59 problems

Let $M$ be a compact immersed constant mean curvature surface without boundary in $\mathbb{R}^3$. Hopf's conjecture. The surface $M$ is the round sphere. The conjecture is a classi…

An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold containing a closed minimal surface homeomorphic to a fixed closed oriented surface of genus at least two,…

Properness and existence conjecture. For any $H\geq\frac12\operatorname{Ch}(X)$, every complete, connected $H$-surface embedded in $X$ with positive injectivity radius or finite to…

Let $M$ be a complete surface of finite genus embedded in $\mathbb{R}^3$ with constant mean curvature. Say that $M$ has at most cubical area growth if there is a constant $C$ depen…

Meromorphic invariant potential conjecture. Every constant mean curvature surface in $\mathbb{R}^3$ associated with $M$ can be obtained from an invariant meromorphic potential.

Extension conjecture. Theorem 3 should hold without the hypothesis $H\geq 0$.

Let $M$ be a connected compact embedded $H$-surface in $\mathbb{R}^3$ whose boundary is a round circle. Sphericality conjecture. The surface $M$ is spherical. In Euclidean space th…

The constructions under consideration produce CMC immersions of the punctured sphere with ends. Asymptotic half-Delaunay conjecture. The ends of the immersions constructed in the s…

Spherical isoperimetric-surface conjecture. Isoperimetric surfaces in ${\mathrm{Nil}\,}$ are homeomorphic to spheres.

Isoperimetric-sphere conjecture. The constant-mean-curvature spheres of revolution in ${\mathrm{Nil}\,}$ are isoperimetric surfaces.

Fixed-axis moduli conjecture. In this case, $\Mk^{0}$ is also a real analytic manifold of dimension $k$.

The paper considers the moduli space of embedded, complete, constant mean curvature surfaces in $\mathbb{R}^3$ with $k$ ends. Korevaar–Kusner's conjecture. There is a good moduli s…

Hauswirth–Pérez–Romon–Ros conjecture. The only solutions of the isoperimetric problem in $\mathbb{T}^2\times\mathbb{R}$ are spheres, cylinders, and pairs of horizontal planes.

Let $c$ be a screw motion geodesic in a homogeneous space with parameters $c\kappa$ and $c\tau$, and let a tube invariant by translations along $c$ have constant mean curvature $H$…

Let $\Sigma$ be a compact surface in $\mathbb{R}^3$ with non-empty circular boundary and nonzero constant mean curvature. Spherical cap conjecture. If either (i) $\Sigma$ has genus…

Let $u:\Sigma^2\to N^3$ be a branched CMC $h$-immersion, where $N^3$ has strictly positive sectional curvature, and let $g$ denote the genus of $\Sigma$. Write $i_{\mathcal{A}_h}$…

Positivity conjecture for $I_a$. $I_a>0$ for all $a\in(0,\frac12)$.

Let $M$ be an embedded free boundary constant-mean-curvature annulus in a geodesic ball of one of the spaces $\mathbb{R}^3$, $\mathbb{S}^3$, or $\mathbb{H}^3$. A spherical curvatur…

Let $S_1$ be the non-trivial spin structure on a torus, and consider conformal classes $[g_{0,b}]$ with $0<b<1$. The source describes non-flat metrics associated with degree-zero m…

Positive-curvature constant-mean-curvature sphere conjecture. For every such metric $g$ and every $c\in\mathbb{R}_{>0}$, there exists an embedded sphere in $S^3$ with constant mean…

Let $\omega\in((4\pi)^{-1},\pi/6)$ be the fixed volume fraction, and let $c$ and $K$ be the phase-field parameters. An unduloid is a constant-mean-curvature surface of revolution i…

Let $\Omega$ be a smooth domain in $\mathbb{R}^3$, and let $u\in C^2(\overline{\mathbb{D}},\mathbb{R}^3)$ be an immersed constant-mean-curvature surface with boundary contained in…

Rosenberg–Smith's conjecture. For every such $H$ and $g$, there exists an embedding of $S^2$ into $S^3$ with constant mean curvature $H$.

Let $M$ be a closed $3$-manifold and let $S$ be an embedded surface in $M$ with nontrivial homology. Define $\widehat{C}_{[S]}$ from the width $w([S])$ of the homology class and th…

Let $M$ be a closed $3$-manifold, and let $S$ be an embedded surface in $M$ with nontrivial homology. Let $\widehat{C}_{[S]}$ be the lower bound defined using the width of the homo…