Dynamical stability of algebraic Ricci solitons

Abstract

We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain interpolation properties that allow the use of maximal regularity theory and the application of a stability theorem of Simonett. With this, we derive two stability theorems, one for a class of Einstein metrics and one for a class of non-Einstein Ricci solitons. Using linear stability results of Jablonski, Petersen, and the first author, we obtain dynamical stability for many specific Einstein and Ricci soliton metrics on simply connected solvable Lie groups.

18 pages, updated reference and statement of Theorems 1.3 and 2.1, revised introduction and fixed typos, final version to appear in Journal für die reine und angewandte Mathematik

References in corpus (2)

• Stability of solutions of certain extended Ricci flow systems
• Stability of symmetric spaces of noncompact type under Ricci flow

Cited by in corpus (2)

• Dynamic instability of ${\mathbb{CP}}^N$ under Ricci flow
• Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity