Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polygons and multi-product of eigenfunctions

Emmett L. Wyman, Yakun Xi, Yi Zhang

TL;DR

The paper studies the $L^2$-mass distribution of multi-products of Laplace-Beltrami eigenfunctions by expanding the product spectrally and linking the resulting coefficients to geometric configurations of $(k+1)$-gons with prescribed side lengths given by the frequencies. It introduces the notions of $k$-good and $k$-bad cones via polygon inequalities and uses Fourier integral operator methods to obtain sharp asymptotics for the joint spectral measure, revealing that the main term is governed by the Leray volume of the configuration space $F^{-1}(\tau)$ of closed polygons. The key contributions are (i) a rapid decay result in the $k$-bad regime and (ii) a precise leading-term description in the $k$-good regime, with a scaling law $\operatorname{vol}F^{-1}(r\tau)=r^{d}\operatorname{vol}F^{-1}(\tau)$ where $d= k(n-1)-1$. The flat-torus example demonstrates the sharpness of the polygon-based cutoff and underscores the geometric-analytic bridge between spectral coefficients and polygon configurations, with potential implications for understanding high-frequency behavior of eigenfunction products on manifolds.

Abstract

Let $M$ be a compact Riemannian manifold without boundary, with $L^2$-normalized Laplace-Beltrami eigenfunctions $\{e_j\}_j$, which satisfy $Δ_g e_j = -λ_j^2 e_j$. We study the following inner product of eigenfunctions \[ \langle e_{i_1} e_{i_2} \ldots e_{i_k}, e_{i_{k+1}} \rangle = \int e_{i_1} e_{i_2}\ldots e_{i_k} \overline{e_{i_{k+1}}} \, dV. \] We show that, after a mild averaging in the frequency variables, the main $\ell^2$-concentration of this inner product is determined by the measure of a set of configurations of $(k+1)$-gons whose side lengths are the frequencies $λ_{i_1}, λ_{i_2}, \dots, λ_{i_{k+1}}$. We prove that a rapidly vanishing proportion of this mass lies in the regime where $λ_{i_1}, λ_{i_2}, \dots, λ_{i_{k+1}}$ cannot occur as the side lengths of any $(k+1)$-gon.

Polygons and multi-product of eigenfunctions

TL;DR

The paper studies the -mass distribution of multi-products of Laplace-Beltrami eigenfunctions by expanding the product spectrally and linking the resulting coefficients to geometric configurations of -gons with prescribed side lengths given by the frequencies. It introduces the notions of -good and -bad cones via polygon inequalities and uses Fourier integral operator methods to obtain sharp asymptotics for the joint spectral measure, revealing that the main term is governed by the Leray volume of the configuration space of closed polygons. The key contributions are (i) a rapid decay result in the -bad regime and (ii) a precise leading-term description in the -good regime, with a scaling law where . The flat-torus example demonstrates the sharpness of the polygon-based cutoff and underscores the geometric-analytic bridge between spectral coefficients and polygon configurations, with potential implications for understanding high-frequency behavior of eigenfunction products on manifolds.

Abstract

Let be a compact Riemannian manifold without boundary, with -normalized Laplace-Beltrami eigenfunctions , which satisfy . We study the following inner product of eigenfunctions We show that, after a mild averaging in the frequency variables, the main -concentration of this inner product is determined by the measure of a set of configurations of -gons whose side lengths are the frequencies . We prove that a rapidly vanishing proportion of this mass lies in the regime where cannot occur as the side lengths of any -gon.
Paper Structure (8 sections, 12 theorems, 132 equations, 2 figures)

This paper contains 8 sections, 12 theorems, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Polygons
  4. Main results
  5. Order of the main term in Theorem \ref{['main theorem k+1-good']}
  6. Proof of Theorem \ref{['main theorem k+1-gon-bad']}
  7. The proof of Theorem \ref{['main theorem k+1-good']}
  8. A flat-torus example

Key Result

Theorem 1.2

For each $\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\epsilon, N}$ such that for any positive numbers $\tau_{1},\tau_{2},\ldots,\tau_{k}$, we have where $\lambda_{i_j} \leq \tau_j$ for $1 \leq j \leq k$.

Figures (2)

  • Figure 1: An example of $\tau\notin\Gamma_g$.
  • Figure 2: A pentagon with side lengths $a_1, \ldots, a_5$.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1: Scaling of $F$
  • proof
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['main theorem k+1-gon-bad']}
  • proof : Proof of Lemma \ref{['lem:kplus1-eigenfunction-product']}
  • Proposition 4.1
  • Lemma 4.2
  • ...and 12 more