12 problems

Let $h_0$ and $h_1$ be the positive heights from part (2) of Theorem, where the ordering of the symmetric and antisymmetric second Neumann eigenvalues changes as the height varies.…

Let $M^n_\kappa$ be the simply connected $n$-dimensional Riemannian manifold of constant sectional curvature $\kappa\in\{-1,1\}$, and let $\Omega\subset M^n_\kappa$ be a bounded co…

Let $P$ be a path graph with $n$ vertices, and let $\lambda_{n-2}(P)$ denote its largest Neumann eigenvalue. Path-graph Neumann eigenvalue conjecture. The largest Neumann eigenvalu…

Ba{n}uelos and Burdzy's conjecture. If $\mu_1(\Omega)$ has multiplicity $2$, then

Neumann monotonicity with volume scaling. The function

Higher-dimensional Neumann monotonicity conjecture. The function

Let $\Omega$ be a convex domain in $\mathbb{R}^2$, let $L(\Omega)$ denote its perimeter, and let $\mu_2(\Omega)$ be its first nonzero Neumann eigenvalue. Laugesen's conjecture. … E…

Let $\Omega$ be a symmetric domain in $\mathbb{R}^2$ with a line of symmetry. Let $\mu_2$ be the second Neumann Laplacian eigenvalue, and suppose that all modes associated with…

Let $\Omega$ be a convex domain in $\mathbb{R}^2$ such that $\kappa_1=\mu_2$. Here $\kappa_1$ denotes the first eigenvalue of the variational problem defined in the paper, $\mu_2$…

Let $\Omega\subset\mathbb{R}^2$ be a planar convex domain, let $P(\Omega)$ denote its perimeter, and let $\mu_1(\Omega)$ denote the second Neumann eigenvalue of the Laplacian. Laug…

Let $\Omega$ be a convex domain of given diameter $D$, and let $0=\mu_1(\Omega)<\mu_2(\Omega)\leq\mu_3(\Omega)\leq\cdots$ be the Neumann Laplacian eigenvalues. Disk minimization co…

Polar-dual Neumann eigenvalue sum conjecture. The scale-invariant quantity