Gaussian upper bounds for the heat kernel on evolving manifolds
Gaussian upper bounds for the heat kernel on evolving manifolds
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DOI：
10.1112/jlms.12793
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发表时间：
2020-07
期刊：
影响因子：
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通讯作者：
R. Buzano;Louis Yudowitz
中科院分区：
文献类型：
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作者：
R. Buzano;Louis Yudowitz
本文证明了由内禀几何流演化的封闭流形上加权热算子的热核的一个一般的、相当灵活的上界。该证明基于对数Sobolev不等式和沿流加权算子的超收缩性估计，这是Davies （Amer）之前使用的一种方法。[j] .数学学报，2009(3):319-334。这一结果直接暗示了热核在演化距离函数的一定边界下的高斯型上界；特别是在具有有界曲率或正Ricci曲率的流形上，我们找到了高斯热核界的新证明。对于一类其他几何流，我们也得到了类似的热核边界。
In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method that was previously used by Davies (Amer. J. Math. 109 (1987) 319–334) in the case of a non‐evolving manifold. This result directly implies Gaussian‐type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.