Characterization of hypersurfaces via the second eigenvalue of the Jacobi operator
In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of the Jacobi operator among all closed immersed orientable surfaces of with genus bigger than zero. After, we characterize the slices of the warped product , under a suitable hypothesis on the warping function , as the only hypersurfaces which saturate a certain integral inequality involving the second eigenvalue of the Jacobi operator. As a consequence, we obtain that if is a closed immersed hypersurface of , then the second eigenvalue of the Jacobi operator of satisfies and the slices are the only hypersurfaces which satisfy .
Let be a closed immersed orientable surface of the Euclidean -dimensional space . It is well-known that the linearization of the mean curvature of in , in the normal direction, is given by the Jacobi operator of ,
where is the Laplace operator of with respect to the induced metric from and is the second fundamental form of in . Motivated by some physical issues concerning the Allen-Cahn equation, Alikakos and Fusco  conjectured that the round -spheres are the most stable closed surfaces for the operator , that is, they proposed that has at least two negative eigenvalues unless is a round -sphere, in which case has exactly one negative eigenvalue and its second eigenvalue is equal to zero. This conjecture was solved by Harrell and Loss  in the general case of closed immersed orientable hypersurfaces of for .
Using a very powerful tool based on the works of Hersch  and Li and Yau , El Soufi and Ilias  generalized the result of Harrell and Loss to the case of submanifolds of arbitrary codimension of the simply connected space forms , , and of dimension . Among other things, they proved that if is a closed immersed submanifold of of dimension , where denotes the Euclidean space , the unit sphere or the hyperbolic space , depending on or , respectively, then the operator
has at least two negative eigenvalues unless is a geodesic -sphere, when the second eigenvalue of is equal to zero and, of course, the first eigenvalue of is negative. So they proved that the maximum value for the second eigenvalue of is equal to zero and the geodesic -spheres are the only closed immersed submanifolds of which attain the maximum. It is important to note that they did not make any assumption on the orientability of .
Our first goal in this work is to find the maximum value for the second eigenvalue of on closed immersed orientable surfaces of of genus bigger than zero and characterize the surfaces which attain the maximum. So our first result is the following.
Let be a closed immersed orientable surface of of genus bigger than or equal to . Then the second eigenvalue of satisfies and the equality holds if and only if is congruent to the Clifford torus.
In our second result, we characterize all closed immersed hypersurfaces of which saturate the maximum value for the second eigenvalue of the Jacobi operator, , where is the Ricci tensor of and is a local unit normal on .
Let be a closed immersed hypersurface of . Then the second eigenvalue of satisfies and the equality holds if and only if is a slice of .
More generally, we can extend the result of Harrell and Loss, in an appropriate sense, to a wide class of warped products, including the de Sitter-Schwarzschild and Reissner-Nordström manifolds (see ).
Consider a smooth positive function defined on the interval such that
Then define the warped product as the Riemannian manifold ,
where is the standard metric on induced from . Observe that corresponds to the cases where , when , or .
Consider the slice of and denote by the Jacobi operator of , that is,
where is the Ricci tensor of . Let be a closed immersed hypersurface of and denote by the Jacobi operator of , i.e.
where is a local unit normal on . Also, denote by the projection on the first factor of , that is, for . Our result is the following.
Let be a closed immersed hypersurface of . Then the second eigenvalue of satisfies
and the equality holds if and only if is a slice of .
2. Proof of Theorem 1.1
This proof is mainly based on the work of Souam .
Let be the first eigenfunction of . It is well-known that does not change sign, so we can assume that . Let denote the immersion in consideration. It follows from an argument presented by Hersch  and by Li and Yau  that there exists a conformal diffeomorphism such that
Then we can use the coordinate functions of as test functions for the second eigenvalue of . Thus
for each . It follows from the above inequality that
since . Here denotes the area of . Now, denoting by and the principal curvatures of in , the Gauss equation says that
where is the Gaussian curvature of . Then
where is the Euler characteristic of . Above we have used the Gauss-Bonnet theorem. Let denote and consider the second fundamental form and the mean curvature of with respect to . It is well-known that
is constant on the conformal class of . Then, using the Gauss equation and the Gauss-Bonnet theorem, we obtain
Therefore, since we are assuming that , we obtain .
If , then all above inequalities must be equalities. In particular, is minimal and . The equality in (2) says that
On the other hand, the equality in (5) says that . Thus
Let be a meromorphic map of degree 2, which exists because is topologically a -torus. Using again the argument of Hersch and Li and Yau, we can assume that . Then, using the coordinate functions of as test functions for the eigenvalue of , we obtain
for each . Now, if satisfies , then
for all . This implies that for all satisfying . Then, for each , there exists a constant such that
Defining , we obtain that
On the other hand, observing that is harmonic, since it is meromorphic, we have
From these last two equations we obtain
where is the canonical inner product of . Since , there exists such that , which implies that because . This gives that . Thus
then is a nonconstant meromorphic map from a -torus to such that the operator has index one. But such a map does not exist (see ). Therefore must be an embedding. Thus, from the solution of the Lawson’s conjecture by Brendle , we obtain that is congruent to the Clifford torus.
Conversely, it is well-known that the Clifford torus satisfies (see ).
Let be a closed immersed orientable minimal surface of which is not totally geodesic. It follows from a result due to Almgren  that has genus bigger than zero. Fix a unit normal on and define for each , where is the standard inner product of . It is not difficult to prove that , where is the Jacobi operator of . Then is an eigenvalue of . It follows from an argument due to Urbano  that has dimension equal to 4. So the index of is at least , since the first eigenvalue of is simple. Thus, if has index 5, then . In this case, by Theorem 1.1, is congruent to the Clifford torus. This gives an alternative proof for the Urbano’s theorem .
3. Proof of Theorem 1.3
Lemma 3.1 (El Soufi-Ilias).
Let be a Riemannian manifold of dimension such that admits a conformal immersion , where is endowed with the canonical metric. If is a closed immersed submanifold of of dimension , then
where is the mean curvature vector of in and
for , where is an orthonormal basis of and is the sectional curvature of .
Take and into the El Soufi-Ilias’ lemma, and observe that the Gauss equation implies that
where is the scalar curvature of . Then
Above we have used that . On the other hand, observe that the Ricci tensor and the scalar curvature of are given by
In particular, the hypothesis (1) on implies that if is a unit vector, then at and the equality holds if and only if . Then
A simple calculation gives that
then , in particular is umbilic, and on , which implies that on . This gives that is constant on , say on . Then . Because is closed (and connected), and is simple connected, we obtain that , Q.E.D.
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