Related papers: On K\"ahler-Einstein surfaces with edge singularit…

We present new constructions of Kaehler metrics with constant scalar curvature on complex surfaces, in particular on certain del Pezzo surfaces. Some higher dimensional examples are provided as well.

We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic…

We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex…

An I-surface $S$ is an algebraic surface of general type with $K_S^2 = 1$ and $p_g(S) = 2$. Recent research has centered on trying to give an explicit description of the KSBA compactification of the moduli space of these surfaces. The…

In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K \"ahler-Einstein surface. We show the relation between the maximum of the…

We develop estimates for the equation of scalar curvature of singular metrics with cone angle $\beta>1$, in a big and semi-positive cohomology class on a K\"ahler manifold. We further derive the Laplacian estimate for the scalar curvature…

We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with…

We determine the complete list of anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. We prove that many of these admit a K\"ahler-Einstein metric and most of them do not have tigers.

We give an example of a family of smooth complex algebraic surfaces of degree $6$ in $\mathbb{CP}^3$ developing an isolated elliptic singularity. We show via a gluing construction that the unique K\"ahler-Einstein metrics of Ricci curvature…

This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/

It is shown that if a minimal ruled surface admits a K\"ahler Yamabe minimizer, then this metric must be generalized K\"ahler-Einstein and the underlying holomorphic vector bundle of the ruled surface must be quasi-stable.

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when…

This is the continuation of our paper \cite{GS}, to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background K\"ahler metric of…

We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature K\"ahler metric for every K\"ahler class. We show that there are only two such rational surfaces,…

In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a K\"ahler-Einstein surface. We prove that the blow-up flow $\Sigma_s^\infty$ at a singular point $(X_0, T_0)$ of…

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$.…

We show that a general class of singular K\"ahler metrics with Ricci curvature bounded below define K\"ahler currents. In particular the result applies to singular K\"ahler-Einstein metrics on klt pairs, and an analogous result holds for…

Over a compact K\"ahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar…

There are many known examples of scalar-flat K\"ahler ALE surfaces, all of which have group at infinity either cyclic or contained in ${\rm{SU}}(2)$. The main result in this paper shows that for any non-cyclic finite subgroup $\Gamma…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…