Heat trace asymptotics on equiregular sub-Riemannian manifolds

2017/06/08 by Yuzuru Inahama, Inahama, Yuzuru, Setsuo Taniguchi +1
Computer Science · Mathematics · #35K08 #41A60 #53C17 #58J65 #60H07 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR) #math.DG #math.PR #msc:35K08 #msc:41A60 #msc:53C17 #msc:58J65 #msc:60H07

paper · pdf · doi:10.48550/arxiv.1706.02450

Final version. To appear in J. Math. Soc. Japan. 44 pages. Several minor errors were fixed

openalex publication_date 2017/06/08 · arxiv created 2020/04/14 · arxiv updated 2020/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a "div-grad type" sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spectral geometric meaning when Popp's measure is considered. Our proof is probabilistic. In particular, we use S. Watanabe's distributional Malliavin calculus.

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