by a Theorem of Hounie-Leite in [7], it is known that, when H_{r+1} = 0, then L_{r} is elliptic if and only if rank(B) > r. In the following, without loss of generality, we will fix P_{r} > 0. Also, the eigenvalues of the operator \sqrt{P_{r}}B appear naturally and we will denote them by θ_{1}(r), θ_{2}(r), ..., θ_{n}(r).

In [1] Alencar, do Carmo and Elbert gave sufficient conditions for a regular domain on a r-minimal hypersurface of the Euclidean space to be r-stable. Their general result assumes that the quotient \frac{|H_{n}|}{∥\sqrt{P_{r}}B∥^{2}} is constant. In this case the hypersurface is said to be r-special.

Theorem A (Theorem 1.3 of [1] ). Let x : M^{n} → ℝ^{n+1} be an oriented r-special hypersurface with H_{r+1} = 0 and H_{n} ≠ 0 everywhere. Let D ⊂ M be a regular domain such that the area of g(D) ⊂ S^{n} is smaller than the area of a spherical cap whose first eigenvalue for the spherical Laplacian is

τ = max_{i,D} (\frac{∑_{j} θ_{j}^{2}(r)}{θ_{i}^{2}(r)}).

Then D is r-stable.

This Theorem is a generalization of a classical result to minimal surfaces of ℝ^{3} due to Barbosa and do Carmo in [3]. We point out that the hypothesis of M to be r-special occurs naturally when r = n−2. In fact, a computation shows that \frac{|S_{n}|}{∥\sqrt{P_{n−2}}B∥^{2}} = \frac{1}{n}.

In this paper we are interested to improve the condition on the Gauss-Kronecker curvature considering points on M for which H_{n} = 0. A simple example, like a cylinder over a plane curve, shows that H_{n} cannot be identically zero. In [1], remark 4.2, was conjectured that Theorem A holds if the set of zeros of H_{n} is contained in a submanifold of codimension d ≥ 2. Here we answer this conjecture affirmatively as a consequence of a more general result. We will consider hypersurfaces with H_{n} = 0 on a subset of capacity zero (see below the definition of capacity). It is known (see [8] §2, p. 35) that submanifolds of codimension d ≥ 2 have capacity zero. In section 2 we will give definitions and develop some facts about capacity. To state our result we will denote by A the set A = {p ∈ M : H_{n}(p) = 0} and by λ_{1}(D) the first eigenvalue for the Laplacian on D. We also point out that, since symmetrization of domains in the sphere does not increase eigenvalues, the hypothesis on the first eigenvalue of the spherical image of D in Theorem A implies that λ_{1}(g(D)) ≥ τ. For simplicity we will use this condition on our result below.

Theorem 1.1. Let M be an oriented r-minimal hypersurface of ℝ^{n+1}, which is r-special on M \ A. Let D ⊂ M be a regular domain such that λ_{1}(g(D)) ≥ τ. Then, if Cap(D ∩ A) = 0, D is r-stable.

The idea of the proof is to use a relation between the eigenvalues of domains from which we remove a subset and the capacity of the removed subset. Actually, we need just a comparison between the first eigenvalues of P_{r} on D and on D \ A. This relation is well known for the Laplacian and we can find in [8], for domains of the Euclidean space, and in [4], for domains of a closed Riemannian manifold. Here we obtain such results for an elliptic operator L in divergence form on a bounded domain D of a n-dimensional Riemannian manifold M. Given A ⊂ D, let λ_{k}(D) and λ_{k}(D\A) denote the k-th eigenvalue of the Dirichlet problem of L on D and on D\A, respectively. We have the following result that will be proved in the next section

Theorem 1.2. For an elliptic operator L in divergence form there exist positive constants ε_{k} and C_{k}, such that if Cap A ≤ ε_{k}, then

λ_{k}(D) ≤ λ_{k}(D \ A) ≤ λ_{k}(D) + C_{k} Cap A^{\frac{1}{2}} .

In particular, λ_{k}(D) = λ_{k}(D \ A) if Cap A = 0.

2 The Spectrum of Domains and the Capacity

In this section we introduce the notion of capacity and recall some spectral properties of domains in Riemannian manifolds. In this section we also prove Theorem 1.2.

Let (M^{n}, ⟨ , ⟩) be a smooth Riemannian manifold and D ⊂ M a bounded domain. As usual, we define H_{0}^{1}(D) as the closure of C_{0}^{∞}(D) with respect to the norm H^{1}:

|u|_{H^{1}}^{2} := ∫_{D} u^{2}dM + ∫_{D} |∇u|^{2}dM, for u ∈ C_{0}^{∞}(D)

where ∇ is the gradient, |.| is the norm of a vector, and dM is the volume element with respect to the metric ⟨ , ⟩. Now, given A ⊂ D, we set

ℌ(D; A) = {u ∈ H_{0}^{1}(D) : ∃ U ⊂ D open, A ⊂ U and u = 1 in U}

and H(D; A) as the closure of ℌ(D; A) with respect to H^{1}. With this notation we define the Capacity of A as

Cap A = inf { ∫_{D} |∇u|^{2}dM : u ∈ H(D; A)}.

Below we have some well known consequences of this definition (see, e.g. [5], §4.7).

Proposition 2.1. For any A ⊂ D we have,

i) Cap A = inf{Cap U : U open, A ⊂ U}; ii) If A_{1} ⊂ ⋯ ⊂ A_{k} ⊂ … are compact subsets of D, then

lim_{k→∞} Cap A_{k} = Cap ( ∩_{k} A_{k}).

We will use the notation V^{ε}(A) to mean the tubular neighborhood of A of radius ε, that is, V^{ε}(A) = {x ∈ M : dist(x, A) < ε}, where dist( , ) stands for the distance function on M. Using this notation we have the following consequence of the above proposition:

Corollary 2.2. If A is compact, then

lim_{ε→∞} Cap V^{ε}(A) = Cap A.

In the case that A is a submanifold of codimension d ≥ 2 we can say more:

Proposition 2.3. If A is an embedded submanifold of codimension d ≥ 2 then there exists a family of open sets V_{j} ⊃ A such that ∩_{j}V_{j} = A and

Cap V_{j} → 0 as j → ∞.

In particular, Cap A = 0.

This proposition follows from minor modifications of Theorem 3, page 154 of [5].

Now recall that, H_{0}^{1}(D) is a Hilbert space with respect to

⟨u_{1}, u_{2}⟩_{∗} = ∫_{D}⟨∇u_{1}, ∇u_{2}⟩dM.

In fact ⟨ , ⟩_{∗} and ⟨ , ⟩_{H^{1}} yield equivalent norms, as we can see by using the Poincaré inequality

|u|_{L^{2}(D)} ≤ C|∇u|_{L^{2}(D)}.

Hence H(D; A) is a closed (affine) subspace of H_{0}^{1}(D) with respect to ⟨ , ⟩_{∗}. Let u_{A} be the orthogonal projection of 0 on H(D; A). By definition we get |∇u_{A}|_{L^{2}(D)}^{2} = Cap A.

Now we start the proof of Theorem 1.2 following some ideas contained in [4]. Here we will consider nonempty boundary domains and general elliptic operators in the divergence form.

For each x ∈ D, let P_{x} : T_{x}M → T_{x}M be a symmetric, positive (or negative) definite operator. We define, for eachf ∈ C_{0}^{∞}(D),

Lf = div(P ∇f) + qf,

where q : D → ℝ is a bounded function.

We consider the unique extension of L to H_{0}^{1}(D). Then, L is an elliptic operator and let us denote by {λ_{0} < λ_{1} ≤ λ_{2} ≤ … } the spectrum of the Dirichlet problem of L on D, repeated according to its multiplicity.

We recall Courant’s min-max principle for the eigenvalues of L on domains of M:

λ_{k}(D \ A) = min_{E∈E_{k}} max_{f∈E\\{0\}} \frac{∫_{D\A}(⟨P ∇f, ∇f⟩ + qf^{2})dM}{∫_{D\A} f^{2}dM} ,

where E_{k} is the set of k-dimensional subspaces of H_{0}^{1}(D \ A). The quotient above is called the Rayleigh quotient for f.

In order to simplify some calculations below we set

Q(f; D) := ∫_{D}(⟨P ∇f, ∇f⟩ + qf^{2})dM.

So, the Rayleigh quotient for f in D is given by Q(f; D)/|f|_{L^{2}(D)}^{2}.

By using the analogous characterization for λ_{k}(D) we easily obtain the first inequality of Theorem 1.2. In order to obtain the second inequality we choose f_{1}, …, f_{k}, an orthonormal basis of eigenfunctions associated to λ_{1}(D), …, λ_{k}(D), and set F_{k} the space generated by f_{1}, …, f_{k}. Then F_{k} ⊂ H_{0}^{1}(D) and

λ_{k}(D) = max_{f∈F_{k}\\{0\}} \frac{Q(f; D)}{|f|_{L^{2}(D)}^{2}}.

We now define E_{k} = {g = f(1 − u_{A}) : f ∈ F_{k}}. It is clear that E_{k} ⊂ H_{0}^{1}(D \ A) is a finite dimensional subspace. We will see that, when A has small capacity, then E_{k} has dimension equal to k. In fact, the functions g_{j} = f_{j}(1 − u_{A}), j = 1, …, k form a basis for E_{k}. We have that

⟨g_{i}, g_{j}⟩_{L^{2}(D)} = δ_{ij} − 2 ∫_{D} f_{i}f_{j}u_{A}dM + ∫_{D} f_{i}f_{j}u_{A}^{2}dM.

Thus, using the Cauchy-Schwarz and Poincaré inequalities we have

|⟨g_{i}, g_{j}⟩_{L^{2}(D)} − δ_{ij}| ≤ 2| ∫_{D} f_{i}f_{j}u_{A}dM| + | ∫_{D} f_{i}f_{j}u_{A}^{2}dM|

≤ 2( ∫_{D}(f_{i}f_{j})^{2}dM)^{\frac{1}{2}} ( ∫_{D} u_{A}^{2}dM)^{\frac{1}{2}} + max_{D}(f_{i}f_{j}) ∫_{D} u_{A}^{2}dM

≤ C_{k}( ∫_{D} u_{A}^{2}dM)^{\frac{1}{2}} + C′_{k} ∫_{D} u_{A}^{2}dM

≤ B_{k}(Cap A^{\frac{1}{2}} + Cap A),

where B_{k} is a positive constant depending only on vol D and max_{i=1,...,k} |f_{i}|_{L^{∞}(D)}.

On the other hand, we may choose ε_{k} > 0 sufficiently small such that, if Cap A < ε_{k} then

B_{k}(Cap A^{\frac{1}{2}} + Cap A) < min\{⟨g_{i}, g_{j}⟩_{L^{2}(D)} − δ_{ij} : i, j = 1, …, k\}.

For such ε_{k} we have that g_{1}, …, g_{k} form an orthonormal basis and consequently E_{k} has dimension equal to k as we claimed. Now we look for estimates of the numerator of the Rayleigh quotient for g = f(1 − u_{A}) ∈ E_{k}, where f ∈ F_{k} and |f|_{L^{2}(D)} = 1. We first observe that for any closed subset A ⊂ D.

|g|_{L^{2}(D)}^{2} = 1 − 2 ∫_{D} f^{2}u_{A}dM + ∫_{D} f^{2}u_{A}^{2}dM ≥ 1 − B′_{k} Cap A^{\frac{1}{2}} . (1)

Now we have,

Q(g; D) = ∫_{D} ([⟨P ∇f, ∇f⟩ + qf^{2}] − 2⟨P ∇f, ∇f⟩u_{A} + ⟨P ∇f, ∇f⟩u_{A}^{2})dM +

+ ∫_{D} (⟨P ∇u_{A}, ∇u_{A}⟩f^{2} − 2⟨P ∇f, ∇u_{A}⟩f(1 − u_{A}) − 2f^{2}u_{A} + f^{2}u_{A}^{2})dM

≤ λ_{k}(D) + B′′_{k}(Cap A + Cap A^{1/2}),

where B′′_{k} depends only on volD, max_{i=1,...,k} |f_{i}|_{L^{∞}(D)}, max_{i=1,...,k} |∇f_{i}|_{L^{∞}(D)} and max_{D} ||P ||.

Therefore, choosing ε_{k} > 0 such that |g|_{L^{2}(D)}^{2} > 0 in (1), we have

\frac{Q(g; D)}{|g|_{L^{2}(D)}^{2}} ≤ λ_{k}(D) + \frac{λ_{k}(D)B′_{k} Cap A^{1/2} + B′′_{k}(Cap A + Cap A^{1/2})}{1 − B′_{k} Cap A^{1/2}}.

We observe that, if Cap A < ε_{k}, then we can estimate the second term above

\frac{λ_{k}(D)B′_{k} Cap A^{1/2} + B′′_{k}(Cap A + Cap A^{1/2})}{1 − B′_{k} Cap A^{1/2}} ≤ \frac{λ_{k}(D)B′_{k} Cap A^{1/2}}{1 − B′_{k}ε_{k}^{1/2}} +

\frac{(B′′_{k} Cap A^{1/2} + B′′_{k}) Cap A^{1/2}}{1 − B′_{k}ε_{k}^{1/2}} ≤ \frac{(λ_{k}(D)B′_{k} + B′′_{k}ε_{k}^{1/2} + B′′_{k}) Cap A^{1/2}}{1 − B′_{k}ε_{k}^{1/2}}

= C_{k} Cap A^{1/2}.

So,

\frac{Q(g; D)}{|g|_{L^{2}(D)}^{2}} ≤ λ_{k}(D) + C_{k} Cap A^{1/2}.

This implies that

λ_{k}(D \ A) := max_{g∈E_{k}\\{0\}} \frac{Q(g; D)}{|g|_{L^{2}(D)}^{2}} ≤ λ_{k}(D) + C_{k} Cap A^{1/2},

and we conclude the proof of Theorem 1.2.

Corollary 2.4. Given a closed subset A ⊂ D with Cap A = 0 let {V_{j}} be the family of open sets given in Proposition 2.3 and let λ_{k}(D \ V_{j}) be the k-th eigenvalue of the Dirichlet problem of the operator L in D \ V_{j}. Then

λ_{k}(D \ V_{j}) → λ_{k}(D) as j → ∞.

In particular, if λ_{k}(D \ V_{j}) ≥ 0 for all j = 1, 2, …, then λ_{k}(D) ≥ 0.

3 Proof of Theorem 1.1

We start by presenting an equivalent condition for r-stability. A simple computation shows that

trace(B^{2}P_{r}) = ||\sqrt{P_{r}}B||^{2} := ∑_{i=1}^{n} θ_{i}^{2}(r).

On the other hand, Lemma 2.1 of [2] says that

trace(B^{2}P_{r}) = S_{1}S_{r+1} − (r + 2)S_{r+2}.

Thus, for r-minimal immersions, the Jacobi operator can be written as

T_{r} = L_{r} + trace(B^{2}P_{r}) = L_{r} + ||\sqrt{P_{r}}B||^{2}.

Using integration by parts, we have that

I_{r}(f, f) = ∫_{D}(⟨P_{r}∇f, ∇f⟩ − ||\sqrt{P_{r}}B||^{2}f^{2})dM
= ∫_{D}(|\sqrt{P_{r}}∇f|^{2} − ||\sqrt{P_{r}}B||^{2}f^{2})dM.

So, in the case P_{r} > 0, to check that a regular domain D ⊂ M is rstable we just need to show that the last term above is always nonpositive or nonnegative for all f ∈ C_{0}^{∞}(D). Similarly in the case P_{r} < 0.

Now we fix ε > 0 and denote D^{ε} = D \ V^{ε}(A). Then, the Gauss map g is a local diffeomorphism on D^{ε}. Let φ : g(D^{ε}) → ℝ be the positive first eigenfunction of the spherical Laplacian Δ̃ on g(D^{ε}), that is,

\{ \begin{matrix}Δ̃φ + λ_{1}^{ε}φ = 0 & in g(D^{ε}), \\ φ > 0 & in g(D^{ε}), \\ φ = 0 & on ∂g(D^{ε}), \end{matrix}

where λ_{1}^{ε} = λ_{1}(D^{ε}) is the first eigenvalue of Δ̃ on g(D^{ε}). Recall that, since D^{ε} ⊂ D, we have λ_{1}^{ε} ≥ λ_{1}.

In the following we will consider the pull back metric s̃ by g on D^{ε} and denote by ˜∇ the gradient, by [ ] the norm of a vector, and by dS the volume element in this metric. By Lemma 2.9 in [1] one have ˜∇f = B^{−2}∇f, for smooth functions f on M and a simple computation gives [X] = |BX|, for any tangent vector X. Also we point out that, in the metric s̃, the Gauss map g : D^{ε} → S^{n} is a local isometry. Now let ψ = φ∘g defined in D^{ε}. Then ψ is positive and satisfies Δ̃ψ + λ_{1}^{ε}ψ = 0 in D^{ε}. Thus, by Corollary 1 in [6], we have that the first eigenfunction of the operator Δ̃ + λ_{1}^{ε} is nonnegative:

0 ≤ inf { ∫_{D^{ε}}([ ˜∇f]^{2} − λ_{1}^{ε}f^{2})dS : f ∈ C_{0}^{∞}(D^{ε}), ∫_{D^{ε}} f^{2}dS = 1}.  (2)

Now, since det g = S_{n} and the immersion is r-special on D^{ε}, we have dS = |S_{n}|dM = c||\sqrt{P_{r}}B||^{2}dM, where c is positive constant. Also, by hypothesis, λ_{1}^{ε} ≥ λ_{1} ≥ τ. Thus

∫_{D^{ε}}([ ˜∇f]^{2} − λ_{1}^{ε}f^{2})dS ≤ c ∫_{D^{ε}}([ ˜∇f]^{2} − τf^{2})||\sqrt{P_{r}}B||^{2}dM.  (3)

Observe now that ρ_{i} = \frac{θ_{i}^{2}(r)}{∑_{j} θ_{j}^{2}(r)}, i = 1, …, n, are the eigenvalues of the

operator \frac{\sqrt{P_{r}}B}{||\sqrt{P_{r}}B||}. Therefore, by definition

τ := max_{i, D} \frac{1}{ρ_{i}^{2}} = max_{D} |(\frac{\sqrt{P_{r}}B}{||\sqrt{P_{r}}B||})^{−1}|^{2} .

So,

[ ˜∇f]^{2} ≤ |(\frac{\sqrt{P_{r}}B}{||\sqrt{P_{r}}B||})^{−1}|^{2} [(\frac{\sqrt{P_{r}}B ˜∇f}{||\sqrt{P_{r}}B||} )]^{2} ≤ τ [(\frac{\sqrt{P_{r}}B ˜∇f}{||\sqrt{P_{r}}B||} )]^{2} .

Using this on (3), we have

c ∫_{D^{ε}}([ ˜∇f]^{2} − τf^{2})||\sqrt{P_{r}}B||^{2}dM ≤ cτ ∫_{D^{ε}} (\frac{[\sqrt{P_{r}}B ˜∇f]^{2}}{||\sqrt{P_{r}}B||^{2}} − f^{2})||\sqrt{P_{r}}B||^{2}dM
= cτ ∫_{D^{ε}} ([\sqrt{P_{r}}B ˜∇f]^{2} − ||\sqrt{P_{r}}B||^{2}f^{2})dM.

Now, we recall that ˜∇f = B^{−2}∇f and [X] = |BX| to find

[\sqrt{P_{r}}B ˜∇f]^{2} = |\sqrt{P_{r}}∇f|^{2}.

We conclude that, for any f ∈ C_{0}^{∞}(D^{ε}) with ∫_{D^{ε}} f^{2}dS = 1,

0 ≤ cτ ∫_{D^{ε}} (|\sqrt{P_{r}}∇f|^{2} − ||\sqrt{P_{r}}B||^{2}f^{2})dM.

This implies that the first eigenvalue of the Jacobi operator T_{r} is nonnegative on D^{ε} and thus, by Corollary 2.4, we have that first eigenvalue of the Jacobi operator T_{r} is nonnegative in all D. This shows that D is r-stable.

Remark: In the case P_{r} < 0 we may set Q_{r} = −P_{r} in the above computations to conclude that

0 ≤ cτ ∫_{D^{ε}} (|\sqrt{Q_{r}}∇f|^{2} − ||\sqrt{Q_{r}}B||^{2}f^{2})dM.

On the other hand the Jacobi operator is given by

T_{r}f = div(P_{r}∇f) + trace(B^{2}P_{r})
= −div(Q_{r}∇f) + trace(B^{2}Q_{r}).

We conclude that the first eigenvalue of T_{r} is nonpositive on D and therefore we obtain r-stability according to our definition.

Acknowledgments: The author would like to express his gratitude to Prof. Manfredo P. do Carmo for suggestions and much encouragement and also wishes to thank Alexander Arbieto and Carlos Matheus for valuable discussions on the material of this paper. Finally, the author is thankful to IMPA-Rio de Janeiro and his staff.

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Marcos Petrúcio de Almeida Cavalcante ( petrucio@impa.br ) Instituto de Matemática - Universidade Federal de Alagoas Maceió - AL -Brazil.