20 problems

For a first-order deformation determined by the inhomogeneous Jacobi field $u=\mathrm{i}v_{\rho}+\mathrm{i}\delta v_0$, let $\psi_j$, $j=0,1,\ldots$, be the sequence of $1$-forms o…

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

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 $i_{\mathcal{A}_h}$…

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,…

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 $M$ 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…

Let $M$ be a compact three-dimensional manifold with positive Ricci curvature. Suppose that $S_1$ and $S_2$ are embedded constant mean curvature (CMC) spheres in $M$ that kiss at a…

Let $M$ be a three-dimensional Riemannian manifold, and let $S$ be a constant mean curvature (CMC) surface in $M$ whose touching set is one-dimensional. Geodesic touching-set conje…

Let $M$ be a three-dimensional Riemannian manifold with positive Ricci curvature, and let $S$ be a self-touching constant mean curvature (CMC) surface in $M$. Finiteness conjecture…

Let $N$ be a complete flat $3$-manifold, and let $M$ be a complete connected $H$-surface embedded in $N$ with finite genus. Let $W$ denote its closed mean-convex complement when…

Let $H>0$ and $g\in\mathbb{N}\cup\{0\}$. Consider the moduli space of non-congruent, connected, closed $H$-surfaces of genus at most $g$ in a fixed flat $3$-torus. Finiteness conje…

Let $N$ be a flat $3$-torus, and let $M_n$ be a sequence of closed constant mean curvature surfaces converging to a limit surface $M_\infty$. Let $q\in N$ be a singular point of co…

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…

A Lawson symmetric CMC surface is a compact constant-mean-curvature surface in $\mathbb{S}^3$ of genus $g\geq 1$ with a cyclic symmetry of order $g+1$ having four fixed points. For…

Symmetry conjecture. The pair $(\hat{Y}^{(n)},\mu^*\hat{\rm I}^{(2n+2)})$ is a Frobenius system. Moreover, each Jacobi field $\textnormal{A}^k$, $k\leq n$, has an associated locall…

Let $\mathrm{Sol}_3$ be the three-dimensional Sol geometry, and let $H>0$. A constant mean curvature (CMC) $H$ sphere is an immersed sphere in $\mathrm{Sol}_3$ with constant mean c…

Let $M\subset \mathbb{R}^3$ satisfy the hypotheses of the Minimal Element Theorem, and let $\Sigma\in\mathcal{T}(M)$ be a minimal element. For $n\in\{1,2,3\}$, consider the area an…