谱几何与渐近锥收敛
Spectral Geometry and Asymptotically Conic Convergence
摘要：
本文定义了一种新的收敛性，称为“渐近锥收敛”，它是指固定紧致流形上的光滑黎曼度量族退化为具有孤立锥奇异性的度量族。我们的结果是：几何Laplacian谱的收敛性和相应热核的一致收敛性，以及存在对所有时间具有一致收敛性的完全渐近展开.技术包括：使用一个新的“分辨率爆破”和微局部分析使用伪微分算子演算流形上的圆锥奇点的决议与各种（标准和非标准）爆破构造的角落。
英文摘要：
In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are: convergence of the spectrum of the geometric Laplacians and uniform convergence of the corresponding heat kernels and existence of a full asymptotic expansion with uniform convergence for all time. Techniques include: the resolution of a conic singularity using a new "resolution blowup" and microlocal analysis using pseudodifferential operator calculi on manifolds with corners constructed by various (standard and non-standard) blowups.
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