A sharp lower bound on the small eigenvalues of surfaces

Let S be a compact hyperbolic surface of genus g≥ 2 and let I(S) = 1/Vol(S)∫S 1/Inj(x)² 1 dx, where Inj(x) is the injectivity radius at x. We prove that for any k∈ \1,…, 2g-3\, the k-th eigenvalue λₖ of the Laplacian satisfies {equation*} λ_k ≥ c k^2/I(S) g^2 \, , {equation*} where $c>0$ is some universal constant. We use this bound to prove the heat kernel estimate {equation*} 1/Vol(S) ∫_S | p_t(x,x) -1/Vol(S) | ~dx ≤ C √ I(S)/t ∀ t ≥ 1 \, , {equation*} where C<∞ is some universal constant. These bounds are optimal in the sense that for every g≥ 2 there exists a compact hyperbolic surface of genus g satisfying the reverse inequalities with different constants.