Paper · Oct 30, 2024

Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds

Dasong Li, John Man Shun Ma

math.DGWider flowsarXiv:2410.22904Calc. Var. 64, 236 (2025)
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Abstract

In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow, including the mean curvature flow, inverse mean curvature flow, Gauss curvature flow and so on.

Mean curvature flowOther curvature flowsInverse mean curvature flowGauss curvature flowStability & uniquenessBoundaries & non-compact

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    Topics Boundaries & non-compact, Stability & uniqueness · Shared terms “backwards uniqueness”, “geometric flows”, “compact hypersurfaces”

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    Topics Other curvature flows · Shared terms “extrinsic geometric”, “geometric flows”

  3. Evolution of the Torsional Rigidity under Geometric Flows

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    Topics Inverse mean curvature flow, Other curvature flows, Stability & uniqueness · Shared terms “geometric flows”, “ambient manifold”

  4. Inverse mean curvature flow with outer obstacle

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    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Shared terms “uniqueness theorem”

  5. Regularity of the -Gauss curvature flow with flat side

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    Topics Gauss curvature flow, Other curvature flows

  6. The asymptotic behaviour of -capacitary potentials in Asymptotically Conical manifolds

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows PDF

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Shared terms “Riemannian manifolds”

  7. The chord Gauss curvature flow and its chord Minkowski problem

    Jinrong Hu, Yong Huang, Jian Lu et al.Wider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  8. A Gauss curvature flow related to the Orlicz-Aleksandrov problem

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    Topics Gauss curvature flow, Other curvature flows

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974)

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.