Parametric inequalities and Weyl law for the volume spectrum

TL;DR

The work develops a parametric framework to derive Weyl-type asymptotics for the volume spectrum of a compact manifold by reducing to parametric isoperimetric and coarea inequalities. It provides low-dimensional proofs, establishes approximation and Almgren-type isomorphism results for families of flat cycles, and proves the Weyl law for 1-cycles in 3-manifolds, linking nonlinear min-max widths to classical geometric measure bounds. The approach unifies Morse-theoretic min-max theory with parametric geometric inequalities, enabling a path from variational problems on cycle spaces to quantitative asymptotics driven by volume. These insights advance understanding of the distribution of generalized minimal submanifolds and offer new tools for studying the density and equidistribution of geometric objects in manifolds.

Abstract

We show that the Weyl law for the volume spectrum in a compact Riemannian manifold conjectured by Gromov can be derived from parametric generalizations of two famous inequalities: isoperimetric inequality and coarea inequality. We prove two such generalizations in low dimensions and obtain the Weyl law for 1-cycles in 3-manifolds. We also give a new proof of the Almgren isomorphism theorem.

Figures (3)

• Figure 1:
• Figure 2: $F(x_1)$ and $F(x_2)$ are two cycles with $x_1, x_2 \in C \cap X(q)_0$. We use coarea inequality to cut them in $\Omega \setminus \Omega_{\varepsilon}$. Then we project the part sticking out outside of $\Omega_\varepsilon$ onto $\Sigma = \partial \Omega_{\varepsilon}$
• Figure 3:

Theorems & Definitions (52)

• Theorem 1.1
• Conjecture 1.3
• Conjecture 1.4
• Conjecture 1.6
• Remark 1.7
• Lemma 2.1
• proof
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4