Volume properties and rigidity on self-expanders of mean curvature flow

Saul Ancari; Xu Cheng

Paper · Dec 6, 2020

Volume properties and rigidity on self-expanders of mean curvature flow

Saul Ancari, Xu Cheng

math.DGWider flowsarXiv:2012.03364v2

View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the $L$ -stability operator and discuss the $L$ -stability of some special self-expanders. Besides, we prove that the surfaces $Γ \times R$ with the product metric are the only complete self-expander surfaces immersed in $R^{3}$ with constant scalar curvature, where $Γ$ is a complete self-expander curve (properly) immersed in $R^{2}$ .

View PDF View HTML arXiv page

Submitted: Dec 6, 2020
Last revised: Dec 23, 2020 (v2)
Subjects: math.DG

Comments: 29 pages. In this new version, we have made some changes

Mean curvature flow Self-shrinkers & translators Scalar curvature Stability & uniqueness Eigenvalues & spectrum

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

Spectral properties and rigidity for self-expanding solutions of the mean curvature flows
Xu Cheng and Detang Zhou2017Wider flows

Topics Eigenvalues & spectrum, Self-shrinkers & translators, Stability & uniqueness · Same author Xu Cheng · Shared terms “self expanders”, “drifted Laplacian”, “stability operator”
On the spectrum and index of expanding and translating solitons of the mean curvature flow in $R^{3}$
Hilário Alencar and Gregório Silva Neto2022Wider flows

Topics Eigenvalues & spectrum, Self-shrinkers & translators, Stability & uniqueness · Shared terms “self expanders”, “weighted volume”, “stability operator”
Volume growth, entropy and stability for translating solitons
Qiang Guang2016Wider flows

Topics Eigenvalues & spectrum, Self-shrinkers & translators, Stability & uniqueness · Shared terms “volume growth”, “stability operator”, “properly”
The rigidity theorems for complete self-expander of mean curvature flow
Zhi Li and Guoxin Wei2026Wider flows

Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “complete self”, “self expanders”, “Euclidean space”
Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow
Debora Impera and Michele Rimoldi2023Wider flows

Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self expanders”, “immersed self”, “weighted volume”
Self-expanders of the mean curvature flow
Knut Smoczyk2020Wider flows

Topics Self-shrinkers & translators, Scalar curvature · Shared terms “self expanders”
On $n$ -dimensional complete self-similar solutions to the mean curvature flow in $R^{n + 1}$ with nonnegative constant scalar curvature
Yong Luo, Linlin Sun and Jiabin Yin2021Wider flows

Topics Self-shrinkers & translators, Scalar curvature · Shared terms “complete self”, “self expanders”, “constant scalar”
On knotted self expanders of the mean curvature flow in $R^{3}$
Alexander Mramor and Shengwen Wang2026

Topics Self-shrinkers & translators · Shared terms “self expanders”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

[1] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[2] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

[3] 6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Volume properties and rigidity on self-expanders of mean curvature flow

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters