Geodesics in the space of Kähler cone metrics, I

Abstract

Abstract. In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics H β ; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space defined in Donaldson [29] to the case of Kähler manifolds with boundary; moreover we introduce a subspace H C of H β which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of C 1,1 β geodesics whose boundary values lie in H C . Moreover, we prove that such geodesic is the limit of a sequence of C 2,α β approximate geodesics under the C 1,1 β -norm. As a geometric application, we prove the metric space structure of H C .

Construction of constant scalar curvature Kähler cone metrics

“…But we can see that the higher order spaces are more complicated, since the geometry of the background metric is a priori involved. The Hölder space and are introduced in and further detailed computations can be found in . The idea is that we first define the local model Hölder spaces in the cone charts, and then extend it to the whole manifold using a global background Kähler cone metric.…”

“…The model metric has rich geometric information. The detailed computation could be found in . We then compare the general Kähler cone metrics with the growth of the model metric and use the following definitions (see also Definition 2.9 for Christoffel symbols and 2.13 for curvature tensors in ).…”

The Dirichlet and the weighted metrics for the space of Kähler metrics

“…The C 2,α -estimate follows from the Evans-Krylov estimate, and the precise statement is specified in Proposition 4.6 in Calamai-Zheng [7]. In the following we discuss the general properties of the distance function, K -energy and the G-functional in any Kähler class.…”

“…Donaldson [14] conjectured that H endowed with the L 2 metric is geodesically convex and is a metric space and pointed out the intensive relation between the geodesics of H and the existence (through geodesic stability), uniqueness (through convexity along the geodesics) of the extremal metrics. In the paper [7], we find geometric conditions on the Dirichlet boundary values with less regularity which assure the existence and uniqueness of C 1,1 geodesic segments. Many beautiful papers on the L 2 metrics could be found in [1,3,5,11,12] and the references therein.…”

“…ω n = V.Lemma 5 7. When ω φ belongs to the first Chern class, then we haveP = −λφ + h ω for f 0 = − log ω n ϕ 0 ω n + h ω − λϕ 0 .…”