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The subelliptic heat kernel on the CR sphere

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Abstract

We study the heat kernel of the sub-Laplacian \(L\) on the CR sphere \(\mathbb{S }^{2n+1}\). An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub-Laplacian \(-L+n^2\) that was obtained by Geller (J Differ Geom 15:417–435, 1980), and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration \(\mathbb{S }^{2n+1} \rightarrow \mathbb{CP }^n\).

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Notes

  1. We call north pole the point with complex coordinates \(z_1=0,\ldots , z_{n+1}=1\), it is therefore the point with real coordinates \((0,\ldots ,0,1,0)\).

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Fabrice Baudoin & Jing Wang

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  1. Fabrice Baudoin
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  2. Jing Wang
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Correspondence to Fabrice Baudoin.

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F. Baudoin supported in part by NSF Grant DMS 0907326.

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Baudoin, F., Wang, J. The subelliptic heat kernel on the CR sphere. Math. Z. 275, 135–150 (2013). https://doi.org/10.1007/s00209-012-1127-4

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  • DOI: https://doi.org/10.1007/s00209-012-1127-4

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