AI 中文总结
该研究分析紧流形及主丛的阿贝尔覆盖上热核的长时间渐近，建立三类互补渐近展开，将结果推广到主丛水平热核，结合弗洛凯理论与博雷尔-韦伊分解完成证明。
AI 中文摘要
我们研究紧流形的阿贝尔覆盖及主丛的阿贝尔覆盖上热核的长时间渐近行为。对于阿贝尔覆盖，我们建立了三种互补的渐近展开：描述热半群相关性的分布展开、紧子集上热核的局部逐点展开，以及在自然扩散尺度下有效的全局展开，该展开揭示了支配热传播的高斯轮廓。随后，我们在自然和乐与曲率假设下，将这些结果推广到主丛上的水平热核。在此设定下，我们证明主导渐近行为完全由底层阿贝尔覆盖的几何决定，而非平凡纤维模因存在一致谱隙而呈指数快速衰减。证明结合了阿贝尔覆盖上的弗洛凯理论与主丛上的博雷尔-韦伊分解。
英文摘要
We study the long-time asymptotics of heat kernels on Abelian covers of compact manifolds and on Abelian covers of principal bundles. For Abelian covers, we establish three complementary asymptotic expansions: a distributional expansion describing correlations of the heat semigroup, a local pointwise expansion of the heat kernel on compact subsets, and a global expansion valid on the natural diffusive scale, revealing the Gaussian profile governing heat propagation. We then extend these results to horizontal heat kernels on principal bundles under natural holonomy and curvature assumptions. In this setting, we show that the leading asymptotics are entirely determined by the geometry of the underlying Abelian cover, while the nontrivial fiber modes decay exponentially fast due to a uniform spectral gap. The proof combines Floquet theory on Abelian covers with the Borel--Weil decomposition on principal bundles.
Comments43 pages
