Two flow approaches to the Loewner-Nirenberg problem on manifolds

Gang Li

Paper · Jan 8, 2021

Two flow approaches to the Loewner-Nirenberg problem on manifolds

Gang Li

math.DGWider flowsarXiv:2101.03005v2

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Abstract

We introduce two flow approaches to the Loewner–Nirenberg problem on comapct Riemannian manifolds $(M^{n}, g)$ with boundary and establish the convergence of the corresponding Cauchy–Dirichlet problems to the solution of the Loewner–Nirenberg problem. In particular, when the initial data $u_{0} \in C^{4, α} (M)$ is a solution or a strict subsolution to the equation $(1.1)$ , the convergence holds for both the direct flow $(1.3) - (1.5)$ and the Yamabe flow $(1.10)$ . Moreover, when the background metric satisfies $R_{g} \geq 0$ , the convergence holds for any positive initial data $u_{0} \in C^{2, α} (M)$ for the direct flow; while for the case the first eigenvalue $λ_{1} < 0$ for the Dirichlet problem of the conformal Laplacian $L_{g}$ , the convergence holds for $u_{0} > v_{0}$ where $v_{0}$ is the largest solution to the homogeneous Dirichlet boundary value problem of $(1.1)$ and $v_{0} > 0$ in $M^{\circ}$ (the interior of $M$ ). We also give an equivalent description between the existence of a metric of positive scalar curvature in the conformal class of $(M, g)$ and $u \in C^{1} (M), u \neq \equiv 0 on \partial M in f Q (u) > - \infty$ , where $Q$ is the energy functional (see $(1.8)$ ) of the second type Escobar-Yamabe problem.

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Submitted: Jan 8, 2021
Last revised: Sep 10, 2021 (v2)
Subjects: math.DG

Comments: We rewrite the introduction, add more details in the proof, and correct some typos. Comments are welcome!

Yamabe flow Scalar curvature Homogeneous spaces Eigenvalues & spectrum

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

[1] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[2] 11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.

[3] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

Two flow approaches to the Loewner-Nirenberg problem on manifolds

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