6 problems

Chen's properness conjecture. A constant scalar curvature Kähler metric exists if and only if the $K$-energy is proper with respect to geodesic distance.

Let $M$ be a compact Kähler manifold and fix a Kähler class. Let $Ca$ denote the Calabi energy and ${\mathbf E}$ the K-energy functional. Lower-bound conjecture for the K-energy. I…

Chen's regularity conjecture. Any $C^{1,1}$ function which attains the absolute minimum of the K-energy or the modified K-energy is actually smooth.

Let $(M,g)$ be a $(2n+1)$-dimensional Sasaki manifold with transverse Kähler form … Suppose that $M$ admits a transverse Sasaki–Einstein metric. Let $K$ be a maximal compact subgro…

Let $\mu_k$ be the closed positive twisting data used to define the functional $J_{\mu_k}$, and let … be the corresponding twisted K-energy functional. A twisted cscK metric is a s…

Let $(M,[\omega])$ be a polarized Kähler manifold, and let $E$ denote the K-energy functional. The maximal invariant, totally geodesic subspace induced by the automorphism group is…