We study minimal graphic functions on complete Riemannian manifolds $\Sigma$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with such growth via splitting for tangent cones of $\Sigma$ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville type theorem under a certain non-radial Ricci curvature decay condition on $\Sigma$. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section.

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration with smooth central fiber.

Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric \omega with cusp singularities along a divisor D, we show the L^ 2 projective embedding of (X, D) defined by L^ k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial one being the neck region around D. This is the first step towards understanding the algebro-geometric stability of extremal Khler metrics with singularities.

We prove that a pair (X, D) with X Fano and D a smooth anti-canonical divisor is K-unstable for negative angles, and K-semistable for zero angle.

We prove some structure results for transversely reducible Sasaki manifolds. In particular, we show a Sasaki manifold with positive Ricci curvature is transversely irreducible, and so join (product) construction cannot be generalized to irregular SasakiEinstein manifolds, as opposed to the quasi-regular case done by WangZiller and BoyerGalicki. As an application, we classify compact Sasaki manifolds with nonnegative transverse bisectional curvature, which can be viewed as the generalized Frankel conjecture in Sasaki geometry.