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A Weyl's Law for Singular Riemannian Foliations with Applications to Invariant Theory

Samuel Lin, Ricardo A. E. Mendes, Marco Radeschi

TL;DR

The paper develops a Weyl-type asymptotic for the basic spectrum of closed singular Riemannian foliations with basic mean curvature, proving N(t) ~ ${\operatorname{vol}(X)\omega_m\over (2\pi)^m} t^{m/2}$ for the leaf space X = M/\mathcal{F} (m = dim X). In the special case M = \mathbb{S}^n, it yields a formula expressing Vol(\mathbb{S}^n/\mathcal{F}) as a rational multiple of Vol(\mathbb{S}^m$, linked to the algebra of basic polynomials via the Laplacian algebra A and its Hilbert series H(z). The approach fuses averaging over leaves, slice-theoretic local models, Hörmander asymptotics, heat-kernel analysis, and Tauberian theorems, with an invariant-theory backbone showing A is Cohen–Macaulay and finitely generated, and connecting spectral data to Hilbert-series coefficients. The paper also provides explicit, representation-theoretic analogues and examples (e.g., Clifford foliations, Hopf fibrations) and discusses generalizations to manifold submetries. This work sharpens the understanding of spectral geometry on singular foliations and their quotient spaces, with implications for invariant theory and geometric analysis on singular spaces.

Abstract

We prove a version of Weyl's Law for the basic spectrum of a closed singular Riemannian foliation $(M,\mathcal{F})$ with basic mean curvature. In the special case of $M=\mathbb{S}^n$, this gives an explicit formula for the volume of the leaf space $\mathbb{S}^n/\mathcal{F}$ in terms of the algebra of basic polynomials. In particular, $\operatorname{Vol}(\mathbb{S}^n/\mathcal{F})$ is a rational multiple of $\operatorname{Vol}(\mathbb{S}^m)$, where $m=\dim (\mathbb{S}^n/\mathcal{F})$.

A Weyl's Law for Singular Riemannian Foliations with Applications to Invariant Theory

TL;DR

The paper develops a Weyl-type asymptotic for the basic spectrum of closed singular Riemannian foliations with basic mean curvature, proving N(t) ~ for the leaf space X = M/\mathcal{F} (m = dim X). In the special case M = \mathbb{S}^n, it yields a formula expressing Vol(\mathbb{S}^n/\mathcal{F}) as a rational multiple of Vol(\mathbb{S}^m$, linked to the algebra of basic polynomials via the Laplacian algebra A and its Hilbert series H(z). The approach fuses averaging over leaves, slice-theoretic local models, Hörmander asymptotics, heat-kernel analysis, and Tauberian theorems, with an invariant-theory backbone showing A is Cohen–Macaulay and finitely generated, and connecting spectral data to Hilbert-series coefficients. The paper also provides explicit, representation-theoretic analogues and examples (e.g., Clifford foliations, Hopf fibrations) and discusses generalizations to manifold submetries. This work sharpens the understanding of spectral geometry on singular foliations and their quotient spaces, with implications for invariant theory and geometric analysis on singular spaces.

Abstract

We prove a version of Weyl's Law for the basic spectrum of a closed singular Riemannian foliation with basic mean curvature. In the special case of , this gives an explicit formula for the volume of the leaf space in terms of the algebra of basic polynomials. In particular, is a rational multiple of , where .
Paper Structure (26 sections, 21 theorems, 94 equations, 1 figure)

This paper contains 26 sections, 21 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular Riemannian foliations
  4. Basic definitions and facts.
  5. Invariant Theory.
  6. Principal part.
  7. Tubular neighborhoods and the Slice Theorem.
  8. Averaging operator.
  9. Hörmander's estimate
  10. Basic Laplacian
  11. Heat Kernels
  12. Tauberian and Abelian Theorem
  13. Geometric Estimate
  14. Reductions
  15. Sasaki metrics
  16. ...and 11 more sections

Key Result

Theorem 2

BF02391913 Let $p$ be the principal symbol of an elliptic differential operator $P$ of order $k$ and let $d\xi$ be the measure induced by $d\mu$ on each fiber of the cotangent bundle $T^{*}\Omega$. Define where $B_x=\{\xi\in T_{x}^{*}\Omega\,|\, p(\xi)<1\}$. Then on every compact subset of $\Omega$, $R(x, \lambda)=O(\lambda^{-1/m})$ uniformly as $\lambda \to \infty$.

Figures (1)

  • Figure 1:

Theorems & Definitions (56)

  • Example 1
  • Theorem 2
  • Lemma 3
  • Remark 4
  • Proposition 5
  • proof
  • Proposition 6
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 46 more