Optimal coercivity estimates and existence of cscK metrics on Kähler manifolds

Zak Sjöström Dyrefelt

created by daniele on 08 Sep 2019 modified on 04 Dec 2019

5 dec 2019
-- 14:30

Aula 1, DiMaI, Firenze

Abstract.

Existence of constant scalar curvature Kähler (cscK) metrics on
compact Kähler manifolds is a central question in complex geometry.
Following the variational approach pioneered by Mabuchi in the 1980's it
was recently proven (by X.X. Chen and J. Cheng) that existence of cscK
metrics is equivalent to coercivity of a certain energy functional on the
space of Kähler metrics. In this talk I will present new coercivity
estimates directly related to this problem. In particular, I will present
an explicit formula for the optimal coercivity constants appearing in the
variational picture mentioned above, thus characterizing explicitly
solvability of Donaldson's J-equation (an equation related both to the
cscK equation and to the Hermitian Yang Mills equation in geometry and
physics). We will focus in this talk on the applications of our estimates
to the cscK problem; in particular we obtain new and improved existence
and non-existence criteria this way.