25 problems

Kusner–McGrath's conjecture. The first Steklov eigenspace of $S$ coincides with the span of the coordinate functions of the embedding.

Let $\mathcal{C}_{N}$ be the class of convex polygons in $\mathbb{R}^2$ with at most $N$ sides and perimeter $2\pi$, and let $\Omega^*$ be a maximizer of the first nonzero Steklov…

Existence and non-degeneracy conjecture. The supremum is attained by a convex polygon $\Omega^*$ with exactly $N$ sides. In particular, the maximizer cannot be a degenerate polygon…

Monotonicity conjecture. The first non-zero Steklov eigenvalue $\sigma_1(\Omega_{\epsilon})$ is a monotone decreasing function of $\epsilon$ on $[0,1-\delta)$.

Wang–Xia's conjecture. The eigenvalue $\xi_1$ satisfies

Let $\Omega=\mathbb{D}$ be the unit disk, let $\Gamma=\partial\Omega=\mathbb{S}^1$, and let $\mathcal{I}_\alpha^n\subset\mathbb{S}^1$ be the union of $n$ evenly spaced intervals of…

Let $(\Sigma,g)$ be a compact Riemannian surface with boundary that is not diffeomorphic to the disk. Let $\sigma_1(\Sigma,[g])$ denote its first normalized Steklov eigenvalue, and…

Let $\mathbb{D}$ be the disk, let $h_t^+$ denote the corresponding Steklov data for the critical ellipse, let $E_t=\{x^2+t y^2=1\}$, and let $co(E_t)=\{x^2+t y^2\leq 1\}$. Equip…

Let $N_{\gamma,1}$ and $N_{\gamma,2}$ be the basic reflection surfaces and let $\Gamma_{12}$ and $\Gamma_{12}'$ be the reflection groups defined in the source. For a free boundary…

Ellipse-hole eigenvalue monotonicity conjecture. The eigenvalues $\sigma_1(\Omega_{(t_1,t_2)})$ and $\mu_2(\Omega_{(t_1,t_2)})$ decrease with respect to the distance between the ce…

Ball-hole eigenvalue monotonicity conjecture. The first and second nonzero Steklov eigenvalues $\sigma_1(\Omega_{(t_1,t_2)})$ and $\sigma_2(\Omega_{(t_1,t_2)})$ decrease as the dis…

Odd-diameter almost seesaw conjecture.

Hyperbolic Weinstock conjecture. Is it true that

Balanced-tree conjecture. For sufficiently large $\ell$, the tree $T_b^*(\ell,D)$ attains the maximum $\lambda_2$ among all trees in $\mathcal{T}_S(\ell,D)$.

Let $N^2\subset\mathbb{B}^n$ be a free boundary branched minimal immersion by first Steklov eigenfunctions, with $\partial N$ having at least two components. Steklov point-separati…

Let $N$ be an orientable surface with two boundary components, and let $\Psi:(N,g)\to\mathbb{B}^3$ be a free boundary minimal embedding by $\sigma_1(N,g)$-eigenfunctions, with mult…

Rectangular minimizer conjecture. For every $0<C\leq 4\sigma_1(\mathbb{S})$, the solution of this minimization problem is given by a rectangle.

Nonattainment conjecture. The displayed minimization problem has no solution. In particular, every minimizing sequence $\Omega_\epsilon$ must be of the form of collapsing rectangle…

Blaschke–Santaló bounds. For every such $\Omega$,

Let $\lambda_{k,\varepsilon}$ and $\lambda_{k,0}$ denote the $k$-th Steklov eigenvalues, respectively, for the rapidly oscillating boundary weight $\rho_\varepsilon$ and its homoge…

Let $\widehat \sigma_1^*$ denote the supremum of the first nonzero Steklov eigenvalue multiplied by the boundary length, over planar domains of area $\pi$, and let…

Let $(M^n,g)$, $n\geq 2$, be a connected compact smooth Riemannian manifold with boundary. Assume that $\operatorname{Ric}_g\geq 0$ and that the principal curvatures of the boundar…

Extremal Steklov domain conjecture. The maximizer $\Omega^{p\star}$ of $\Lambda_p(\Omega)$ is unique up to dilations and rigid transformations, has $p$-fold symmetry and an axis of…

Let $\Sigma$ be a compact surface with boundary. Write $m_1(\Sigma)$ for the maximal multiplicity of the first Steklov eigenvalue $\sigma_1(\Sigma,g,\rho,\gamma)$ as the metric $g$…

Let $M$ be a compact surface with boundary, and let $\sigma_1(M)$ denote its first nonzero Steklov eigenvalue. Define $\mathrm{Chr}(M,\partial M)$ to be the number of vertices of t…