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 .

The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kähler (cscK) metrics, when the cone angle is less than π. We introduce a new Hölder space called C 4,α,β to study the regularities of this fourth order elliptic equation, and prove that any C 2,α,β conic cscK metric is indeed of class C 4,α,β . Finally, the reductivity is established by a careful study of the conic Lichnerowicz operator. A new Hölder spaceLet (X, ω 0 ) be a compact Kähler manifold with complex dimension n, and D is a simple smooth divisor on X. Suppose (L D , h) is the line bundle induced by the divisor D with a metric h, and s is a non-trivial holomorphic section of it.For arbitrary point p ∈ D, we can introduce a local coordinate chart (ζ, z 2 , · · · , z n ) in an open neighborhood U centered at p, such that the divisor is defined by the zero locus of ζ in U. And we call it as the z-coordinate. Writing ζ = ρe iθ in polar coordinate, we can introduce another local coordinate as (ξ, w 2 , · · · , w n ), where ξ := re iθ , r = ρ β ,

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In this paper, we prove Matsushima's theorem for Kähler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler-Einstein cone metrics by the continuity method. Moreover, our method provides an existence theorem of Kähler-Einstein cone metrics with respect to conic Ding functional.Remark 1.2. For the klt -pair, Chen-Donaldson-Sun [13] proved that the automorphism group is reductive. However, they required the uniqueness of weak Kähler-Einstein metrics in their proof.Remark 1.3. In [10], Cheltsov-Rubinstein also announced a result for extremal cone metrics, but their method is based on an expansion formula for edge metrics, which is very different from ours.Based on this reductivity result, we can extend Bando-Mabuchi's celebrated work [1] to conic setting and prove the uniqueness of Kähler-Einstein cone metris by applying the continuity path, which connects the Kähler-Einstein cone metric ω ϕ to a given Kähler cone metric ω. I.e. for any t ∈ [0, 1],And we proved the following Theorem 1.4. The Kähler-Einstein cone metric is unique up to automorphisms.The way to prove uniqueness is first to establish a continuity path connecting a general Kähler cone metric to our target, i.e. a Kähler-Einstein cone metric. The difficulties are to prove openness and closedness along the path in Donaldson's C 2,α β space: here openness on [0, 1) follows from a Bochner type formula with contradiction argument. Thus we are able to carry on the implicit function theorem on [0, 1) and the apriori estimates on [0, τ ] for a small fixed τ > 0.Meanwhile, in order to prove closedness on [τ, 1], everything is boiled down to prove the zero order estimate (Section 4.3.1) and the higher order estimates (Section 4.3.2). We first show that the zero order estimate of the continuity path on [τ, 1] requires only the uniform bound