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 ${\mathbb{H}}^{n+1},$ 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 $t\in (0,\infty )$ and all $r\in [0,\infty ).$ 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 L^p of the Riesz transform $\nabla {(\Delta -n^2/4+{\lambda }^2)}^{-1/2},$ for $\lambda \in (0,n/2],$ on such manifolds, for p satisfying $|p^{-1}-2^{-1}|<\lambda /n.$ For $\lambda =n/2$ (the standard Riesz transform $\nabla {\Delta }^{-1/2}$), this was previously shown by Lohoué in a more general setting.

Keywords:

• Asymptotically hyperbolic manifolds
• heat kernel
• resolvent
• Riesz transform

Acknowledgment

The authors would like to thank Pierre Portal, Colin Guillarmou, Hong-Quan Li, András Vasy, Xuan Thinh Duong and Michael Cowling for various illuminating conversations. The first author is also grateful to Jun Li and Jiaxing Hong for their continuous encouragement and support.

Notes

1 We will denote the Laplacian on $X^{°}$ by Δ_X, even though ${\Delta }_{X^{°}}$ would be more accurate.

2 In terms of our parametrization of the spectrum, they showed a meromorphic continuation except at $\lambda =im/2$ where $m=1,2,\dots ;$ the resolvent may have essential singularities at these points unless the metric is even at x = 0, as shown by Guillarmou [28].

3 We suggest the reader consult the papers by Mazzeo-Melrose [5] or by Mazzeo [6] for details of the blow-up.

4 The proof in [26] is only claimed for metrics close to the hyperbolic metric. However, it applies verbatim to any asymptotically hyperbolic Cartan-Hadamard manifold.

5 This is related to the ζ parameter of Mazzeo-Melrose by $\lambda =-i(\zeta -n/2)$

6 Here we regard these kernels as functions on $X_0^2$ rather than half-densities on $X_0^2$ as in [5]. To regard as a half-density we simply multiply by the Riemannian half-density on each factor of X.

7 See [5, Section 4, particularly (4.12)] for the precise sense in which this is true.

8 This was not explicitly addressed in [5]. See for example the paper of Patterson-Perry [30].

9 In [21], ‘microlocalized’ estimates are proved. However, when the manifold is Cartan-Hadamard, the microlocalizing operators Q_i are not required, which implies (2.6(2.6) $$\left|d{E}_P\right(\lambda \left)\right(z,z\prime \left)\right|\le \{\begin{array}{cc}C{\lambda }^2,\hfill & \mathit{\text{if}}\hspace{1em}\lambda \le 1;\hfill \\ C{\lambda }^n\hfill & \mathit{\text{if}}\hspace{1em}\lambda \ge 1.\hfill \end{array}$$(2.6) ).

10 Details of the method can be found in the book of Erdélyi [32, p.39-40].

11 This is related to work of Clerc-Stein [34], who showed that a necessary condition for L^p boundedness of functions $F({\Delta }_{{\mathbb{H}}^{n+1}}-n^2/4)$ is that F extends to a holomorphic function in a strip; thus ${({\Delta }_{{\mathbb{H}}^{n+1}}-n^2/4)}^{-1/2}$ cannot act boundedly on L^p, but ${({\Delta }_{{\mathbb{H}}^{n+1}}-n^2/4+{\lambda }^2)}^{-1/2}$ does for some range of p if $\lambda >0.$ See also Taylor [22].

Additional information

Funding

The first author was supported by the general financial grant (Grant No. 2016M591591) from the China Postdoctoral Science Foundation as well as NSFC Grant No.11701094. The second author was supported by Discovery Grants DP150102419 and DP160100941 from the Australian Research Council.