A sharp lower bound on the small eigenvalues of surfaces

Let $S$ be a compact hyperbolic surface of genus $g\ge 2$ and let $I(S)=\frac{1}{\mathrm{V}\mathrm{o}\mathrm{l}(S)}{\int }_S\frac{1}{\mathrm{I}\mathrm{n}\mathrm{j}(x{)}^2\wedge 1}dx$, where $\mathrm{I}\mathrm{n}\mathrm{j}(x)$ is the injectivity radius at $x$. We prove that for any $k\in \{1,\dots ,2g-3\}$, the $k$-th eigenvalue ${\lambda }_k$ of the Laplacian satisfies \begin{equation*} λ_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}{$\mathrm{V}\mathrm{o}\mathrm{l}$(S)} \int_S \Big| p_t(x,x) -\frac{1}{$\mathrm{V}\mathrm{o}\mathrm{l}$(S)} \Big | ~dx \leq C \sqrt{ \frac{I(S)}{t}} \qquad \forall t \geq 1 , , \end{equation*} where $C<\mathrm{\infty }$ is some universal constant. These bounds are optimal in the sense that for every $g\ge 2$ there exists a compact hyperbolic surface of genus $g$ satisfying the reverse inequalities with different constants.