Index Estimates for Surfaces with Constant Mean Curvature in $3$-dimensional Manifolds

TL;DR

This paper establishes lower bounds on the index of constant mean curvature surfaces in 3D manifolds, linking geometric properties like genus to stability measures, with implications for understanding surface stability in geometric analysis.

Contribution

It provides new index estimates for CMC surfaces in 3D manifolds, especially when mean curvature is large, connecting index bounds to genus.

Findings

Index of CMC surfaces bounded below by a linear function of genus.

Proved index estimates for surfaces with large mean curvature.

Applicable to surfaces with free boundary conditions.

Abstract

We prove index estimates for closed and free boundary CMC surfaces in certain $3$-dimensional submanifolds of some Euclidean space. When the mean curvature is large enough we are able to prove that the index of a CMC surface in an arbitrary $3$-manifold is bounded below by a linear function of its genus.

Equations

$${\int }_{\Sigma }\sum _{i=1}^2\left(|I{I}_M(e_i,\xi ){|}^2+|I{I}_M(e_i,\star \xi ){|}^2\right)-(R_M+H_{\Sigma }^2)|\xi {|}^{2}d{V}_{\Sigma }<2\eta {\int }_{\Sigma }|\xi {|}^{2}d{V}_{\Sigma }.$$

$$\#\{\text{eigenvalues of L~Σ that are strictly smaller than}\eta \}\ge \frac{q}{2d}.$$

$${\int }_{\Sigma }\sum _{i=1}^2\left(|I{I}_M(e_i,\xi ){|}^2+|I{I}_M(e_i,\star \xi ){|}^2\right)-R_M|\xi {|}^{2}d{V}_{\Sigma }<{\int }_{\Sigma }{H}_{\Sigma }^2|\xi {|}^{2}d{V}_{\Sigma }.$$

$$\text{Index}(\Sigma )\ge \frac{g}{d}.$$

$${\int }_{\Sigma }$$

$$-{\int }_{\Sigma }(R_M+H_{\Sigma }^2)|\xi {|}^{2}d{V}_{\Sigma }-2{\int }_{\partial \Sigma }{H}_{\partial M}|\xi {|}^{2}d{V}_{\partial \Sigma }<2\eta {\int }_{\Sigma }|\xi {|}^{2}d{V}_{\Sigma }.$$

$$\#\{\text{eigenvalues of L~Σ that are smaller than}\eta \}\ge \frac{q-d}{2d}.$$

Full text

1. Introduction

A closed hypersurface of constant mean curvature (CMC) may be variationally characterized a critical point of the area functional under variations that preserve enclosed volume. In a similar way, a free boundary constant mean curvature (free boundary CMC) hypersurface is an extremal solution of the same problem where, in addition, the boundary is restricted inside a closed hypersurface. If such a hypersurface minimizes area for small perturbations then it is stable for the corresponding problem. For example, solutions to the isoperimetric problem, that is, the hypersurface with or without boundary that has least area for a fixed enclosed volume, are in particular stable CMC hypersurfaces.

In [barbosa-docarmo1984, barbosa-docarmo-eschenburg1988] Barbosa-do Carmo and Barbosa-do Carmo-Eschenburg classify stable closed CMC hypersurfaces of Euclidean spaces, spheres and hyperbolic spaces. A similar result was obtained by Souam [souam1997] for stable free boundary CMC hypersurfaces in a hemisphere and more recently for closed CMC surfaces in $S^{2}\times$ and $\mathbb{H}^{2}\times$. Other classification results for stable free boundary CMC surfaces were obtained by Ros-Vergasta [ros-vergasta1995] and later improved by Nunes [nunes2017]. It is also natural to study CMC hypersurfaces of higher index. That is, those that have some small pertubations that decrease area with fixed enclosed volume.

In [torralbo-urbano2012] Torralbo-Urbano make use of isometric embeddings of homogenous $3$-manifolds into Euclidean space to study stable closed CMC surfaces and the isoperimetric problem in Berger spheres. The authors also use coordinates of hamornic vector fields to construct test functions for the second variation of area.

In the case of minimal surfaces there has been multiple results establishing a connection between the topology of the surface and its index. For example, do Carmo-Peng [docarmo-peng1982], Fischer-Colbrie-Schoen [fischer-colbrie-schoen1980] and Pogorelov [pogorelov1981] have independently proved that stable two-sided minimal surfaces in 3 are planes.

In [ros2006] Ros proves that the index of a minimal surface in 3 is bounded below by a linear function of its genus, which was later improved by Chodosh-Maximo [chodosh-maximo2016, chodosh-maximo2018:arxiv]. A corresponding result was shown by Savo [savo2010] for minimal hypersurfaces in $n$-spheres. In both situations the authors use harmonic vector fields to construct test functions but their construction is fundamentally distinct. Ros uses the coordinates of the harmonic vector field with respect to the usual basis of 3 which works seamlessly since the ambient space is flat. On the other hand, Savo uses the coordinate with respect to a vector field parametrized by two unit vectors and later takes the average of the second variation for all such test functions. This allows to overcome the extra term in the second variation given by the curvature of the ambient space.

Using a similar method to Savo’s, Sargent [sargent2017] proved a corresponding index estimate for free boundary minimal hypersurfaces in convex bodies of Euclidean space. In an outstanding work, Ambrozio-Carlotto-Sharp [ambrozio-carlotto-sharp2018.1] used both Ros’ and Savo’s approach to relate the index of closed minimal hypersurface and its first betti number in an arbitrary ambient manifold that can be suitably embedded into some Euclidean space. The authors later did the same for free boundary minimal hypersurfaces in $2$-convex domains [ambrozio-carlotto-sharp2018.2].

Our main results in this paper follow the natural generalization of Torralbo-Urbano for higher index CMC surfaces, similar to Ambrozio-Carlotto-Sharp’s approach. The main difference is that we are only allowed to use admissible functions. Fortunately, the coordinates of harmonic vector fields are admissible in the closed case. In the free boundary case we need the extra condition that these vector fields are tangential along the boundary. In either case we have to use both the harmonic vector field and its Hodge dual, so it only makes sense in the case of CMC surfaces. It is not clear whether or not Savo’s test functions preserve enclosed volume, so the generalization to higher dimensions seem to not be straightforward.

A surprising difference between the CMC case and the minimal case, at least in the case of surfaces, is that the extra term involving the non-zero mean curvature allows us to have a wider variety of applications. More specifically, some of the ambient spaces that satisfy the conditions of our theorems for non-zero mean curvature do not work in the case of minimal surfaces, for example, flat ambient spaces.

Our main theorem for closed CMC surfaces is:

Theorem LABEL:main_theorem_closed.

Let $(M,g)$ be a $3$-dimensional Riemannian manifold without boundary isometrically embedded in d and $\Sigma$ a closed two-sided immersed CMC surface in $M$. Assume there exists a real number $\eta$ and a $q$-dimensional vector space $\mathbb{V}^{q}$ of harmonic vector fields on $\Sigma$ such that any non-zero $\xi\in\mathbb{V}^{q}$ satisfies

[TABLE]

Then

[TABLE]

And the corresponding index estimates:

Corollary LABEL:index_estimates.

Let $(M,g)$ be a $3$-dimensional Riemannian manifold without boundary isometrically embedded in d and $\Sigma$ a closed two-sided immersed CMC surface of genus $g$ in $M$. Suppose that every non-zero $\xi\in\mathcal{H}^{1}(\Sigma)$ satisfies

[TABLE]

Then

[TABLE]

As mentioned above, the free boundary case is slightly different since, a priori, only harmonic vector fields that are tangential along the boundary provide admissible test functions. However, it is still possible to find tangential vector fields whose dual vector, despite not being tangential any more, has zero average along the boundary. This can be done as long as the dimension of the space of tangential harmonic vector fields is sufficiently large. As a consequence the index estimates are fairly weaker.

The respective results for free boundary CMC surfaces are:

Theorem LABEL:main_theorem_boundary.

Let $(M,\partial M,g)$ be a $3$-dimensional Riemannian manifold with boundary isometrically embedded in d and $\Sigma$ a compact two-sided immersed free boundary CMC surface in $M$. Assume there exists a real number $\eta$ and a $q$-dimensional vector space $\mathbb{W}^{q}$ of harmonic vector fields on $\Sigma$ that are tangential on the boundary $\partial\Sigma$, such that any non-zero $\xi\in\mathbb{W}^{q}$ satisfies

[TABLE]

Then

[TABLE]