30 problems

Let $\mathbb{B}^3$ be the Euclidean unit ball, and let an embedded free boundary minimal annulus be an embedded minimal annulus whose boundary meets $\partial\mathbb{B}^3$ orthogon…

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 $\Sigma^3\subset B^4\subset\mathbb{R}^4$ be a capillary minimal hypersurface in the unit ball; the free boundary case corresponds to contact angle $\pi/2$. Assume that … for al…

Let $R\in[0,\frac{\pi}{2}]$ and let $\mathbb{M}^3_{\kappa_R}$ be the space form of curvature $\kappa_R=\sin^2 R$. An embedded free boundary minimal annulus in…

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…

Let an ellipsoid be a smooth ellipsoidal domain in $\mathbb{R}^3$, and let $\mathbb{D}_2$ denote the dihedral symmetry group acting on it. A free boundary minimal surface is a mini…

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 $B$ be any geodesic ball in $\mathbb{S}^3_+$ or $\mathbb{H}^3$. Geodesic-ball realization conjecture. The ball $B$ admits a free boundary minimal surface of every prescribed or…

Let $N_{\gamma,k}$ be the compact orientable surface of genus $\gamma$ with $k$ boundary components. Embedding-growth conjecture. The number of distinct embeddings of…

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…

Let $\mathcal{A}(\gamma,k)$ be the minimum area of a free boundary minimal surface in $\mathbb{B}^3$ with genus $\gamma$ and $k$ boundary components. Critical catenoid area conject…

Let $N'$ be a degeneration realized by a free boundary minimal embedding $N'\to \mathbb{B}^3$ by first eigenfunctions, with boundary given by the intersection of $\mathbb{S}^2$ and…

Let $N$ be a compact orientable surface with boundary, and let $\bar{\sigma}_1$ denote its first normalized Steklov eigenvalue. The source considers the possibility that a degenera…

Let $\Sigma$ be an immersed free boundary minimal surface in a three-dimensional geodesic ball in either $\mathbb S^n_+$ or $\mathbb H^n$, and let $\operatorname{Ind}(\Sigma)$ deno…

Let a three-dimensional geodesic ball be a geodesic ball in either $\mathbb S^n_+$ or $\mathbb H^n$, and let a critical spherical catenoid be the corresponding rotational free boun…

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…

Let $\mathbb B^3$ be the Euclidean unit $3$-ball, and let $\mathbb K$ denote the critical catenoid. The second least area means the second value in the area spectrum of immersed fr…

Let $\Sigma$) be the free boundary minimal surface in the Euclidean unit $3$-ball supplied by Theorem 1.1: it is embedded, has genus $0$ or $1$, Morse index $4$ or $5$, and area in…

Let $b\geq3$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Maximal-are…

Let $n\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $g\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $g\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $k$ and $l$ be relatively prime integers with $k>l>0$, and let $\widetilde{IFS}_{k,l}$ denote the orientable cover of the Fraser-Sargent free-boundary minimal surface…

Let $\mathbb B^4$ be the $4$-dimensional Euclidean ball, and let the critical Möbius band be the embedded non-orientable free boundary minimal surface in $\mathbb B^4$ whose index…

For any $T\geqslant T_1$, let $\mathbb{A}_T=[0,T]\times\mathbb{S}^1$ and let $\mathcal{C}_T$ be the conformal class containing the flat metric on $[0,T]\times\mathbb{S}^1$. Let…