9 problems

Consider partitions of the sphere into three cells, with the objective given by the sum of their first Laplace--Beltrami eigenvalues. Let the $Y$ partition be the partition consist…

Let $k$ be a positive integer, let $\mathcal{Z}\mathcal{V}_k$ be the coset-type-constant subspace, let $\mathfrak{p}_\lambda$ be the $b$-deformed class vector for $\lambda\vdash k$…

Let $M$ be a connected manifold, let $\mathcal{R}(M)$ be the set of Riemannian metrics on $M$, and for each metric let $\Delta_g$ denote its Laplace--Beltrami operator. The strong…

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with piecewise smooth boundary, or a compact Riemannian surface, with or without boundary. Let $\Delta$ be the Laplace–Beltram…

Let $S\subset\mathbb{R}^3$ be a closed surface. Let $u_1$, $u_2$, and $u_3$ be the three first eigenfunctions of the Laplace–Beltrami operator associated with the three first nonva…

Let $N$ be a smooth manifold equipped with a sub-Riemannian structure that is rank-varying or non-equiregular on an embedded hypersurface, and let $\Delta$ denote its Laplace-Beltr…

An almost-Riemannian structure (ARS) is a Riemannian metric that is singular on an embedded smooth hypersurface and smooth on its complement. Let $M$ be the nonsingular part of an…

Let $(\mathbb S^2,g)$ be a Riemannian manifold diffeomorphic to the two-sphere. Denote the discrete spectrum of the Laplace-Beltrami operator by … For $k\geq 1$, define the extrema…

Let $M$ be a closed, smooth, simply connected surface with no holes, and let the second eigenfunction of its Laplace–Beltrami operator be the continuous analogue of a Fiedler vecto…