6 problems

Milnor's conjecture. Either the Gauss curvature $K$ changes sign on $S$, or $K=0$.

Let $\Sigma\subset\mathbb{R}^3$ be an immersed connected closed surface. Topping's conjecture. … This conjecture asks for the optimal lower bound for the total absolute mean curvat…

Let $\operatorname{Imm}_{A}(S^2,\mathbb R^3)$ be the space of smooth Alexandrov immersed $2$-spheres, and let…

Simple loop conjecture. If $F_*$ is not injective, then there is an essential simple loop in $\Sigma$ that represents an element of the kernel of $F_*$. This generalizes the Loop T…

Virtual embedding conjecture. Every non-geometric closed aspherical $3$-manifold contains a closed essentially immersed subsurface which is not virtually embedded.

Let $\vec{\Phi}:T^2\to\mathbb{R}^m$ be an immersion, and let $\vec{H}_{\vec{\Phi}}$ and $d\operatorname{vol}_{\vec{\Phi}^*g_{\mathbb{R}^m}}$ denote its mean curvature vector and in…