8 problems

BTP-to-BAS conjecture. Under these assumptions, $g$ must be BAS.

Let $S$ be a compact $\eta$-Einstein Sasakian manifold that admits a Sasakian immersion into a sphere. For a simply-connected compact homogeneous Hodge manifold $M$, let…

A regular $3$-Sasakian manifold is a $3$-Sasakian manifold whose associated Reeb foliation is regular. Homogeneity conjecture. Every regular $3$-Sasakian manifold is homogeneous. T…

Let $X$ be a homogeneous three-manifold diffeomorphic to $\mathbb{R}^3$, and let $\Sigma$ be a constant mean curvature sphere immersed in $X$. Embeddedness conjecture. $\Sigma$ is…

Let $X$ be a homogeneous three-manifold diffeomorphic to $\mathbb{R}^3$, and let $V>0$. An isoperimetric domain in $X$ is a domain enclosing volume $V$ and minimizing boundary area…

Let $\operatorname{Nil}_3=\mathbb{R}^2\rtimes_A\mathbb{R}$ be the Heisenberg group with its standard metric, let $\Sigma\subset\operatorname{Nil}_3$ be the complete embedded stable…

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…

Let $M$ be a compact homogeneous manifold of a reductive Lie group. A flag manifold is a compact homogeneous manifold of a compact semisimple Lie group admitting a Kähler structure…