A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

Franziska Borer; Peter Elbau; Tobias Weth

Paper · Jan 27, 2023

A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

Franziska Borer, Peter Elbau, Tobias Weth

math.APWider flowsarXiv:2301.12015v2

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Abstract

On a closed Riemannian surface $(M, g ˉ)$ with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume $A > 0$ and the property that their Gauss curvatures $f_{λ} = f + λ$ are given as the sum of a prescribed function $f \in C^{\infty} (M)$ and an additive constant $λ$ . Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results. In contrast to previous work, our approach does not require any sign conditions on $f$ . Moreover, we exhibit conditions under which the function $f_{λ}$ is sign changing and the standard prescribed Gauss curvature flow is not applicable.

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Submitted: Jan 27, 2023
Last revised: Sep 18, 2023 (v2)
Subjects: math.AP

Other curvature flows Gauss curvature flow Surfaces Limit spaces & RCD

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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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
11A.12
Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles
Uniformization by Ricci flow, and Hamilton’s entropy.
6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

[1] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[2] 11A.12
Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles
Uniformization by Ricci flow, and Hamilton’s entropy.

[3] 6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

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