Non-scale-invariant inverse curvature flows in Euclidean space

By Claus Gerhardt. Calculus of Variations and Partial Differential Equations, 2011

86 citations recorded by the source corpus.

We consider the inverse curvature flows \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=F^{-p}\nu $$\end{document} of closed star-shaped hypersurfaces in Euclidean space in case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p\not =1$$\end{document} and prove that the flow exists for all time and converges to infinity, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p<1,$$\end{document} while in case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1,$$\end{document} the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to the unit sphere.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2c8cf7e8cd973e3a77f9c0185b7cfe54d2a89794

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In Applied Mathematics

1. Haodi Chen
2. Qi-Rui Li
3. Kyeongsu Choi
4. S. Brendle
5. Yan-nan Liu
6. Jian Lu
7. P. Daskalopoulos
8. Hong-wei Xu

In Astronomy and Astrophysics

1. S. Brendle
2. P. Daskalopoulos
3. Jian Lu
4. Kyeongsu Choi
5. Haodi Chen
6. Qi-Rui Li
7. Yan-nan Liu
8. Burkhard Wilking

In Geometry and Topology

1. Haodi Chen
2. Qi-Rui Li
3. Kyeongsu Choi
4. S. Brendle
5. P. Daskalopoulos
6. Jian Lu
7. Yan-nan Liu
8. Burkhard Wilking

Citations in the corpus

• The Lp dual Minkowski problem and related parabolic flows (2021, background): “[29] Gerhardt, C.: Non-scale-invariant inverse curvature flows in Euclidean space.”
• Asymptotic behavior of flows by powers of the Gaussian curvature (2016, methodology): “In particular, Gerhardt [15] studied convex hypersurfaces moving outward with speed Kν, where α<0.”
• A flow method for the dual Orlicz–Minkowski problem (2020, background): “And the induced metric and the second fundamental form of M can also be computed according to the radial function, see for example [24].”
• Curvature flows in the sphere (2013, unknown): “Expanding flows, or inverse curvature flows, have been considered in Euclidean and hyperbolic space [8, 11, 12, 18].”
• A unified flow approach to smooth, even Lp-Minkowski problems (2016, unknown): “C. Gerhardt, “Curvature problems.””