AI 中文总结
本文证明 Marques-Neves 关于节点集扫除渐近最优性的猜想在任意维度均不成立，但引入较宽松的节点扫除概念，证明其过滤公式可恢复 Gromov 体积谱，并提出了相关问题。
AI 中文摘要
设 $(M^{n+1},g)$ 为闭光滑黎曼流形，令 $0=\lambda_0\leq\lambda_1\leq\ldots$ 为 Laplace-Beltrami 算子的谱，对应的实特征函数为 $\{\phi_j\}_{j\geq0}$。设 $\{\omega_p(M,g)\}_{p\geq1}$ 表示 Gromov 体积谱。2014 年，Marques-Neves 猜想由 $\phi_0,\ldots,\phi_p$ 的节点集生成的扫除在 $p\to\infty$ 时对 $\omega_p(M,g)$ 渐近最优。我们证明该猜想在每一维度均不成立，即使对于正曲率实解析度量亦然。然而，我们证明一种较不严格的节点扫除概念可恢复体积谱。令 $E_N(M,g):=\operatorname{span}\{\phi_0,\ldots,\phi_N\}$，$a_N\in H^1\bigl(\mathbb P(E_N(g));\mathbb Z_2\bigr)$ 为生成元，并对 $p\geq1$ 定义 $\mathcal N_p^N(g):= \left\{ \Psi \mid X\text{ 为有限复形},\\ \Psi\in C\bigl(X,\mathbb P(E_N(g))\bigr),\\ \Psi^*(a_N^p)\neq0 \right\}$。我们定义节点 $(p,N)$-宽度为 $\nu_p^N(M,g):= \inf_{\Psi\in\mathcal N_p^N(g)} \sup_{x\in\operatorname{dmn}(\Psi)} \mathcal H_g^{n}\bigl(\{\Psi_x=0\}\bigr)$。我们证明以下过滤公式 $\omega_p(M,g) = \lim_{N\to\infty}\nu_p^N(M,g)$ 对所有 $p\geq1$ 成立。我们提出若干关于节点几何与极小极大理论的问题。
英文摘要
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.
Comments36 pages. Comments are welcome!
