Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary

Abstract

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n " 3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

Sharp eigenvalue bounds and minimal surfaces in the ball

“…By the maximum principle both points lie on ∂ B 3 , and it follows that the density of at x is 1/2 as is the density of at x . By the boundary maximum principle (see [10]) we get a contradiction. Therefore we conclude that ∞ is connected.…”

“…By Allard-type minimal surface regularity theorems (see [15]), a boundary point x is a smooth point if and only if the density at x is equal to 1/2. We use the argument of [10] to first show that ∞ is connected. Indeed if we have two connected components and , then we can find nearest points x ∈¯ and x ∈¯ .…”

A Compactness Theorem of the Space of Free Boundary f-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

“…On the other hand, in a recent beautiful work [12], Fraser and Li proved a compactness theorem for the space of compact properly embedded minimal surfaces with free boundary in compact three-dimensional manifold with nonnegative Ricci curvature and strictly convex boundary, which is a free boundary version of the classical compactness theorem by Choi and Schoen [6]. In this paper, we consider the following natural problem: The compactness property for the space of compact properly embedded f -minimal surfaces with free boundary in a compact three-dimensional smooth metric measure space with nonempty boundary, i.e., a free boundary version of the result in Li-Wei [5].…”

“…The proof is by using a similar argument as in Meeks-Simon-Yau [14,Sect. 8], with an argument in the proof of Theorem 2.11 in [12]. In Sect.…”

“…Then we prove our main theorem using a standard argument as in [6,12,16] with some modification. As in [5], one of the key ingredients in the proof of Theorem 1.3 is the observation that an f -minimal hypersurface in (M, g) is a minimal hypersurface in (M,g) with the conformal changed metricg = e − 2 n−1 f g. However, Theorem 1.3 does not directly follow from Fraser-Li's [12] compactness theorem for free boundary minimal surfaces in three-dimensional smooth metric measure space with nonnegative Ricci curvature and strictly convex boundary. In fact, the Ricci curvature of the conformal changed metricg may not have a sign (see [5,Sect.…”