22 problems

Berger's conjecture. The flat equilateral torus is the maximizer of the first Laplace eigenvalue among metrics on the torus.

Let $\mathcal M$ denote the parameter space of conformal classes represented by the flat metrics $g_{a,b}$, and let $g_{a,b}^{1,1,0}$ be the rotationally symmetric critical metric…

Quarter-sphere spectral conjecture. One has

Let $\Pi:{\cal{P}}\to{\Bbb S}^2$ be a degree-two branched covering ramified at six points, and let ${\cal{P}}$ be the resulting Bolza surface. Let $g_0$ be the pullback of the roun…

Let $X$ be a compact hyperbolic surface, with first positive Laplace eigenvalue denoted by … and volume by $\operatorname{Vol}(X)$. Buser's conjecture. As $\operatorname{Vol}(X)$ t…

Let $\mathbb{S}^n$ be the $n$-dimensional round sphere with round metric $h$, let $\Lambda_k(\mathbb{S}^n,[h])$ denote the supremum of the normalized $k$-th Laplace eigenvalue over…

Let $B$ denote the Bolza surface, and for a closed hyperbolic surface $S$ of genus $2$, let $m_1(S)$ be the multiplicity of its first positive Laplace eigenvalue. Strohmaier's conj…

Let $g_0$ be a round metric of curvature one on $\mathbb{S}^2$. Given $k\geq 2$, let $\mu$ be an admissible measure such that $\lambda_k(\mathbb{S}^2,\mu)=2$. Then there exists a c…

Consider the family of ellipses … For each such ellipse, consider the multiplicities of the eigenvalues of its Laplacian. Generic multiplicity-two conjecture. For a generic class o…

Rayleigh's spectral conjecture. Among all open sets $\Omega\subset\mathbb R^2$ with a fixed area, the disk minimizes $\lambda_{1,2}(\Omega)$.

Let $M$ be a closed orientable Riemannian surface of genus $\gamma$, and let $\Lambda_1(M)$ denote the supremum of the normalized first Laplace eigenvalue over all Riemannian metri…

Let $(\Sigma^2,g)$ be a closed Riemannian surface of genus $\gamma$, and let $\lambda_k(\Sigma^2,g)$ denote its $k$th Laplace eigenvalue and $\operatorname{Vol}_g(\Sigma^2)$ its vo…

Let $\mathcal{D}$ range over planar $n$-polygons of fixed area $\mathcal{A}$, and let $\lambda_1^{\mathrm{D}}(\mathcal{D})$ denote the first Dirichlet eigenvalue. Pólya's regular p…

Let $\Lambda_k(\mathbb{RP}^2)$ denote the supremum of the normalized $k$-th Laplace eigenvalue over metrics on the real projective plane. A sequence of metrics is said to degenerat…

Let $\mathbb{KL}$ be the Klein bottle, and let $\tilde{\tau}_{3,1}$ denote the bipolar Lawson surface equipped with its induced metric. For a Riemannian metric $g$ on $\mathbb{KL}$…

Jin–Li–Liu–Nadirashvili conjecture. There exists a metric with $6$ conical singularities on the Bolza surface that achieves $\Lambda_1(2)$. The conjecture concerns whether an extre…

Let $M$ be an orientable surface. For a conformal class of Riemannian metrics on $M$, consider the supremum of the first nonzero Laplace eigenvalue, with the usual scale-invariant…

Let $\Omega$ be a simply connected plane domain with area $A$, perimeter $L$, and first Dirichlet Laplace eigenvalue $\lambda_1$. Let $j_{0,1}$ be the first positive zero of the Be…

Let $\Omega$ be a bounded convex plane domain of area $A$ and moment of inertia $I$, and let $\mu_2,\ldots,\mu_n$ be its positive Neumann Laplace eigenvalues. Neumann sum conjectur…

Let $\Omega$ be a bounded convex plane domain, with area $A$, moment of inertia $I$, and first Dirichlet Laplace eigenvalue $\lambda_1$. Convex-domain bounds conjecture. … Equality…

Let $D$ be the square with vertices $(\pm1,0)$ and $(0,\pm1)$, and let $T$ be a piecewise linear map obtained from linear maps $T_+$ and $T_-$ on the upper and lower halfplanes tha…

Let $\Omega$ be a convex plane domain of area $A$ and moment of inertia $I$, and let $\lambda_1$ be its first Dirichlet Laplace eigenvalue. The normalized quantity $\lambda_1A^3/I$…