The heat kernel on asymptotically hyperbolic manifolds

Abstract

Upper and lower bounds on the heat kernel on complete Riemannian manifolds were obtained in a series of pioneering works due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. However, these estimates do not give a complete picture of the heat kernel for all times and all pairs of points — in particular, there is a considerable gap between available upper and lower bounds at large distances and/or large times. Inspired by the work of Davies-Mandouvalos on Hn+1, we study heat kernel bounds on Cartan-Hadamard manifolds that are asymptotically hyperbolic in the sense of Mazzeo-Melrose. Under the assumption of no eigenvalues and no resonance at the bottom of the continuous spectrum, we show that the heat kernel on such manifolds is comparable to the heat kernel on hyperbolic space of the same dimension (expressed as a function of time t and geodesic distance r), uniformly for all t∈(0,∞) and all r∈[0,∞). In particular our upper and lower bounds are uniformly comparable for all distances and all times. The corresponding statement for asymptotically Euclidean spaces is not known to hold, and as we argue in the last section, it is very unlikely to be true in that geometry. As an application, we show boundedness on Lp of the Riesz transform ∇(Δ−n2/4+λ2)−1/2, for λ∈(0,n/2], on such manifolds, for p satisfying |p−1−2−1|<λ/n. For λ=n/2 (the standard Riesz transform ∇Δ−1/2), this was previously shown by Lohoué in a more general setting.

Locations

• Communications in Partial Differential Equations - View
• arXiv (Cornell University) - View - PDF