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A sharp lower bound on the small eigenvalues of surfaces

Renan Gross, Guy Lachman, Asaf Nachmias

TL;DR

This work proves a sharp, universal lower bound for the small Laplacian eigenvalues on compact hyperbolic surfaces in terms of the genus $g$ and the geometric bottleneck quantity $I(S)$. Central to the approach is a geometric-extremal-length framework: by conformally stretching collars to cylinders and bounding the extremal length between small discs via a reciprocal-injectivity-weight, the authors bound the Dirichlet energy of a spectral kernel and derive a quantitative eigenvalue bound $\lambda_k \ge \frac{c k^2}{I(S) g^2}$. This eigenvalue control yields a global heat-kernel estimate on average, $\frac{1}{\mathrm{Vol}(S)} \int_S |p_t(x,x) - 1/\mathrm{Vol}(S)|dx \le C \sqrt{\frac{I(S)}{t}}$ for all $t\ge 1$, with universal constants; both results are shown to be optimal in the sense that reversed inequalities hold with different constants on certain surfaces. The paper blends extremal-length techniques with spectral kernel methods and thin-thick geometry to connect injectivity-radius bottlenecks to spectral data, yielding robust, geometry-driven bounds applicable across all genera.

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 $λ_k$ 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{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.

A sharp lower bound on the small eigenvalues of surfaces

TL;DR

This work proves a sharp, universal lower bound for the small Laplacian eigenvalues on compact hyperbolic surfaces in terms of the genus and the geometric bottleneck quantity . Central to the approach is a geometric-extremal-length framework: by conformally stretching collars to cylinders and bounding the extremal length between small discs via a reciprocal-injectivity-weight, the authors bound the Dirichlet energy of a spectral kernel and derive a quantitative eigenvalue bound . This eigenvalue control yields a global heat-kernel estimate on average, for all , with universal constants; both results are shown to be optimal in the sense that reversed inequalities hold with different constants on certain surfaces. The paper blends extremal-length techniques with spectral kernel methods and thin-thick geometry to connect injectivity-radius bottlenecks to spectral data, yielding robust, geometry-driven bounds applicable across all genera.

Abstract

Let be a compact hyperbolic surface of genus and let , where is the injectivity radius at . We prove that for any , the -th eigenvalue of the Laplacian satisfies \begin{equation*} λ_k \geq \frac{c k^2}{I(S) g^2} \, , \end{equation*} where 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 is some universal constant. These bounds are optimal in the sense that for every there exists a compact hyperbolic surface of genus satisfying the reverse inequalities with different constants.
Paper Structure (14 sections, 15 theorems, 108 equations, 3 figures)

This paper contains 14 sections, 15 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Discrete analogue
  4. Extremal length
  5. Geometric preliminaries
  6. Extremal length and the inverse injectivity radius
  7. A conformal change of metric
  8. Proof of
  9. Proof of the eigenvalues lower bound
  10. Proof of
  11. Proof of the gradient upper bound
  12. Heat kernel bound
  13. Proof of
  14. Sharpness

Key Result

Theorem 1

There exists a universal constant $c>0$ such that for any compact hyperbolic surface ${S}$ of genus $g\geq 2$ and every $k\in \{1, \ldots, 2g-3\}$,

Figures (3)

  • Figure 1: The domain $D$ as a subset of $\tilde{{S}}$. Note that the uniform thickness in the middle part is guaranteed by the injectivity radius lower bound of $\tilde{{S}}$. The paths $\gamma_q$ start at distances in $[-\alpha r_x/\widehat{\mathrm{Inj}}\left({x} \right), \alpha r_x/\widehat{\mathrm{Inj}}\left({x} \right)]$ from $x$, fan out to a spread of $\alpha$, then fan back towards $y$.
  • Figure 2: The domain $D$ as a subset of $\tilde{{S}}$. In this case there is no middle part corresponding to the main term $\mathrm{dist}_w\left(x, y \right)$ in \ref{['eq:main_extremal_length']}.
  • Figure 3: A schematic of $S$ for $n=6$.

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Remark
  • Definition 3: Extremal length
  • Definition 4: Reciprocal-injectivity weight
  • Theorem 5
  • Remark 6
  • Lemma 7: Collar lemma
  • Proposition 8: Injectivity radius estimate
  • proof
  • ...and 23 more