480 problems

Let $\mathbb{B}^3$ be the unit ball in $\mathbb{R}^3$. A free boundary minimal annulus is a genus-zero free boundary minimal surface with two boundary components. Two such surfaces…

Let $T\subset S^3$ be an embedded minimal torus, and let $T_{\mathrm{Cl}}$ denote the Clifford torus in $S^3$. Lawson's conjecture. There exists an isometry $F$ of $S^3$ such that…

Let $\xi_{g,1}\subset S^3$ be the Lawson minimal surface of genus $g$ indexed by $(g,1)$, and consider its stereographic projection to $\mathbb{R}^3$. Kusner's conjecture. For surf…

Let $G$ be a split real semisimple Lie group, let $X$ be its associated symmetric space, and let $\rho$ lie in a Hitchin component of representations of $\pi_1(\Sigma_g)$ into $G$.…

Let $g>0$, and consider surfaces of genus $g$ in $\mathbb{CP}^2$. The Clifford torus conjecture asserts that the Willmore functional is minimized by the Clifford torus … This is th…

A minimal graph is a hypersurface given by the graph of a function from Euclidean space to the real line, and such graphs are automatically locally area-minimizing. Bernstein's con…

Let $M$ be a compact, embedded minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$, and let $\lambda_1$ denote the first non-trivial eigenvalue of its Laplacian. Yau's conje…

Let $M$ be an embedded minimal surface in the unit sphere $\mathbb{S}^n$, and let $\lambda_1(M)$ denote its first non-zero Laplacian eigenvalue. Yau's conjecture. One has … The con…

Let $M$ be a complete embedded $δ$-stable minimal surface in a complete flat three-manifold, where $δ$-stability means that … for every $\varphi\in C_0^\infty(M)$. Meeks–Pérez–Ros…

Let a strongly irreducible Heegaard surface lie in a Riemannian three-manifold. Pitts–Rubinstein conjecture. It is either isotopic to a minimal surface of index at most $1$, or iso…

Let $S$ be a complete minimal surface embedded in $\mathbb{R}^3$ and of finite total curvature. If $S$ is neither a plane nor a catenoid, let $n$ be its number of ends and let…

Let $M$) be a closed surface minimally immersed in $\mathbb{S}^N$ such that its image is not contained in any hyperplane of $\mathbb{R}^{N+1}$. Set $S=|h|^2$, where $h$ is the seco…

Sullivan's and Schoen–Yau's conjectures. Every proper minimal surface in $\mathbb{R}^3$ having finite topology, as conjectured by Sullivan, or having a proper projection into a pla…

Let $M\subset\mathbb{R}^3$ be a connected, complete embedded minimal surface of finite genus with compact, possibly empty, boundary. Embedded Calabi–Yau conjecture for finite genus…

Let $\mathbb S^3$ be the 3-sphere, and let a rotational torus mean a torus invariant under a rotational symmetry of $\mathbb S^3$. Pinkall–Sterling conjecture. The rotational torus…

Ros's conjecture. Any open orientable surface can be properly embedded in ${\mathbb H}^2\times {\mathbb R}$ as a minimal surface.

Let $(M^3,g)$ be a Riemannian three-manifold with scalar curvature $R_M\geq 0$, and let $\Sigma\subset M$ be a two-sided two-torus. A surface is locally area-minimizing if it minim…

Lawson–Osserman conjecture. Every Lipschitz weak solution of the minimal graph system should satisfy the full geometric Euler–Lagrange condition of graph stationarity.

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 $A$ be a subregion of the conformal boundary of an Anti-de Sitter spacetime, let $\text{CFT}$ denote the associated conformal field theory, and let $\Sigma$ be the minimal surf…

Let $n\geq 3$. A minimal hypersurface is a hypersurface in Euclidean space $\mathbb{R}^n$ whose mean curvature vanishes. A minimal hypersurface is complete when its induced Riemann…

For $d=1$, minimal surfaces on the infinite domain $Lambda_\infty=\mathbb{Z}$ are analogous to bigeodesics in first-passage percolation. Bigeodesic nonexistence conjecture. No bige…

Let $(M^3,g)$ be a closed three-dimensional Riemannian manifold. Yau's conjecture. $M$ contains infinitely many immersed minimal surfaces. In recent years, the development of min-m…

Let $M$ be a complete non-compact 3-manifold with scalar curvature $R\geq 6$, and let $f:M\to\mathbb R$ be the proper Morse function whose level-set components have controlled area…

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