Self-shrinkers Of Mean Curvature Flow Research Articles

Closed Embedded Self-shrinkers of Mean Curvature Flow

In this article, we show the existence of closed embedded self-shrinkers in {mathbb {R}}^{n+1} that are topologically of type S^1times M, where Msubset S^n is any isoparametric hypersurface in S^n for which the multiplicities of the principle curvatures agree. This yields new examples of closed self-shrinkers, for example self-shrinkers of topological type S^1times S^ktimes S^ksubset {mathbb {R}}^{2k+2} for any k. If the number of distinct principle curvatures of M is one, the resulting self-shrinker is topologically S^1times S^{n-1} and the construction recovers Angenent’s shrinking doughnut (Angenent in Shrinking doughnuts, Birkhäuser, Boston, pp 21–38).

Complete Lagrangian self-shrinkers in $${\mathbf {R}}^4$$

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space \({\mathbb {R}}^4\) with constant squared norm of the second fundamental form.

A Survey of Closed Self-Shrinkers with Symmetry

In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide numerical evidence for the existence of new examples.

The stability of self-shrinkers of mean curvature flow in higher co-dimension

We generalize Colding and Minicozzi’s work (2012) on the stability of hypersurface self-shrinkers to higher co-dimension. The first and second variation formulae of the F F -functional are derived and an equivalent condition to the stability in general co-dimension is found. We also prove that R n \mathbb R^n is the only stable product self-shrinker and show that the closed embedded Lagrangian self-shrinkers constructed by Anciaux are unstable.

Lower volume growth estimates for self-shrinkers of mean curvature flow

We obtain a Calabi-Yau type volume growth estimate for complete noncompact self-shrinkers of the mean curvature flow. More precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.