(\cF\)-functional and geodesic stability

We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this f...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Author He, Weiyong
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.06.2016
Subjects
Critical point
Gradient flow
Lower bounds
Manifolds
Pictures
Stability
Online AccessGet full text

Cover

More Information
Summary:We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this functional. As an application, we prove that Kahler-Ricci soliton, if exists, maximizes Perelman's \(\mu\)-functional without extra assumptions. Second we consider a conjecture proposed by S.K. Donaldson in terms of \(\cK\)-energy. Our simple observation is that \(\cF\)-functional, as \(\cK\)-energy, also integrates Futaki invariant. We then restate geodesic stability conjecture on Fano manifolds in terms of \(\cF\)-functional. Similar pictures can also be extended to Kahler-Ricci soliton and modified \(\cF\)-functional.
ISSN:2331-8422