Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold Mⁿ×ℝ

Ya Gao; Yi-Juan Gong; Jing Mao

Paper · Jan 27, 2020

Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold $M^{n} \times R$

Ya Gao, Yi-Juan Gong, Jing Mao

math.DGWider flowsarXiv:2001.09860v2

View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold $M^{n} \times R$ , where $M^{n}$ is an $n$ -dimensional ( $n \geq 2$ ) complete Riemannian manifold with nonnegative Ricci curvature, and $R$ is the Euclidean $1$ -space.

View PDF View HTML arXiv page

Submitted: Jan 27, 2020
Last revised: Jan 28, 2020 (v2)
Subjects: math.DG

Comments: 22 pages. A related paper [Y. Gao, J. Mao, C.-L. Song, Existence and uniqueness of solutions to the constant mean curvature equation with nonzero Neumann boundary data in product manifold $M^{n} \times R$ , preprint.] will be put on arXiv very soon. Comments are welcome. Several revisions have been made in Section 1 of v1

Mean curvature flow Self-shrinkers & translators Free boundary & capillary

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

Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space $R_{1}^{n + 1}$
Ya Gao and Jing Mao2021Wider flows

Topics Free boundary & capillary · Same authors Ya Gao, Jing Mao · Shared terms “Neumann boundary”
An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space $R_{1}^{n + 1}$
Ya Gao and Jing Mao2021Wider flows

Topics Free boundary & capillary · Same authors Ya Gao, Jing Mao · Shared terms “Neumann boundary”
An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane $R_{1}^{2}$
Ya Gao, Chenyang Liu and Jing Mao2021Wider flows

Topics Free boundary & capillary · Same authors Ya Gao, Jing Mao · Shared terms “Neumann boundary”
Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold $M^{2} \times R$
Li Chen, Dan-Dan Hu, Jing Mao et al.2018Wider flows

Topics Free boundary & capillary, Self-shrinkers & translators · Same author Jing Mao
Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone
Ya Gao and Jing Mao2024Wider flows

Topics Free boundary & capillary · Same authors Ya Gao, Jing Mao · Shared terms “Neumann”, “Euclidean”
Translating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz-Minkowski plane $R_{1}^{2}$
Ya Gao, Jinghua Li and Jing Mao2021Wider flows

Topics Self-shrinkers & translators · Same authors Ya Gao, Jing Mao · Shared terms “translating solutions”
Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space $R_{1}^{n + 1}$
Ya Gao and Jing Mao2021Wider flows

Topics Free boundary & capillary · Same authors Ya Gao, Jing Mao · Shared terms “Neumann boundary”
Mean Curvature Flows of Graphs with Neumann Boundary condition
Jinju Xu2016Wider flows

Topics Free boundary & capillary · Shared terms “Neumann boundary”

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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

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] 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.

Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold $M^{n} \times R$

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters

Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold Mn×R

In the guidebook

Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold $M^{n} \times R$