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

作者：Barbosa Ezequiel; Wei Yong^*

来源：Journal of Geometric Analysis, 2016, 26(3): 1995-2012.

DOI：10.1007/s12220-015-9616-4

• Compactness
• f-Minimal
• Free boundary
• Bakry-Emery Ricci curvature

Let (M-3, g, e(-f), d mu(M)) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry-Emery Ricci curvature and the boundary partial derivative M is strictly f-mean convex. We prove that there exists a properly embedded smooth f-minimal surface Sigma in M with free boundary partial derivative Sigma on partial derivative M. If we further assume that the boundary partial derivative M is strictly convex, then we prove that M-3 is diffeomorphic to the 3-ball B-3, and a compactness theorem for the space of properly embedded f-minimal surfaces with free boundary in such (M-3, g, e(-f) d mu(M)), when the topology of these f-minimal surfaces is fixed.