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Heat flows for extremal Kähler metrics

Santiago R. Simanca (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let ( M , J , Ω ) be a closed polarized complex manifold of Kähler type. Let G be the maximal compact subgroup of the automorphism group of ( M , J ) . On the space of Kähler metrics that are invariant under G and represent the cohomology class Ω , we define a flow equation whose critical points are the extremal metrics,i.e.those that minimize the square of the L 2 -norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its...

Heat kernel estimates for critical fractional diffusion operators

Longjie Xie, Xicheng Zhang (2014)

Studia Mathematica

We construct the heat kernel of the 1/2-order Laplacian perturbed by a first-order gradient term in Hölder spaces and a zero-order potential term in a generalized Kato class, and obtain sharp two-sided estimates as well as a gradient estimate of the heat kernel, where the proof of the lower bound is based on a probabilistic approach.

Heat kernel on manifolds with ends

Alexander Grigor’yan, Laurent Saloff-Coste (2009)

Annales de l’institut Fourier

We prove two-sided estimates of heat kernels on non-parabolic Riemannian manifolds with ends, assuming that the heat kernel on each end separately satisfies the Li-Yau estimate.

Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time

Alexei Lozinski, Jacek Narski, Claudia Negulescu (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter , and this for fixed...

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