A sharp lower bound on the small eigenvalues of surfaces

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.