[Paper Review] Stability index jump for cmc hypersurfaces of spheres

Óscar Perdomo, Aldir Brasil|arXiv (Cornell University)|Feb 9, 2012

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper proves that the weak stability index of any non-totally umbilical compact constant mean curvature (CMC) hypersurface in the $n+1$-sphere cannot be in the set $\{1, 2, \dots, n\}$, establishing a sharp jump from 0 to at least $n+1$. The result is derived using spectral analysis of the Jacobi operator and constructing a test subspace of functions involving the Gauss map and position vector projections, leveraging geometric identities and the Cauchy-Schwarz inequality on the second fundamental form.

ABSTRACT

It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface M\subset S^{n+1} with cmc cannot take the values 1,2,3... n.

Motivation & Objective

• To determine the possible values of the weak stability index for compact CMC hypersurfaces in $S^{n+1}$.
• To resolve the gap in the stability index spectrum by showing that values from 1 to $n$ are impossible for non-totally umbilical hypersurfaces.
• To extend the known result that only totally umbilical hypersurfaces have weak stability index 0, by proving a sharp lower bound of $n+1$ for all other CMC hypersurfaces.
• To provide a general structural result on the spectrum of the Jacobi operator for CMC hypersurfaces in spheres, independent of additional symmetry or curvature assumptions.

Proposed method

• Define the weak stability index $\mathrm{ind}_T(M)$ as the maximal dimension of a subspace of $C^\infty(M)$ where functions integrate to zero and the quadratic form $\int_M f J(f) < 0$.
• Construct a test subspace $V$ of smooth functions $h_u = f_u + \frac{\sqrt{1+H^2}-1}{H} l_u$ with $\int_M h_u = 0$, where $f_u = \langle \nu, u \rangle$, $l_u = \langle \phi, u \rangle$, and $u \in \mathbb{R}^{n+2}$.
• Use the known Laplacian identities $\Delta l_u = -n l_u + nH f_u$ and $\Delta f_u = -|A|^2 f_u + nH l_u$ to compute the action of the Jacobi operator $J$ on $h_u$.
• Apply Lemma 2.1 to express $\int_M |A|^2 f_u l_u$ in terms of $\int_M f_u l_u$, $\int_M f_u^2$, and $\int_M l_u^2$, enabling substitution in the quadratic form.
• Use the inequality $|A|^2 \geq nH^2$ from the Cauchy-Schwarz inequality applied to the shape operator and identity matrix to bound the quadratic form from above.
• Show that $\int_M h_u J(h_u) < 0$ for all non-zero $u$ by proving it is bounded above by $-n \int_M (f_u + H l_u)^2 < 0$, which is strictly negative due to non-umbilical and non-Clifford assumptions.

Experimental results

Research questions

• RQ1 What values can the weak stability index $\mathrm{ind}_T(M)$ take for compact CMC hypersurfaces $M \subset S^{n+1}$ that are not totally umbilical?
• RQ2 Is there a gap in the spectrum of the weak stability index, excluding values from 1 to $n$, for non-totally umbilical CMC hypersurfaces in $S^{n+1}$?
• RQ3 Can the weak stability index be bounded below by $n+1$ for all non-totally umbilical CMC hypersurfaces in $S^{n+1}$, regardless of symmetry or curvature constraints?
• RQ4 Does the absence of eigenfunctions proportional to $l_u$ and $f_u$ in non-Clifford, non-umbilical cases allow for a uniform lower bound on the weak stability index?

Key findings

• The weak stability index $\mathrm{ind}_T(M)$ of any compact CMC hypersurface $M \subset S^{n+1}$ that is not totally umbilical satisfies $\mathrm{ind}_T(M) \geq n+1$, ruling out all values from 1 to $n$.
• The construction of the test subspace $V$ of dimension at least $n+1$ ensures that the quadratic form $\int_M f J(f)$ is negative definite on $V$, proving the lower bound.
• The strict negativity of $\int_M h_u J(h_u)$ is guaranteed by the non-vanishing of $f_u + H l_u$ under the non-umbilical and non-Clifford assumptions, as established in [2].
• The proof relies on the identity $\int_M |A|^2 f_u l_u = n \int_M f_u l_u - nH \int_M f_u^2 + nH \int_M l_u^2$, derived via integration by parts and the Laplacian identities.
• The inequality $|A|^2 \geq nH^2$ is essential in bounding the quadratic form and ensuring the final expression is negative definite.
• The result generalizes Barbosa and Do Carmo’s theorem that $\mathrm{ind}_T(M) = 0$ if and only if $M$ is totally umbilical, by showing the next possible index is at least $n+1$.