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Asymptotics of complete Kahler metrics of finite volume on quasiprojective manifolds
Frédéric Rochon,Zhou Zhang +1 more
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In this article, the Ricci flow converges to a complete Kahler-Einstein metric at the boundary of a quasiprojective manifold, and the associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion.Abstract:
Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of $X$ by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume on X. When these metrics evolve according to the Ricci flow, we prove that such asymptotic behaviors persist at later time by showing the associated potential function is smooth up to the boundary on the compactification $\tX$. However, when the divisor $\bD$ is smooth with $K_{\bX}+[\bD]>0$ and the Ricci flow converges to a Kahler-Einstein metric, we show that this Kahler-Einstein metric has a rather different asymptotic behavior at infinity, since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary.read more
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Kahler-Einstein metrics with edge singularities
TL;DR: In this paper, the existence and regularity of a compact Kahler manifold M with edge singularities with cone angle 2 along a smooth divisor D was studied and it was shown that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coecients along D for all 2 < 2.
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Ricci flow on surfaces with conic singularities
TL;DR: In this article, the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and 2π, was shown to exist and converge to a soliton.
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Smooth and singular Kahler-Einstein metrics
TL;DR: In this paper, the authors present a survey of the development of smooth Kahler-Einstein metrics and their applications in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts.
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Ricci flow and the determinant of the Laplacian on non-compact surfaces
TL;DR: On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a constant point.
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Pseudodifferential operators on manifolds with fibred corners
TL;DR: In this article, a pseudodifferential calculus generalizing the Phi-calculus of Mazzeo and Melrose is proposed. But it is based on the assumption that a stratified pseudomanifold can be resolved into a manifold with fibred corners.
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The Bergman kernel and biholomorphic mappings of pseudoconvex domains
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Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature
Rafe Mazzeo,Richard B. Melrose +1 more
TL;DR: In this paper, the Schwartz kernel of the resolvent of the Laplacian on complete Riemannian manifolds with negative sectional curvature near infinity is described.
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