Soliton-type metrics and Kähler-Ricci flow on symplectic quotients

Gabriele La Nave, Gang Tian

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

In this paper, we first show an interpretation of the Kähler-Ricci flow on a manifold X as an exact elliptic equation of Einstein type on a manifold M of which X is one of the (Kähler) symplectic reductions via a (non-trivial) torus action. There are plenty of such manifolds (e.g. any line bundle on X will do). Such an equation is called V-soliton equation, which can be regarded as a generalization of Kähler-Einstein equations or Kähler-Ricci solitons. As in the case of Kähler-Einstein metrics, we can also reduce the V-soliton equation to a scalar equation on Kähler potentials, which is of Monge-Ampère type. We then prove some preliminary results towards establishing existence of solutions for such a scalar equation on a compact Kähler manifoldM. One of our motivations is to apply the interpretation to studying finite time singularities of the Kähler-Ricci flow.

Original languageEnglish (US)
Pages (from-to)139-166
Number of pages28
JournalJournal fur die Reine und Angewandte Mathematik
Volume2016
Issue number711
DOIs
StatePublished - Feb 1 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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