A sharp lower bound on the small eigenvalues of surfaces

TL;DR

This paper establishes a sharp lower bound on small eigenvalues of the Laplacian for hyperbolic surfaces, linking eigenvalues to geometric properties, and derives related heat kernel estimates, demonstrating optimality through constructed examples.

Contribution

It provides a novel universal lower bound on small Laplacian eigenvalues of hyperbolic surfaces based on geometric invariants, with implications for heat kernel behavior.

Findings

Lower bound on eigenvalues proportional to k^2 / (I(S) g^2)

Heat kernel deviation bounded by a constant times sqrt(I(S)/t)

Existence of surfaces where bounds are tight and optimal

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.