12 problems

Hyperbolic Carathéodory conjecture. The least number of quadratic points on a generic compact non-degenerate hyperbolic surface is $8$.

The quadrant-wise fundamental forms suggest gluing analytic patches across the sectors. In particular, the restrictions to $\Omega_1\cup\Omega_3$ agree with the positive alignable…

Continuous-family conjecture. A minimal surface with flat points is still contained in a continuous family of twice differentiable constant-ratio-of-principal-curvatures surfaces i…

Let $M$ be a negatively curved surface in Euclidean space, and suppose that $M$ forms the interior of an annular surface $\overline M$ bounded by convex curves in planes tangent to…

Ivanov's meta-conjecture. When $g \geq 3$, every such object has $Mod^{\displaystyle \pm}(S)$ as its group of automorphisms, with a similar claim for $g=2$.

Envelope characterization. The surface $\Phi$ contains the envelope of the family $C_t$ if and only if one of the following conditions holds:

Bloch-type conjecture. If this cup product map is zero, then the intersection product map

Index bound conjecture. The index of the cubic form is always at most $1$, without assuming that the cubic form is semi-homogeneous.

A regular surface in $\mathbb{R}^3$ is a surface defined by an embedding. Let $\tilde{S}_{\tilde{C}_0}$, with $\tilde{C}_0>0$, denote the reference family of complete non-constant-…

Uniqueness conjecture. The only simply connected complete non-constant-mean-curvature biconservative surfaces in $\mathbb{R}^3$ are those given by that theorem. This is proposed as…

Let $M^2$ be a biconservative surface in $\mathbb{R}^3$, with mean-curvature function $f$, and let $U$ be a non-empty open subset of $M$. Uniqueness conjecture. If … on $U$, then ……

A Weingarten surface in $\mathbb{R}^3$ is an immersed surface whose principal curvatures satisfy a functional relation. An integrable Weingarten surface is one whose governing equa…