Stability of complex hyperbolic space under curvature-normalized Ricci flow

TL;DR

This paper proves the stability of complex hyperbolic space under curvature-normalized Ricci flow for both closed and complete noncompact manifolds, using advanced regularity theory and analysis of the Lichnerowicz Laplacian.

Contribution

It establishes the first stability results for complex hyperbolic space under Ricci flow in both compact and noncompact settings, employing maximal regularity and weighted function spaces.

Findings

Stability of complex hyperbolic space on closed manifolds.

Stability of complex hyperbolic space on complete noncompact manifolds.

Development of weighted Hölder spaces for noncompact analysis.

Abstract

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little H\"{o}lder spaces on a complete noncompact manifold and establish their interpolation properties.