Volume growth, eigenvalue and compactness for self-shrinkers

Ding, Qi; Xin, Y. L.

doi:10.4310/ajm.2013.v17.n3.a3

Cited by 101 publications

(90 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…where the constant C 1 = 2 l C = 2 l e Remark 4.1. Theorem 1.2 extends the known results in [9] and [15]. Moreover, the order l of the polynomial volume growth estimate is optimal.…”

Section: Upper Estimate Of Volume Growthsupporting

confidence: 77%

“…It is known that the volume of a properly immersed minimal hypersurface in R m has at least Euclidean growth and may grow exponentially. But, Ding-Xin [15] proved that a complete properly immersed self-shrinker hypersurface in R m has polynomial volume growth. Further, in [11], the first and third authors of the present paper, proved that for a complete self-shrinker in R m , properness of immersion, polynomial volume growth and finiteness of weighted volume are equivalent to each other.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

Cheng

Vieira

Zhou

2019

International Mathematics Research Notices

View full text Add to dashboard Cite

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if the ambient manifold is of bounded geometry, we prove that such a submanifold must have at least linear volume growth. In particular, we show that a properly immersed complete noncompact hypersurface in the Euclidean space with bounded Gaussian weighted mean curvature must have polynomial volume growth and at least linear volume growth.

show abstract

“…where the constant C 1 = 2 l C = 2 l e Remark 4.1. Theorem 1.2 extends the known results in [9] and [15]. Moreover, the order l of the polynomial volume growth estimate is optimal.…”

Section: Upper Estimate Of Volume Growthsupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

Cheng

Vieira

Zhou

2019

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…Remark We would like to mention that Ding-Xin [8] derived for any complete embeddedselfshrinker M n in R n+1 an estimate for the first eigenvalue, in particular, it is positive. In view of (1.4), one may obtain a similar corollary as above.…”

Section: Introductionmentioning

confidence: 94%

Existence and Liouville theorems for V -harmonic maps from complete manifolds

2012

View full text Add to dashboard Cite

Abstract. We establish existence and uniqueness theorems for V -harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps and Finsler maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V -harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the BakryEmery Ricci condition.

show abstract

“…Together with the result that for a self-shrinker, proper immersion, Euclidean volume growth, polynomial volume growth and finite weighted volume are equivalent each other (cf. [8,9]), their theorem can be stated as follows, Theorem 9.1. [8] Let Σ n be a complete n-dimensional self-shrinker in the Euclidean space R n+p , p ≥ 1.…”

Section: Hamiltonian F-stability Of Complete Lagrangian Self-shrinkersmentioning

confidence: 99%

Hamiltonian F-Stability of Complete Lagrangian Self-Shrinkers

Yang

2015

J Geom Anal

View full text Add to dashboard Cite

Abstract. In this paper, we study the Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers. We prove a characterization theorem for the Hamiltonian F-stability of n-dimensional complete Lagrangian self-shrinkers without boundary, with polynomial volume growth and with the second fundamental form satisfying the condition that there exist constants C0 > 0 and ε < 1 16n such that |A| 2 ≤ C0 +ε|x| 2 . We characterize the Hamiltonian F-stablity by the eigenvalues and eigenspaces of the drifted Laplacian.Mathematics Subject Classification (2000): 53C44 (primary), 53C21 (secondary). IntroductionAn n-dimensional submanifold Σ n of R n+p is called a self-shrinker if it is the time t = −1 slice of a self-shrinking mean curvature flow that disappears at (0, 0), i.e. of a mean curvature flow satisfying Σ t = √ −tΣ −1 . We can also consider a self-shrinker as a submanifold that satisfiesSelf-shrinkers are very important singularities of the mean curvature flow. According to the blow up rate of the second fundamental form, Huisken [11] classified the singularities of mean curvature flows into two types: Type I and Type II. In 1984, Huisken [10] showed that, if the initial hypersurface in R n+1 is strictly convex, then along the mean curvature flow, the surface will be strictly convex at each time, and the mean curvature flow will contract to a point at a finite time T . Moreover, the normalized mean curvature flow will converge to a round sphere. In 1990, Huisken [11] proved that any Type I singularity of the mean curvature flow must be a self-shrinker by using the monotonicity formula. He also proved that the only compact self-shrinkers with nonnegative mean curvature are spheres.In [6], Colding-Minicozzi introduced the concept of F-stability and entropy-stability of a self-shrinker, and gave a classification of self-shrinkers in the hypersurface case. The definitions of many concepts in their paper can be naturally generalized to the higher codimension case (cf. [2,3,13]). Given x 0 ∈ R n+p and t 0 > 0, F x 0 ,t 0 is defined by dµ.

show abstract

Volume growth, eigenvalue and compactness for self-shrinkers

Cited by 101 publications

References 14 publications

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

Existence and Liouville theorems for V -harmonic maps from complete manifolds

Hamiltonian F-Stability of Complete Lagrangian Self-Shrinkers

Contact Info

Product

Resources

About