二次曲线退化与Laplacian行列式
Conic degeneration and the determinant of the Laplacian
摘要：
我们调查的各种谱不变量的行为，特别是决定性的拉普拉斯算子，对家庭的光滑黎曼流形进行圆锥退化，也就是说，以特定的方式收敛到一个流形的圆锥奇点。我们的主要结果是一个渐近公式的行列式消失的退化参数为零。证明过程分为两部分：我们通过参数化构造研究退化流形上热迹的精细结构，然后使用该精细结构分析拉普拉斯算子的zeta函数和行列式。
英文摘要：
We investigate the behavior of various spectral invariants, particularly the determinant of the Laplacian, on a family of smooth Riemannian manifolds which undergo conic degeneration; that is, which converge in a particular way to a manifold with a conical singularity. Our main result is an asymptotic formula for the determinant up to terms which vanish as the degeneration parameter goes to zero. The proof proceeds in two parts; we study the fine structure of the heat trace on the degenerating manifolds via a parametrix construction, and then use that fine structure to analyze the zeta function and determinant of the Laplacian.
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