Paper · Jun 21, 2016

Mean Curvature Flows of Graphs with Neumann Boundary condition

Jinju Xu

math.APWider flowsarXiv:1606.06392
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As on arXiv

Abstract

In this paper, we study the mean curvature flow of graphs with Neumann boundary condition. The main aim is to use the maximum principle to get the boundary gradient estimate for solutions. In particular, we obtain the corresponding existence theorem for the mean curvature flow of graphs.

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Submitted
Jun 21, 2016
Subjects
math.AP

Comments: 11pages. This is an extended parabolic version of arxiv:1406.0046(published in Adv.Math.290(2016),1010-1039.)

Mean curvature flowBoundaries & non-compactFree boundary & capillary

Its neighbourhood

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    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  2. Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

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  3. Neumann boundary value problem for gernera curvature flow with forcing term

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    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  4. Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

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    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”, “gradient estimates”

  5. A remark on gradient estimates for spacelike mean curvature flow with boundary conditions

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    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “Neumann boundary”, “gradient estimates”

  6. Higher order curvature flows of plane curves with generalised Neumann boundary conditions

    James McCoy, Glen Wheeler and Yuhan WuWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  7. Level-set forced mean curvature flow with the Neumann boundary condition

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  8. Singularity formation in axially symmetric mean curvature flow with Neumann boundary

    John Head and Sevvandi KandanaarachchiWider 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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

  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.