12 problems

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

Strengthening of the Thom conjecture. One has

Immersed-surface conjecture. Every closed hyperbolic $3$-manifold contains a number of immersed incompressible surfaces, each of which lifts to an embedding in a finite-sheeted cov…

Let $\mathfrak{m}(X_8,B^2)$ be the component of the immersion complex consisting of immersed annuli of figure-8 type in the disk $B^2$. Figure-8 disk fundamental-group conjecture.…

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…

Let $f:S\rightarrow M$ be a 2-sided immersion of a closed surface $S$ in a 3-manifold $M$, and let $f_{\sharp}:\pi_{1}(S)\rightarrow\pi_{1}(M)$ be the induced homomorphism. The ker…

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…

Alexandrov's conjecture. Any other compact surface $\Sigma$ of genus zero immersed in $\mathbb{R}^3$ whose Gauss map and principal curvatures satisfy the same prescribed-curvature…

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

In Hilbert's theorem, the hyperbolic plane is replaced by a complete immersed surface in $\mathbb{R}^3$ whose Gaussian curvature is not greater than a negative constant. Cohn-Vosse…

Let $F=\langle a,b\rangle$ be a free group of rank $2$, and let $w\in[F,F]$ be a homologically trivial word. Regard $F$ as the fundamental group of the hyperbolic once-punctured to…