Convexity of the 𝐾-energy on the space of Kähler metrics and uniqueness of extremal metrics

Abstract

Abstract. We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen and give some applications in Käh-ler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogenuous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

Uniqueness of constant scalar curvature Sasakian metrics

“…Since our original definition of the Mabuchi K-energy depends on the forth order derivative, we want to rewrite an explicit formula for it, which has an "energy part" and "entropy part" as in the Kähler case(see [1] for the Kähler case). We begin with some notations.…”

“…Now we consider the convexity of the Mabuchi K-energy along the weak geodesics, modifying the method of [1].…”

“…We will introduce a truncation in the following way as in [1]. For a fixed positive number A, we define…”

“…In [1], Berman and Berndtsson proved the convexity of the Mabuchi K-energy along the weak geodesic in the space of Kähler metrics by using Chen's weak C 2 regularity ( [3,4,8]). As an application, they obtained the uniqueness of constant scalar curvature Kähler metric (cscK metric) modulo automorphisms on a compact Kähler manifold in a fixed Kähler class.…”

“…Another Sasakian metric g ′ is compatible with g means that they have the same Reeb vector field, the same transverse holomorphic structure and the same holomorphic structure on the cone C(M ), see section 2 for details. In this paper, by using the weak C 2 regularity in [14] and following the argument of Berman and Berndtsson ( [1]), we prove the uniqueness of constant scalar curvature Sasakian metrics (cscS) up to the action of the identity component of the automorphism group for the transverse holomorphic structure, which is denoted to be G 0 in Definition 4.2. In fact, we obtain the following theorem.…”