44 problems

Let $M$ be a smooth embedded torus in Euclidean three-space, and let its bending energy be … where $H$ is the mean curvature and $k_c$ is the bending rigidity. The Clifford torus i…

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…

Montiel–Urbano conjecture. The Clifford torus achieves the minimum of the functional $\mathcal{W}^-$, and hence of $\mathcal{W}$, either among all tori in $\mathbb{C}P^2$ or among…

Let $\Sigma^k$ be an oriented surface of topological genus $k\geq 1$, let $\iota:\Sigma^k\to\mathbb{S}^3$ be an immersion, and let $\mathcal{W}_{\iota}(\Sigma^k)$ denote its Willmo…

In the Möbius geometry of the 4-sphere, a Willmore surface is a surface that is critical for the Willmore functional, and its Möbius curvature is the curvature of its conformally i…

Taimanov's mNV-flow conjecture. A torus that is nonstationary with respect to the mNV flow cannot be a local minimum of the Willmore functional.

Wintgen's conjecture. Among tori in $\mathbb{R}^4$, the Willmore functional attains its minimum on the Clifford torus.

Let $h>0$ be the distance between two coaxial circular boundary curves, and consider catenoid-type critical immersions of Willmore's functional with parameters $\alpha=0$,…

Let $\mathcal{W}$ be the Willmore functional of an immersed torus, and fix its conformal class. The spectral genus is the genus associated with the torus's spectral curve. Taimanov…

Let $\vec{\Phi}$ be a Willmore immersion obtained by inverting a minimal immersion with $n$ ends, and suppose that $\vec{\Phi}$ has no branch point. Let…

Let $\vec{\Phi}$ be a Willmore immersion, possibly branched, obtained as the inversion of a minimal immersion with $n$ ends, and let $\operatorname{Ind}_W(\vec{\Phi})$ denote its M…

The $16\pi$ conjecture. The number $\beta_0$ is attained by a path $\omega\in C^0([0,1],\operatorname{Imm}(\mathbb{S}^2,\mathbb{R}^3))$ such that both restrictions…

Let $g>1$ be an integer, and consider smooth surfaces of $\mathbb{R}^3$ of genus $g$ and their Willmore energy. Willmore conjecture in codimension 1. The only minimizers are the co…

Let $\mathbb{C}P^2$ be the complex projective plane, let $\mathcal{W}$ be the Willmore functional, and let a complex curve mean a holomorphic curve in $\mathbb{C}P^2$. For each suc…

Let $\mathbb{C}P^2$ be the complex projective plane, let $\mathcal{W}$ be the Willmore functional, and consider the Veronese sphere … Here an immersed sphere has the same self-cros…

Let $(M,g)$ be the asymptotically flat manifold considered in Theorem, and let $\Sigma$ be an area-constrained Willmore sphere with associated quantities $\rho(\Sigma)$ and…

Let an axially symmetric critical disk be a critical disk for the Helfrich energy with elastic boundary that is invariant under an axial symmetry. Palmer–Pampano's conjecture. Axia…

Grand uniqueness conjecture. The genus-$g=0$ or $1$ Canham problem with any isoperimetric ratio constraint $v_0\in(0,1)$ has a unique solution up to homothety. Moreover: (i) for…

Let a closed genus-$1$ surface in $\mathbb{R}^3$ be a minimizer of the Willmore energy subject to a prescribed isoperimetric ratio. Since Möbius transformations produce a one-param…

Let $\e2$ be a conformal immersion satisfying the generalized Willmore Euler–Lagrange equation in an arbitrary codimension, under the regularity hypotheses considered in the codime…

The Chen–Gackstatter immersions are minimal immersions with critical curvature, and fix a genus. Chen–Gackstatter uniqueness conjecture. The Chen–Gackstatter immersions are the onl…

Let ${\vec{\Phi}}_1,\ldots,{\vec{\Phi}}_p,\vec{\Psi}_1,\ldots,\vec{\Psi}_q$ be the Willmore spheres given by the decomposition in equation (12), where the preceding discussion conc…

Let $\beta_0$ be the min-max width associated with a nontrivial loop in the space of immersions of the round sphere $S^2$ into $\mathbb{R}^3$, equivalently the infimum of the maxim…

Let $X:\Sigma\setminus\{p_1,\ldots,p_m\}\to\mathbb{R}^3$ be a complete minimal surface with $m$ embedded planar ends $\{p_1,\ldots,p_m\}$. Let $n_X(p_i)$ denote the normal at the e…

Let $f^b_2$ denote the 2-lobed Delaunay torus associated with the rectangular conformal class $(0,b)$, and let $cmathcal{W}$ denote the Willmore energy. The 2-lobed Delaunay minimi…