Volume estimate about shrinkers

By Xu Cheng, Detang Zhou. 2011

135 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5dacc13ec98c4dcd97bb5a536e3ac7329af1f183

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In Applied Mathematics

1. Xin Zhou
2. Daniel Ketover
3. Xu Cheng
4. Lu Wang
5. Qing Cheng
6. G. Wei
7. Detang Zhou
8. Tito Mejia

In Geometry and Topology

1. Xin Zhou
2. Daniel Ketover
3. Xu Cheng
4. Lu Wang
5. Qing Cheng
6. G. Wei
7. Detang Zhou
8. Tito Mejia

Citations in the corpus

• Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends (2011, background): “2 are implied from the content of Lecture 3 in [23] and the equivalence of Euclidean volume growth and properness of self-shrinkers demonstrated in [9] and [3].”
• Complete $$\lambda $$λ-hypersurfaces of weighted volume-preserving mean curvature flow (2014, background): “By making use of the same assertions as in X. Cheng and Zhou [9] for self-shrinkers, we can prove the weighted area of a complete and non-compact properly immersed λ-hypersurface in the Euclidean space Rn+1 is bounded.”
• Entropy of closed surfaces and min-max theory (2015, methodology): “APPENDIX 35 10.2 F-stationary varifold with bounded entropy has polynomial volume growth In this part, we generalize the polynomial volume growth bound for self-shrinkers with bounded Gaussian volume [CZ] to the varifold setting. Let V ∈ Vn(Rn+k) be a F-stationary n-varifold, i.e. for anyX∈ X c(Rn+k”
• Stability and compactness for complete f -minimal surfaces (2012, background): “Since the first and third authors [3] of the present paper proved that for selfshrinkers, properness, the polynomial volume growth, and finite weighted volume are equivalent, Theorem 3 implies Theorem 4.”
• Complete self-shrinkers of the mean curvature flow (2012, background): “Since M2 is a complete proper self-shrinker, we know that M2 has polynomial volume growth from the result of Ding and Xin [11] or X. Cheng and Zhou [9].”
• Eigenvalues of the drifted Laplacian on complete metric measure spaces (2013, background): “ eigenvalue λ1 of L satisfies that 1−2δ 2 ≤ λ1 ≤ 1 2 . Proof. Since all eigenvalues of Anα, 1 ≤ α≤ p, satisfy sup i Xp α=1 η2 i,α ≤ δ< 1 2 , Ricf ≥ 1−2δ 2 g. By [24], Mhas finite weighted volume. By [8], Mis equivalently properly immersed and with polynomial volume growth. The rest of conclusion of”