The total second variation of Perelman's $\mathcal{W}$-functional

Abstract

We show a very simple and general total second variation formula for Perelman's $\mathcal{W}$-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for Perelman's $\mathcal{W}$-functional with respect to such variations. In this case the main therm in the formula depends strongly on the variation of the complex structure. We discover also convexity of Perelman's $\mathcal{W}$-functional along particular variations over points with non-negative Bakry-Emery-Ricci tensor.

37 pages

References in corpus (2)

• Supremum of Perelman's entropy and Kähler-Ricci flow on a Fano manifold
• On second variation of Perelman's Ricci shrinker entropy

Cited by in corpus (1)

• On the uniqueness of asymptotic limits of the Ricci flow