In this paper we introduce the interpolationdegeneration strategy to study KhlerEinstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By interpolation we show the angles in (0, 2] that admit a conical KhlerEinstein metric form a connected interval, and by degeneration we determine the boundary of the interval in some important cases. As a first application, we show that there exists a KhlerEinstein metric on P 2 with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in (/2, 2]. When the angle is 2/3 this proves the existence of a SasakiEinstein metric on the link of a three dimensional <i>A</i> ₂ singularity, and thus answers a question posed by GauntlettMartelliSparksYau. As a second application we prove a version of Donaldsons conjecture about conical Khler

We prove several formulas related to Hodge theory and the KodairaSpencerKuranishi deformation theory of Khler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on CalabiYau manifolds and also a construction of global canonical family of holomorphic ( n , 0 ) -forms on the deformation spaces of CalabiYau manifolds. Similar constructions are also applied to the deformation spaces of compact Khler manifolds.

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström (2018) and coupled Kähler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).

Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ∂M. Assume that the mean curvature H of the boundary ∂M satisfies H≥(n−1)k>0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ∂M is bounded from above by 1/𝑘 and the upper bound is achieved if and only if M is isometric to an n-dimensional Euclidean ball of radius 1/𝑘.

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