Paper · Dec 27, 2017

Inverse curvature flows in Riemannian warped products

Julian Scheuer

math.DGWider flowsarXiv:1712.09521v2
View PDF View HTML arXiv page

As on arXiv

Abstract

The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower bound on the ambient sectional curvature. The inverse speeds are given by powers of a curvature function satisfying few common properties.

View PDF View HTML arXiv page
Submitted
Dec 27, 2017
Last revised
Jan 24, 2018 (v2)
Subjects
math.DG, math.AP
Journal
J. Funct. Anal. 276, no. 4, (2019), p. 1097-1144
DOI
10.1016/j.jfa.2018.08.021

Comments: 34 pages. Fixed minor errors and typos. Comments welcome

Other curvature flowsInverse mean curvature flowLong-time existence & convergenceRotational symmetry & warped products

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)4Zhou 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)2Mullins 2016 Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature (Heiko Kröner and Julian Scheuer, 2017)8Kröner 2017 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)5Chen 2017 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)6Allen 2017 Locally constrained inverse curvature flows (Julian Scheuer and Chao Xia, 2017)3Scheuer 2017 Self-similar solutions of curvature flows in warped products (Shanze Gao and Hui Ma, 2018)7Gao 2018 Minkowski inequalities and constrained inverse curvature flows in warped spaces (Julian Scheuer, 2020)1Scheuer 2020
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Minkowski inequalities and constrained inverse curvature flows in warped spaces

    Julian ScheuerWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Julian Scheuer · Shared terms “Riemannian warped”

  2. 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

  3. Locally constrained inverse curvature flows

    Julian Scheuer and Chao XiaWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence · Same author Julian Scheuer

  4. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

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

  5. Inverse mean curvature flow inside a cone in warped products

    Li Chen, Jing Mao, Ni Xiang et al.Wider flows

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

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

    Brian AllenWider flows

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

  7. Self-similar solutions of curvature flows in warped products

    Shanze Gao and Hui MaWider flows

    Topics Rotational symmetry & warped products, Other curvature flows · Shared terms “curvature function”

  8. Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

    Heiko Kröner and Julian ScheuerWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Julian Scheuer · Shared terms “expanding curvature”, “curvature function”

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.

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)