# Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

## TL;DR

This paper establishes eigenvalue bounds for differential form Laplacians on Alexandrov spaces and constructs these operators, linking their kernels to intersection homology, thus extending spectral analysis to singular spaces.

## Contribution

It introduces a method to estimate eigenvalues and construct Laplacians on Alexandrov spaces, connecting spectral properties with topological invariants.

## Key findings

- Eigenvalue upper bounds for differential form Laplacians on Alexandrov spaces
- Construction of differential form Laplacians on singular spaces
- Identification of the Laplacian kernel with intersection homology groups

## Abstract

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.