Let $(M,\bar{g}, e^{-f}d\mu)$ 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 $\mathbb{R}^3$ by Colding-Minicozzi.

/pdf/stability-and-compactness-for-complete-f-minimal-surfaces-1h84d3b56v.pdf

Stability and compactness for complete f -minimal surfaces

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $$\mathcal {L}$$
 -operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on polynomial volume growth, which is assumed in Cao and Li [5]. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on polynomial volume growth. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form.

Complete self-shrinkers of the mean curvature flow

In this paper, first we study a complete smooth metric measure space (Mn,g,e−fdv) with the (∞)-Bakry–Emery Ricci curvature Ricf ≥ a 2 g for some positive constant a. It is known that the spectrum of the drifted Laplacian Δf for M is discrete and the first nonzero eigenvalue of Δf has lower bound a 2. We prove that if the lower bound a 2 is achieved with multiplicity k ≥ 1, then k ≤ n, M is isometric to Σn−k × ℝk for some complete (n − k)-dimensional manifold Σ and by passing an isometry, (Mn,g,e−fdv) must split off a gradient shrinking Ricci soliton (ℝk,g can, a 4 |t|2), t ∈ ℝk. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian ℒ = Δ −1 2〈x,∇⋅〉 for properly immersed self-shrinkers in the Euclidean space ℝn+p, p ≥ 1 and show the discreteness of the spectrum of ℒ and a logarithmic Sobolev inequality.

Eigenvalues of the drifted Laplacian on complete metric measure spaces

We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry–Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li–Tam theory, we investigate the topology at infinity of $f$-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted $L^1$-Sobolev inequality for hypersurfaces in Cartan–Hadamard weighted manifolds, satisfying suitable restrictions on the weight function.

Stability properties and topology at infinity of f-minimal hypersurfaces

In this paper, we introduce a special class of hypersurfaces which are called $$\lambda $$
 -hypersurfaces related to a weighted volume preserving mean curvature flow in the Euclidean space. We prove that $$\lambda $$
 -hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $$\lambda $$
 -hypersurfaces with polynomial area growth and $$H-\lambda \ge 0$$
 . The classification result generalizes the results of Huisken (J Differ Geom 31:285–299, 1990) and Colding and Minicozzi (Ann Math 175:755–833, 2012).

Complete $$\lambda $$ λ -hypersurfaces of weighted volume-preserving mean curvature flow

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.

https://www.ams.org/proc/2013-141-02/S0002-9939-2012-11922-7/S0002-9939-2012-11922-7.pdf

Volume estimate about shrinkers

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Volume estimate about shrinkers

Volume estimate about shrinkers