Convergence of heat kernels for degenerating hyperbolic surfaces

Ji, Lizhen

doi:10.1090/S0002-9939-1995-1219726-3

Convergence of heat kernels for degenerating hyperbolic surfaces
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by Lizhen Ji

Proc. Amer. Math. Soc. 123 (1995), 639-646

DOI: https://doi.org/10.1090/S0002-9939-1995-1219726-3

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Abstract:

For a degenerating family of hyperbolic surfaces ${S_l}(l \geq 0)$, we show that the heat kernel of ${S_l}$ converges to the heat kernel of ${S_0}$. The proof consists of two steps. For small time, we use the Brownian motion interpretation of the heat kernels to prove the convergence. Then we use Gaussian type bounds for the heat kernels and their derivatives and a priori bounds for heat equations to finish the proof.

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