f-Minimal Surface and Manifold with Positive m-Bakry–Émery Ricci Curvature

Abstract

In this paper, we first prove a compactness theorem for the space of closed embedded f-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry–Émery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the f-Laplacian on a compact manifold with positive m-Bakry–Émery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the n-sphere, or the n-dimensional hemisphere. Finally, for a compact manifold with positive m-Bakry–Émery Ricci curvature and f-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if and only if the manifold is isometric to a Euclidean ball. Show more