6 problems

Barycenter conjecture. The unique solution of this equation occurs when $x=m$, the barycenter of $P$.

Let $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ be an isometric immersion of a Kähler manifold into Euclidean space, with $2\leq p\leq n-1$. Along any connected component of an open dense…

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…

Let $\phi_{\tau}(z)$ be a solution of the initial value problem for geodesics in the space of Kähler metrics, and let $U_{h^k}(i\tau,z,z)$ denote the corresponding quantity associa…

Let $M$ be a compact Kähler manifold with $c_1(M)>0$, and let $\omega_t$ solve … with $\omega+\sqrt{-1}\partial\bar\partial\varphi>0$. Suppose that $M$ admits no Kähler–Einstein me…

Conjecture. Such a manifold is a bounded symmetric domain of rank one. This means, in particular, that it is a product of complex hyperbolic spaces.