5 problems

Let $M\subset\mathbb{R}^3$ be a connected, complete embedded minimal surface of finite genus with compact, possibly empty, boundary. Embedded Calabi–Yau conjecture for finite genus…

Calabi–Yau property conjecture. The Calabi–Yau property holds for bordered minimal surfaces in any smooth Riemannian manifold $(N,g)$ with $\dim N\ge 3$. Explicitly, for every bord…

Let $\Omega\subset\mathbb{R}^3$ be the particular bounded domain, smooth except at one point, described by Ferrer, Martín, Meeks and Nadirashvili. An open surface with compact, pos…

Let $(M^{4n}, g, I, J)$ be a nondegenerate generalized Kähler manifold, and consider its $\Omega$-Hamiltonian deformation class. Generalized Kähler Calabi–Yau conjecture. There exi…

Let $X$ be a simply-connected homogeneous $3$-manifold, let $H_0(X)$ denote the associated critical mean-curvature value, and let ${\mathcal S}$ be a complete embedded $H$-surface…