Linear stability of compact shrinking Ricci solitons

作者：Cao, Huai-Dong; Zhu, Meng^*

来源：Mathematische Annalen, 2024.

DOI：10.1007/s00208-024-02824-w

• KAHLER
• SINGULARITIES

In this paper, we continue investigating the second variation of Perelman's nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}-entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in Cao and Zhu (Math Ann 353(3):747-763, 2012), as well as the more recent work in Mehrmohamadi and Razavi (arXiv:2104.08343, 2021), and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in Cao et al. (arXiv:math.DG/0404165, 2004) (see also Cao and He in J Reine Angew Math, 2015:229-246, 2015) for positive Einstein manifolds to the compact shrinking Ricci soliton case.