Paper · Aug 8, 2017

Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

Brian Allen

math.DGWider flowsarXiv:1708.02535v2
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Abstract

In this paper we study Inverse Mean Curvature Flow (IMCF) on manifolds that are conformal to a warped product manifold. To this end, we show how the gradient conformal vector field in warped product manifolds is related to the conformal vector field on the conformal metric and use this to gain control of the flow in order to show long time existence. Connections are made to recent results of the author on stability of the positive mass theorem (PMT) and the Riemannian Penrose inequality (RPI) where long time existence of IMCF is an important assumption.

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Submitted
Aug 8, 2017
Last revised
Feb 7, 2018 (v2)
Subjects
math.DG

Comments: 27 pages, Lemma 3.4 corrected, asymptotic estimates added

Mean curvature flowOther curvature flowsInverse mean curvature flowStability & uniquenessLong-time existence & convergenceRotational symmetry & warped productsGeometric inequalitiesPhysics

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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Inverse mean curvature flows in warped product manifolds (Hengyu Zhou, 2016)6Zhou 2016 ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space (Brian Allen, 2016)7Allen 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)5Mullins 2016 Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under L² Metric Convergence (Brian Allen, 2017)1Allen 2017 Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow (Brian Allen, 2017)3Allen 2017 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)8Scheuer 2017 Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem (Brian Allen, 2018)4Allen 2018 Sobolev stability of the PMT and RPI using IMCF (Brian Allen, 2018)2Allen 2018
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under Metric Convergence

    Brian AllenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Same author Brian Allen · Shared terms “Riemannian Penrose”, “IMCF”, “Penrose inequality”

  2. Sobolev stability of the PMT and RPI using IMCF

    Brian AllenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Same author Brian Allen · Shared terms “IMCF”, “RPI”, “PMT”

  3. Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow

    Brian AllenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Same author Brian Allen · Shared terms “Riemannian Penrose”, “IMCF”, “Penrose inequality”

  4. Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Same author Brian Allen · Shared terms “mass theorem”, “positive mass”, “IMCF”

  5. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence

  6. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence

  7. ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Brian Allen · Shared terms “IMCF”

  8. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence

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) 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) 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

  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.