[Paper Review] Convergence of the Ricci flow on asymptotically flat manifolds with integral curvature pinching

Eric Chen|arXiv (Cornell University)|Jul 30, 2019

Geometric Analysis and Curvature Flows23 references4 citations

TL;DR

This paper establishes a curvature pinching condition for the Ricci flow on asymptotically flat manifolds of dimension $n \geq 3$: if the scale-invariant $L^{n/2}$-norm of the Riemann curvature tensor is sufficiently small relative to the inverse of the initial Sobolev constant, then the Ricci flow exists for all time and converges smoothly to flat Euclidean space. The result implies the initial manifold is diffeomorphic to $\mathbb{R}^n$, providing a noncompact analogue of integral curvature pinching theorems for compact manifolds.

ABSTRACT

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to $\mathbb{R}^n$.

Motivation & Objective

• To establish a curvature pinching condition ensuring long-time existence and convergence of the Ricci flow on noncompact, asymptotically flat (AF) manifolds.
• To overcome the lack of uniform Sobolev constant control along the flow by introducing curvature-weighted Sobolev inequalities.
• To prove that under this pinching condition, the Ricci flow converges to flat $\mathbb{R}^n$, implying the initial manifold is diffeomorphic to $\mathbb{R}^n$.
• To provide explicit examples of nontrivial AF manifolds satisfying the curvature pinching condition, unlike previous pointwise pinching results.

Proposed method

• Use Perelman's $\mathcal{W}$-functional to derive a curvature-weighted Sobolev inequality valid along the Ricci flow on AF manifolds.
• Establish a uniform bound on the $L^{\infty}$-norm of the Riemann curvature tensor via the weak maximum principle, relying on the curvature-weighted Sobolev inequality to control blow-up.
• Analyze the evolution of $\|\text{Rm}_{g(t)}\|_{L^q}$ for $q = \frac{n}{2}\frac{n}{n-2}$ to obtain sharp decay estimates.
• Apply Moser iteration with the specific $q$-norm to upgrade pointwise curvature decay to $o(t^{-1})$ decay, essential for convergence to flat space.
• Prove that the $L^{n/2}$-norm of the curvature is monotone decreasing under the flow, enabling decay estimates.
• Use conformal and rotational symmetry reductions to prove lower bounds on the $L^{n/2}$-norm of curvature for manifolds with minimal hyperspheres, showing the pinching condition is non-vacuous.

Experimental results

Research questions

• RQ1 Can a noncompact analogue of integral curvature pinching for the Ricci flow be established on asymptotically flat manifolds?
• RQ2 Does a small $L^{n/2}$-norm of the Riemann curvature tensor, relative to the Sobolev constant, guarantee long-time existence and convergence of the Ricci flow to flat space?
• RQ3 Can curvature-weighted Sobolev inequalities be constructed on noncompact AF manifolds to control curvature blow-up without uniform Sobolev constant bounds?
• RQ4 Are there nontrivial examples of asymptotically flat manifolds satisfying the curvature pinching condition, despite previous results ruling out pointwise pinching?
• RQ5 What is the geometric implication of a small $L^{n/2}$-norm of curvature on the existence of minimal hypersurfaces in asymptotically flat manifolds?

Key findings

• There exists a dimension-dependent $\delta(n) > 0$ such that if $\left(\int_M |\text{Rm}_{g_0}|^{n/2} dV_{g_0}\right)^{2/n} < \delta(n)/C_{g_0}$, then the Ricci flow exists for all $t \in [0, \infty)$.
• The Ricci flow converges in $C^\infty_{-\tau'}(M^n)$ for any $\tau' \in (0, \min(\tau, n-2))$ to the flat metric on $\mathbb{R}^n$, with $\|\text{Rm}_{g(t)}\|_{L^\infty} = o(t^{-1})$.
• The initial manifold $M^n$ is diffeomorphic to $\mathbb{R}^n$, even without assuming this a priori.
• The curvature-weighted Sobolev inequality (Theorem C) holds along the flow under the pinching condition and enables the weak maximum principle argument.
• For rotationally symmetric or conformally flat AF manifolds, the $L^{n/2}$-norm of curvature is bounded below by a positive constant depending only on $n$, showing the pinching condition is non-vacuous.
• The result implies that $\int_M |\text{Rm}|^{n/2} dV$ being small suffices for long-time existence and convergence to flat space, even without a priori Sobolev control.