Collapsing and the Differential Form Laplacian : The Case of a Singular Limit Space

TL;DR

This paper investigates how the p-form Laplacian behaves under collapsing sequences of manifolds with bounded curvature and diameter, leading to insights on eigenvalue bounds in the limit space.

Contribution

It provides a detailed analysis of the limit of the p-form Laplacian during collapse to a singular space, extending previous work to include eigenvalue estimates.

Findings

Eigenvalue bounds depend on sectional curvature and diameter.

Analysis of Laplacian behavior in collapsing sequences.

Results applicable to singular limit spaces.

Abstract

In this paper, which is a sequel to math.DG/9902111, we analyze the limit of the p-form Laplacian under a collapse with bounded sectional curvature and bounded diameter to a singular limit space. As applications, we give results about upper and lower bounds on the j-th eigenvalue of the p-form Laplacian, in terms of sectional curvature and diameter.