Stability and compactness for complete 𝑓-minimal surfaces

Abstract

Let (M, g, e −f dμ) be a complete metric measure space with Bakry-Emery Ricci curvature bounded below by a positive constant. We prove that in M there is no complete two-sided L f -stable immersed f -minimal hypersurface with finite weighted volume. Further, if M is a 3-manifold, we prove a smooth compactness theorem for the space of complete embedded f -minimal surfaces in M with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in R 3 by Colding-Minicozzi.

Essential spectrum of the weighted Laplacian on noncompact manifolds and applications

“…Manifolds with the Bakry-Émery Ricci curvature bounded below have been studied by many authors in recent years, particularly some interesting weighted volume estimates and splitting theorems, which can be found in [30] and [26], for example. Some results about estimates of weighted volume can be also seen in [11] and [12].…”

Dirichlet problem for f-minimal graphs

“…Here, too, ν is a unit normal vector field along Σ, f is a smooth function on N , and∇f denotes its gradient with respect to the Riemannian metric g. Hence the graph of a solution u of (1.1) is an f -minimal hypersurface in M × R. . We refer to [7], [6], [3], [4], [5], [15], and references therein for recent studies on self-shrinkers and f -minimal hypersurfaces. Let us just point out a recent result relevant to our paper.…”

Estimates for the Eigenvalues of the Drifting Laplacian on Some Complete Ricci Solitons

“…During the last two decades, the metric measure space has received a lot of attention in geometric analysis (cf. [1,3,4,7,13,14,27,28,32,35]). In a celebrated work [29], Perelman introduced a functional that involves an integral of the scalar curvature in the sense of the weighted measure, such that the Ricci flow becomes a gradient flow of such a functional.…”