Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge-Laplacian on differential forms is essentially self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to prove bounds on the growth of $L^2$-harmonic forms at the singular set and to prove a Hodge theorem, namely that the space of $L^2$-harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Moreover, we develop an asymptotic expansion for the heat trace near $t = 0$.

中文翻译：

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“Witt”不完全尖端边缘空间的谱和霍奇理论

不完整的尖点边缘模拟 Weil-Petersson 度量在除数内部附近的紧化黎曼模空间上的行为。假设这样的空间是 Witt，我们构造了热方程的基本解，并使用其在奇异集上的渐近行为的精确描述，我们证明了微分形式上的 Hodge-Laplacian 本质上是自伴的，具有离散谱满足外尔渐近线。我们继续证明在奇异集上$L^2$-调和形式的增长的边界并证明霍奇定理，即$L^2$-调和形式的空间自然同构于中变态交集上同调。此外，我们为 $t = 0$ 附近的热迹开发了渐近扩展。