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112 years of listening to Riemannian manifolds

Gustav Mårdby, Julie Rowlett

Abstract

In 1910, Hendrik Antoon Lorentz delved into the enigmatic Laplace eigenvalue equation, also known as the Helmholtz equation, pondering to what extent the geometry in which one solves the equation can be recovered from knowledge of the eigenvalues. Lorentz, inspired by physical and musical analogies, conjectured a fundamental relationship between eigenvalues, domain volume, and dimensionality. While his conjecture initially seemed insurmountable, Hermann Weyl's groundbreaking proof in 1912 illuminated the deep connection between eigenvalues and geometric properties. Over the ensuing 112 years, mathematicians and physicists have continued to decipher the intricate interplay between eigenvalues and geometry. From Weyl's law to Milnor's example of isospectral non-isometric flat tori, and Kac's inspiring question about hearing the shape of a drum, the field has witnessed remarkable progress, uncovering spectral invariants and advancing our understanding of geometric properties discernible through eigenvalues. We present an overview of this field amenable to both physicists and mathematicians.

112 years of listening to Riemannian manifolds

Abstract

In 1910, Hendrik Antoon Lorentz delved into the enigmatic Laplace eigenvalue equation, also known as the Helmholtz equation, pondering to what extent the geometry in which one solves the equation can be recovered from knowledge of the eigenvalues. Lorentz, inspired by physical and musical analogies, conjectured a fundamental relationship between eigenvalues, domain volume, and dimensionality. While his conjecture initially seemed insurmountable, Hermann Weyl's groundbreaking proof in 1912 illuminated the deep connection between eigenvalues and geometric properties. Over the ensuing 112 years, mathematicians and physicists have continued to decipher the intricate interplay between eigenvalues and geometry. From Weyl's law to Milnor's example of isospectral non-isometric flat tori, and Kac's inspiring question about hearing the shape of a drum, the field has witnessed remarkable progress, uncovering spectral invariants and advancing our understanding of geometric properties discernible through eigenvalues. We present an overview of this field amenable to both physicists and mathematicians.

Paper Structure

This paper contains 15 sections, 16 theorems, 166 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Weyl's asymptotic law
  3. Preliminaries
  4. Weyl's law
  5. Listening to flat tori
  6. One cannot hear the shape of a Riemannian manifold but can one hear the shape of a drum?
  7. Geometric preliminaries
  8. What could Kac hear?
  9. Hot preliminaries
  10. Kac's principle of not feeling the boundary and the leading order term
  11. Kac's principle of feeling the boundary and the $t^{-1/2}$-term
  12. Kac's holes and how to fill them
  13. One cannot hear...
  14. One cannot hear the shape of a drum, but one can hear...
  15. We're still listening...

Key Result

Proposition 2.2

Let $I = (a_1, b_1) \times (a_2, b_2) \subset \mathbb{R}^2$ be a rectangle. Then both the Dirichlet and Neumann eigenvalues on $I$ satisfy

Figures (7)

  • Figure 1: This domain $\Omega$ is a rectangle with side lengths $l_1$ and $l_2$.
  • Figure 2: A quarter ellipse whose radii are proportional to $R$. Each lattice point in its interior corresponds to a unit square. The total area of the squares and the area of the quarter ellipse both grow on the order $R^2$ as $R \to \infty$, while the error is of the order $R$. Hence the number of lattice points in the interior of the quarter ellipse is asymptotically equal to its area as $R \to \infty$.
  • Figure 3: The domain $\Omega$ has an inner and outer covering consisting of finitely many rectangles. The rectangles contained in the interior are marked with a cross.
  • Figure 4: An infinite sector with opening angle $\theta \in (\pi/2, \pi)$.
  • Figure 5: The polygon is split up into sectors which are close to the vertices, and regions close to the boundary but away from the vertices.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5: Weyl's law
  • proof
  • Lemma 2.6
  • proof : Proof of lemma \ref{['lemma:weyl_inequalities']}
  • Theorem 3.1: nilsson2022isospectral, Thm. 2.8
  • ...and 13 more