Paper · Oct 8, 2023

On local rigidity theorems with respect to the scalar curvature

As on arXiv

Abstract

By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^{2}$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an $n$ -dimensional Riemannian manifold satisfies $\int_{V} R (g) d v o l_{g} \geq 0 and I (V) \geq I (R^{n}),$ or $\int_{V} R (g) d v o l_{g} \geq 0 and S (V) \geq S (R^{n}),$ then $g = g_{R^{n}}$ on $V$ , where $R (g)$ is the scalar curvature of $g$ , $R^{n}$ is Euclidean space, $I (V)$ is the isoperimetric constant of $V$ and $S (V)$ is best constant of $L^{2}$ logarithmic Sobolev inequality of $V$ . Moreover,we also obtain the local $R^{n}$ -rigidity about local Perelman's $ν$ -entropy, and local $S^{n}$ -rigidity (resp. $H^{n}$ -rigidity) theorems regarding the cases concerning $R (g) \geq n (n - 1)$ (resp. $R (g) \geq - n (n - 1)$ ), weighted isoperimetric constant and best constant of weighted $L^{2}$ logarithmic Sobolev inequality for the weighted metric $(cos \frac{d _{g} ( p , x )}{2})^{- 4} g$ (resp. $(cosh \frac{d _{g} ( p , x )}{2})^{- 4} g$ ).

Comments: We also study corresponding rigidity theorems in the new version for cases where the scalar curvature is bounded below by $- n (n - 1)$ or $n (n - 1)$ . Precisely, we add Theorem 1.7, Theorem 1.8 and add Section 5 for the proofs

Ricci flow Scalar curvature Stability & uniqueness Geometric inequalities

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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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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

On local rigidity theorems with respect to the scalar curvature

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