13 problems

The study of surfaces in $\mathbb{R}^4$ from the affine viewpoint is related to affine differential geometry. A convex compact surface is a compact convex surface…

Let $M \subset \mathbb{R}^4$ be a locally strictly convex surface homeomorphic to the sphere. Loewner's conjecture. The surface $M$ has at least two inflections. This conjecture co…

Discrete uniformization conjecture. Suppose that $S$ is a genus zero complete hyperbolic surface with countably many ends, all but at most one of which are cusp ends. Then $S$ is i…

A closed convex surface is a convex surface without boundary, and an arbitrarily small piece means a portion of the surface removed with arbitrarily small size. Liebmann's conjectu…

Let $S$ be a compact piece of a smooth surface in $\mathbb{R}^n$, and let $N_\delta(M)$ denote its $\delta$-neighborhood. A family of parallelograms has width at least $\delta$ if…

Let $K$ be a fibered knot in a manifold $M$, and let $\xi$ be a contact structure on $M$. Non-thickenable-tori conjecture. The knot type $K$ admits non-thickenable tori in $\xi$ if…

Circle-type convex-hull uniqueness conjecture. If these two boundaries are isometric, then $X$ and $Y$ differ by a Mf6bius transformation.

Circle-type realization conjecture. There exists a circle type closed set $X\subset\mathbb{C}\cup\{\infty\}$ such that $\Omega$ is isometric to the boundary of the convex hull of…

Let $(S,d)$ be a planar surface with a complete path metric whose curvature is at least $-1$. Weyl realization conjecture. There exists a complete convex surface $Y$ in…

Let $\mathbb{S}^2$ be the sphere at infinity of hyperbolic 3-space. For a circle-type closed set $Y\subset\mathbb{S}^2$, meaning a compact set whose complement is a circle domain,…

Let $M\subset\mathbb R^4$ be a locally strictly convex surface, meaning that at each point $p\in M$ there is a tangent hyperplane with non-degenerate contact that locally supports…

Let $\Sigma\subset\widehat{\mathbb{R}}^n$ be a smooth hypersurface that is the boundary of a centred convex body, and let $\kappa$ denote its Gaussian curvature and…

Let $(S^2,g)$ be an oriented two-dimensional sphere with a smooth Riemannian metric, and let $c:S^2\to\mathbb{R}$ be a smooth positive function. Consider curves satisfying … where…