Paper · Mar 31, 2020

Parabolic approaches to curvature equations

Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

math.DGWider flowsarXiv:2003.14384v2
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Abstract

We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate -Aleksandrov and dual Minkowski problems as well as their counterparts, namely the -Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli-Guan-Ma (Commun. Pure Appl. Math. 60 (2007), 1769–1791). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures.

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Submitted
Mar 31, 2020
Last revised
Oct 24, 2020 (v2)
Subjects
math.DG, math.AP
Journal
Nonlinear Anal. 203 (2021), art. 112174
DOI
10.1016/j.na.2020.112174

Comments: 32 pages

Other curvature flowsGauss curvature flowPhysics

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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 A unified flow approach to smooth, even Lₚ-Minkowski problems (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)2Bryan 2016 Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2017)4Bryan 2017 A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1 (Li Chen, Qiang Tu and Ni Xiang, 2019)5Chen 2019 The Minkowski inequality in de Sitter space (Julian Scheuer, 2019)7Scheuer 2019 Orlicz-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)3Bryan 2020 Christoffel-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)1Bryan 2020 An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems (Bin Chen, Jingshi Cui and Peibiao Zhao, 2022)8Chen 2022 Uniqueness of solutions to a class of isotropic curvature problems (Mohammad N. Ivaki and Emanuel Milman, 2023)6Ivaki 2023
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  1. Christoffel-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Christoffel Minkowski”, “global term”

  2. A unified flow approach to smooth, even -Minkowski problems

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “prescribed curvature”, “Minkowski”, “Guan”

  3. Orlicz-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Minkowski”, “global term”, “Christoffel”

  4. Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Physics, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Sitter space”, “Lorentzian”, “strictly convex”

  5. A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1

    Li Chen, Qiang Tu and Ni XiangWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”, “normal vector”, “strictly convex”

  6. Uniqueness of solutions to a class of isotropic curvature problems

    Mohammad N. Ivaki and Emanuel MilmanWider flows

    Topics Gauss curvature flow, Other curvature flows · Same author Mohammad N. Ivaki · Shared terms “Christoffel Minkowski”, “Minkowski type”

  7. The Minkowski inequality in de Sitter space

    Julian ScheuerWider flows

    Topics Physics, Other curvature flows · Same author Julian Scheuer · Shared terms “Sitter space”, “Minkowski”, “Lorentzian”

  8. An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

    Bin Chen, Jingshi Cui and Peibiao ZhaoWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”, “curvature measure”, “dual Minkowski”

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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) 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)