The upper bounds for eigenvalues of Dirac operators

Abstract

Let $D$ be a Dirac operator on a compact oriented Riemannian manifold $M$ of dimension $2m$. The operator $D$ can be one of the four classical elliptic operators that arise from geometry, or one of the twisted operators of these four operators. Let $\lambda_{k}^{2}$ be the k-th nonzero eigenvalue of the operator $D^{2}$ counting with multiplicity. We show that \[ \lambda_{k}^{2}\leq c(2m)\max\{(\frac{N(a)}{V(M)})^{1/m},(\frac{2^{m-1}(m_{0}+k-1)-2^{-m}m_{0}+1}{|k_{0}|V(M)})^{1/m}} \], where $N(a)$ is an integer determined by the geometry of $M$,$m_{0}$ the dimension of the kernel of $D^{2}$ and $k_{0}$ an integer defined by the operator $D$. These results, in case $M$ being a surface, give a partial answer to a conjecture of Yau.