A sharp lower bound on the small eigenvalues of surfaces

Abstract

Let \(S\) be a compact hyperbolic surface of genus \(g\geq 2\) and let \(I(S) = \frac{1}{\mathrm{Vol}(S)}\int_{S} \frac{1}{\mathrm{Inj}(x)^2 \wedge 1} dx\), where \(\mathrm{Inj}(x)\) is the injectivity radius at \(x\). We prove that for any \(k\in \{1,\ldots, 2g-3\}\), the \(k\)-th eigenvalue \(\lambda_k\) of the Laplacian satisfies \begin{equation*} \lambda_k \geq \frac{c k^2}{I(S) g^2} \, , \end{equation*} where \(c>0\) is some universal constant. We use this bound to prove the heat kernel estimate \begin{equation*} \frac{1}{\mathrm{Vol}(S)} \int_S \Big| p_t(x,x) -\frac{1}{\mathrm{Vol}(S)} \Big | ~dx \leq C \sqrt{ \frac{I(S)}{t}} \qquad \forall t \geq 1 \, , \end{equation*} where \(C<\infty\) is some universal constant. These bounds are optimal in the sense that for every \(g\geq 2\) there exists a compact hyperbolic surface of genus \(g\) satisfying the reverse inequalities with different constants.