Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds
arXiv:1301.6257
Abstract
We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.
arXiv comment16 pages. Revised version. To appear in The Journal of Geometric Analysis