Volume growth, eigenvalue and compactness for self-shrinkers

By Qi Ding, Y. Xin. 2011

138 citations recorded by the source corpus.

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in $\ir{3}$ obtained by Colding-Minicozzi under weaker conditions.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38d34db458b720e70235207fcdbd071fe2e76893

How researchers see this paper

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In Applied Mathematics

1. Qing Cheng
2. Xu Cheng
3. G. Wei
4. Detang Zhou
5. Yejuan Peng
6. Tito Mejia
7. Niels Moller
8. N. Kapouleas

In Geometry and Topology

1. Qing Cheng
2. Xu Cheng
3. G. Wei
4. Detang Zhou
5. Yejuan Peng
6. Tito Mejia
7. Niels Moller
8. N. Kapouleas

In Mathematical Physics

1. Qing Cheng
2. Yejuan Peng
3. Detang Zhou
4. Xu Cheng
5. G. Wei
6. Niels Moller
7. N. Kapouleas
8. S. Kleene

Citations in the corpus

• Complete self-shrinkers of the mean curvature flow (2012, result): “3 Since M3 is a complete proper self-shrinker, we know that M3 has polynomial volume growth from the result of Ding and Xin [11] or Cheng and Zhou [9].”
• Translating solitons of the mean curvature flow (2015, result): “This is in sharp contrast to self-shrinkers [7].”
• Stability and compactness for complete f -minimal surfaces (2012, background): “In [9], Ding-Xin proved that a properly immersed self-shrinker must have the Euclidean volume growth.”
• Complete $$\lambda $$λ-hypersurfaces of weighted volume-preserving mean curvature flow (2014, background): “) ≤ Crn. For n-dimensional complete and non-compact gradient shrinking Ricci solutions, Cao and Zhou [5] have proved geodesic balls have at most polynomial area growth. Forself-shrinkers, Dingand Xin [11] proved that any complete non-compact properly immersed self-shrinker in the Euclidean space has”
• Mean curvature self-shrinkers of high genus: Non-compact examples (2011, background): “a smooth compactness theorem (for closed, xed genus surfaces [CM1]) and a theory of generic singularities of ColdingMinicozzi, including classication of all H0 complete hypersurfaces (see [CM2] and [DX]). See also [LS] and [Wa] for other uniqueness results. The self-shrinker equation is a nonlinear ”