On filtration formula and nodal sets in min-max theory

Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let $$0={\lambda }_0\le {\lambda }_1\le \dots$$ be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{{\phi }_j{\}}_{j\ge 0}$. Let $\{{\omega }_p(M,g){\}}_{p\ge 1}$ denote Gromov’s volume spectrum. In 2014, Marques-Neves conjectured that the sweepout generated by the nodal sets of ${\phi }_0,\dots ,{\phi }_p$ is asymptotically optimal for ${\omega }_p(M,g)$ as $p\to \mathrm{\infty }$. 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 $$E_N(M,g):=\mathrm{span}\{{\phi }_0,\dots ,{\phi }_N\},a_N\in H^1(\mathbb{P}(E_N(g));{\mathbb{Z}}_2)$$ be the generator, and for $p\ge 1$ define $${\mathcal{N}}_p^N(g):=\{\mathrm{\Psi }\mid X\text{ is a finite complex},\mathrm{\Psi }\in C(X,\mathbb{P}(E_N(g))),{\mathrm{\Psi }}^{\ast }(a_N^p)\mathrm{\ne }0\}\mathrm{.}$$ We define the nodal $(p,N)$-width by $${\nu }_p^N(M,g):=\underset{\mathrm{\Psi }\in {\mathcal{N}}_p^N(g)}{\inf }\underset{x\in \mathrm{dmn}(\mathrm{\Psi })}{\sup }{\mathcal{H}}_g^n(\{{\mathrm{\Psi }}_x=0\})\mathrm{.}$$ We prove the following filtration formula $${\omega }_p(M,g)=\underset{N\to \mathrm{\infty }}{\lim }{\nu }_p^N(M,g)$$ for every $p\ge 1$. We state several questions on nodal geometry and min-max theory.