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Harnack inequalities on a manifold with positive or negative Ricci curvature.

Dominique Bakry, Zhongmin M. Qian (1999)

Revista Matemática Iberoamericana

Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established. These estimates are sharp both for small time, for large time and for large distance, and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below.

Heat diffusion on homogeneous trees (Note on a paper by G. Medolla and A. G. Setti)

Wolfgang Woess (2001)

Bollettino dell'Unione Matematica Italiana

Medolla e Setti [6] studiano l'andamento della diffusione del calore generata dal Laplaciano discreto su un albero omogeneo e dimostrano che il calore è asintoticamente concentrato in «anelli» che viaggiano verso l'infinito a velocità lineare e la cui larghezza divisa per t tende all'infinito, dove t è il tempo. Qui si spiega come un risultato più preciso si ottiene come corollario della legge dei grandi numeri e del teorema del limite centrale per la passeggiata aleatoria sull'albero. Inoltre,...

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 and semigroup estimates for sublaplacians with drift on Lie groups.

Nick Dungey (2005)

Publicacions Matemàtiques

Let G be a Lie group. The main new result of this paper is an estimate in L2 (G) for the Davies perturbation of the semigroup generated by a centered sublaplacian H on G. When G is amenable, such estimates hold only for sublaplacians which are centered. Our semigroup estimate enables us to give new proofs of Gaussian heat kernel estimates established by Varopoulos on amenable Lie groups and by Alexopoulos on Lie groups of polynomial growth.

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.

Heat kernel upper bounds on a complete non-compact manifold.

Alexander Grigor'yan (1994)

Revista Matemática Iberoamericana

Let M be a smooth connected non-compact geodesically complete Riemannian manifold, Δ denote the Laplace operator associated with the Riemannian metric, n ≥ 2 be the dimension of M. Consider the heat equation on the manifoldut - Δu = 0,where u = u(x,t), x ∈ M, t > 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t)...

High order regularity for subelliptic operators on Lie groups of polynomial growth.

Nick Dungey (2005)

Revista Matemática Iberoamericana

Let G be a Lie group of polynomial volume growth, with Lie algebra g. Consider a second-order, right-invariant, subelliptic differential operator H on G, and the associated semigroup St = e-tH. We identify an ideal n' of g such that H satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of n'. The regularity is expressed as L2 estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel....

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