Paper · Nov 4, 2023

The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces

Shujing Pan, Bo Yang

math.DGWider flowsarXiv:2311.02364
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Abstract

We consider a locally constrained curvature flow in a static rotationally symmetric space , which was firstly introduced by Hu and Li in the hyperbolic space. We prove that if the initial hypersurface is graphical, then the smooth solution of the flow remains to be graphical, exists for all positive time and converges to a slice of exponentially in the smooth topology. Moreover, we prove that the flow preserves static convexity if the initial hypersurface is close to a slice of in the sense. As applications, we prove a family of weighted geometric inequalities for static convex domains which is close to a slice of in the sense.

Comments: 31 pages. arXiv admin note: text overlap with arXiv:2105.03911 by other authors

Other curvature flowsVolume-preserving & constrained flowsLong-time existence & convergenceRotational symmetry & warped productsGeometric inequalities

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  3. Inverse curvature flows for capillary hypersurfaces in the unit ball

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  4. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

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  5. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

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    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Same author Bo Yang · Shared terms “hyperbolic space”, “smooth topology”, “flow remains”

  6. Capillary quermassintegral inequalities in the unit ball

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    Topics Geometric inequalities, Other curvature flows · Same author Shujing Pan

  7. New quermassintegral and Poincaré type inequalities for non-convex domains

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  8. Mean curvature type flow and sharp Micheal-Simon inequalities

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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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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)