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.
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The authors would like to thank the referee for helpful suggestions.
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Communicated by Ben Andrews.
The research of the authors was supported by NSFC No. 11271214.
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Li, H., Wei, Y. f-Minimal Surface and Manifold with Positive m-Bakry–Émery Ricci Curvature. J Geom Anal 25, 421–435 (2015). https://doi.org/10.1007/s12220-013-9434-5
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DOI: https://doi.org/10.1007/s12220-013-9434-5