Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of $f$ under $\sigma$ is approximately Gaussian. Write $\mu$ for the measure whose density with respect to $\sigma$ is $|\nabla f|^2$. We observe that the value distribution of $f$ under $\mu$ admits a unimodal density attaining its maximum at the origin. Thus, in a sense, the zero set of an eigenfunction is the largest of all level sets. When $M$ is a manifold with boundary, the same holds for Laplace eigenfunctions satisfying either the Dirichlet or the Neumann boundary conditions. Additionally, we prove a monotonicity formula for level sets of solid spherical harmonics, essentially by viewing nodal sets of harmonic functions as weighted minimal hypersurfaces.

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拉普拉斯本征函数的单峰值分布和单调性公式

令 $M$ 是一个紧凑的、连通的黎曼流形，其黎曼体积度量由 $\sigma$ 表示。令 $f: M \rightarrow \mathbb{R}$ 是拉普拉斯算子的非常量本征函数。随机波猜想表明，在某些情况下，$\sigma$ 下的 $f$ 值分布近似为高斯分布。为相对于 $\sigma$ 的密度为 $|\nabla f|^2$ 的度量写出 $\mu$。我们观察到 $\mu$ 下 $f$ 的值分布承认单峰密度在原点达到最大值。因此，从某种意义上说，本征函数的零集是所有水平集中最大的。当 $M$ 是有边界的流形时，同样适用于满足 Dirichlet 或 Neumann 边界条件的拉普拉斯特征函数。此外，我们证明了固体球谐函数的水平集的单调性公式，