Rigidity of self-shrinkers and translating solitons of mean curvature flows. (English) Zbl 1336.53074
Summary: In this paper, we prove that any complete \(m\)-dimensional spacelike self-shrinkers in pseudo-Euclidean spaces \(\mathbb R_n^{m+n}\) must be affine planes, and there exists no complete \(m\)-dimensional spacelike translating soliton in \(\mathbb R_n^{m+n}\). These results are proved by using a new Omori-Yau maximum principle. We also derive a rigidity theorem of self-shrinking hypersurfaces in Euclidean space whose Gauss image lies in a regular ball.
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 53C40 | Global submanifolds |
| 53C43 | Differential geometric aspects of harmonic maps |
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