43 problems

Let $(\Omega^{n+1},g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$. Assume that the Ricci curvature of $\Omega$ satisfies…

Let $\Sigma\subset\mathbb{B}^n$ be an embedded free boundary minimal hypersurface, and let $\sigma_1(\Sigma)$ denote its first Steklov eigenvalue. The coordinate functions show tha…

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…

Karpukhin–Medvedev conjecture. If $T>T_1$, the value of $\Sigma_1(\mathbb{A},[g_T])$ is achieved by an explicitly described rotationally symmetric metric, whereas if $T<T_1$, the…

Let $\Omega\subset\mathbb{R}^{2}$ be a simply connected domain in the plane, and consider the Steklov eigenvalue problem … with domains normalized to have fixed area. Kao and Ostin…

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.

Let $S$ be a surface of genus $\mathfrak{g}$ with $l$ boundary components, and let … where the supremum is over all Riemannian metrics $g$ and strictly positive smooth density func…

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…

Conjecture on ellipse eigenvalue indices and bifurcation branches. One has $k_1(p)=1$ for every $0\leq p\leq 1$, and a new branch of non-planar free-boundary minimal surfaces is cr…

Payne's conjecture. For every smooth exterior domain $U$,

Weinstock's conjecture. The sharp isoperimetric inequality above, and hence the Weinstock inequality

For an integer $p\geq 2$, let $\mathcal{D}_p$ be the class of simply connected planar domains with Lipschitz boundary and rotational symmetry of order $p$, equipped with rotational…

Let $p$ be a positive integer, and consider simply connected planar domains $\Omega$ normalized to have area one. Denote by $\Omega_p^*$ an area-constrained maximizer of the $p$th…