We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on Khler manifolds will be studied by Li and Liu (2014).

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.

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

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