On filtration formula and nodal sets in min-max theory

Abstract

Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \[ 0=λ_0\leqλ_1\leq\ldots \] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{ϕ_j\}_{j\geq0}$. Let $\{ω_p(M,g)\}_{p\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured that the sweepout generated by the nodal sets of $ϕ_0,\ldots,ϕ_p$ is asymptotically optimal for $ω_p(M,g)$ as $p\to\infty$. We show that this conjecture is false in every dimension, even for positively curved real-analytic metrics. Nevertheless, we show that a less rigid notion of nodal sweepouts recovers the volume spectrum. Let \[ E_N(M,g):=\operatorname{span}\{ϕ_0,\ldots,ϕ_N\}, \qquad a_N\in H^1\bigl(\mathbb P(E_N(g));\mathbb Z_2\bigr) \] be the generator, and for $p\geq1$ define \[ \mathcal N_p^N(g) := \left\{ Ψ\mid X\text{ is a finite complex},\ Ψ\in C\bigl(X,\mathbb P(E_N(g))\bigr),\ Ψ^*(a_N^p)\neq0 \right\}. \] We define the nodal $(p,N)$-width by \[ ν_p^N(M,g) := \inf_{Ψ\in\mathcal N_p^N(g)} \sup_{x\in\operatorname{dmn}(Ψ)} \mathcal H_g^{n}\bigl(\{Ψ_x=0\}\bigr). \] We prove the following filtration formula \[ ω_p(M,g) = \lim_{N\to\infty}ν_p^N(M,g) \] for every $p\geq1$. We state several questions on nodal geometry and min-max theory.