We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.

Stability of translating solutions to mean curvature flow

In an article of Cao-Shen-Zhu [C-S-Z], they proved that a complete, immersed, stable minimal hypersurface M of R with n ≥ 3 must have only one end. When n = 2, it was proved independently by do Carmo-Peng [dC-P] and FischerColbrie-Schoen [FC-S] that a complete, immersed, oriented stable minimal surface in R must be a plane. Later Gulliver [G] and Fischer-Colbrie [FC] proved that if a complete, immersed, minimal surface in R has finite index, then it must be conformally equivalent to a compact Riemann surface with finitely many punctures. Fischer-Colbrie actually proved this for minimal surfaces in a complete manifold with non-negative scalar curvature. In any event, a corollary is that if a complete, immersed, oriented minimal surface in R has finite index then it must have finitely many ends. The purpose of this paper is to generalize this result for finitely many ends to higher dimensional minimal hypersurfaces in Euclidean space (see Theorem 5). In fact, we will also show that the first L-Betti number of such a manifold must be finite. The strategy of Cao-Shen-Zhu was to utilize a result a Schoen-Yau [S-Y] asserting that a complete, stable minimal hypersurface of R cannot admit a non-constant harmonic function with finite Dirichlet integral. Assuming that M has more than one end, Cao-Shen-Zhu constructed a non-constant harmonic function with finite Dirichlet integral. This approach very much fits into the scheme studied by the first author and Tam in [L-T]. In fact, the authors showed that the number of non-parabolic ends of any complete Riemannian manifold is bounded above by the dimension of the space of bounded harmonic functions with finite Dirichlet integral. The proof of Cao-Shen-Zhu can be modified to show that each end of a complete, immersed, minimal submanifold must be non-parabolic. Due to this connection

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2002/0009/0001/MRL-2002-0009-0001-a007.pdf

Minimal Hypersurfaces with Finite Index

We introduce a {\it hyperbolic Gauss map\/} into the Poincar\'e disk for any surface in ${\Bbb H}^2\times {\Bbb R}$ with regular vertical projection, and prove that if the surface has constant mean curvature $H=1/2$, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere conformal harmonic map from an open simply connected Riemann surface $\Sigma$ into the Poincar\'e disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on $\Sigma$ can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with $H=1/2$ in ${\Bbb H}^2 \times {\Bbb R}$. A similar result applies to minimal surfaces in the Heisenberg group ${\rm Nil_3}$. Finally, we classify all complete minimal vertical graphs in ${\Bbb H}^2\times {\Bbb R}$.

Harmonic maps and constant mean curvature surfaces in H 2 x R

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

We provide a new topological obstruction for complete stable minimal hypersurfaces in R^n. For $n\geq 4$, we prove that any complete orientable stable hypersurfaces in R^n has only one end. This follows from a more general analytic theorem we prove in the paper.

The structure of stable minimal hypersurfaces in R^n

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces.

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer and Y.L. Xin where they assume that the image of the Gauss map is bounded. We also proved a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces.

We provide a new topological obstruction for complete stable minimal hypersurfaces in R^n. For $n\\geq 4$, we prove that any complete orientable stable hypersurfaces in R^n has only one end. This follows from a more general analytic theorem we prove in the paper.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1997/0004/0005/MRL-1997-0004-0005-a002.pdf

The structure of stable minimal hypersurfaces in $I\!\!R^{n+1}$

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Papers

The structure of stable minimal hypersurfaces in R^n

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

The structure of stable minimal hypersurfaces in $I\!\!R^{n+1}$