$f$-minimal surface and manifold with positive $m$-Bakry-\'{E}mery Ricci curvature

TL;DR

This paper establishes compactness results for $f$-minimal surfaces, eigenvalue bounds, and geometric characterizations of manifolds with positive $m$-Bakry-Émery Ricci curvature, extending classical geometric analysis results.

Contribution

It proves a compactness theorem for $f$-minimal surfaces, derives a Lichnerowicz-type eigenvalue lower bound, and characterizes manifolds with positive $m$-Bakry-Émery Ricci curvature.

Findings

Compactness of $f$-minimal surfaces with fixed topology

Lower bound for the first eigenvalue of the $f$-Laplacian

Characterization of manifolds as spheres, hemispheres, or Euclidean balls

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-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the $n$-shpere, or the $n$-dimensional hemisphere. Finally, for compact manifold with positive $m$-Bakry-\'{E}mery 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 only if the manifold is isometric to an Euclidean ball.