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Index estimates and compactness for constant mean curvature surfaces and Steklov eigenvalues Hong, Han

Abstract

In this work, we focus on three problems. First, we give a relationship between the number of eigenvalues of the Jacobi operator below a certain threshold and the topology of closed constant mean curvature (CMC) surfaces in three-dimensional Riemannian manifolds. We then obtain that the (weak) Morse index of CMC surfaces in an arbitrary 3-manifold is bounded below by a linear function of the genus when the constant mean curvature is greater than a certain nonnegative value. In particular, this implies that stable CMC surfaces are topological spheres. Corresponding results for CMC surfaces with free boundary in 3-manifolds with boundary are obtained as well. Second, we consider the space of embedded free boundary CMC surfaces with bounded topology, bounded area, and bounded boundary length in a 3-manifold N with boundary. We show that this space is almost compact in the sense that any sequence of surfaces in this space has a convergent subsequence that converges to a free boundary CMC surface, graphically and smoothly except on a finite set of singularities. If in addition Ric_N>0 and the boundary of N is convex, then the convergence is at most 2-sheeted. In particular, it is 1-sheeted if the limiting surface is not a minimal surface. Third, we consider the maximization of Steklov eigenvalues in higher dimensions. We show that for compact manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimension 3 and higher.

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