$$C^{1,1}$$ regularity of geodesics in the space of volume forms

Abstract

We prove a C 1,1 estimate for solutions of a class of fully nonlinear equations introduced by Chen-He. As an application, we prove the C 1,1 regularity of geodesics in the space of volume forms.

Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics

“…By a result of Chen [11], with complements of Blocki [9] and Chu-Tossati-Weinkove [16], the boundary value problem ( 9) admits a unique G-invariant solution Φ ∈ C 1,1 ( X, R) such that π * X ω + dd c Φ is a positive current with bounded coefficients, up to the boundary, corresponding to a family of functions (φ t ) t∈[0,1] in the space K 1,1 (X, ω) T of all T-invariant functions φ ∈ C 1,1 (X, R) such that ω φ is a positive current with bounded coefficients. The curve (φ t ) t∈[0,1] ⊂ K 1,1 (X, ω) T is called the weak geodesic segment joining φ 0 , φ 1 ∈ K(X, ω) T .…”

C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays

“…When m = 1, (1.8) is equivalent to the geodesic equation in the space of volume forms on Riemannian manifold with fixed volume (see [22]). In this setting, Chen-He [10] proved the geodesic is C 2 w , and Chu [12] improved this regularity to C 1,1 .…”