# Eigenvalue estimates via H\"{o}mander's $L^2$-method

## TL;DR

This paper uses Hörmander's weighted L^2 technique to derive lower bounds for eigenvalues of Dirac operators under various boundary conditions, linking these bounds to the volume of the manifolds.

## Contribution

It introduces a novel application of Hörmander's L^2 method to estimate Dirac eigenvalues under different boundary conditions, incorporating sharp Sobolev inequalities.

## Key findings

- Lower eigenvalue bounds for Dirac operators established
- Bounds expressed in terms of manifold volume
- Application of Hörmander's technique to geometric analysis

## Abstract

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the sharp Sobolev inequality due to Li and Zhu(\cite{LZ}).