Related papers: Constant scalar curvature metrics on toric surface…
This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold
We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.
We investigate the relationship between stability and the existence of extremal K\"ahler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a…
We present a notion of a random toric surface modeled on a notion of a random graph. We then study some threshold phenomena related to the smoothness of the resulting surfaces.
Total five different types of translation surfaces, based upon planarity of translating curves and the absolute figure, arise in a Galilean 3-space. Excepting the type in which both of translating curves are non-planar we obtain these…
Local estimates of the maximal curvatures of admissible spacelike hypersurfaces in de Sitter space for k-symmetric curvature functions are obtained. They depend on interior and boundary data.
In this paper we prove that some well known K\"ahler surfaces with zero scalar curvature are QCH K\"ahler. We prove that family of generalized Taub-Nut K\"ahler surfaces parametrized by $k\in[-1,1]$ is of orthotoric type for $k\in(-1,1)$…
The aim of this paper is to present examples of Kahler holomorphically pseudosymmetric metrics on the projective space CP^n.
In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.
In this paper we introduce higher extremal Kahler metrics. We provide an example of the same on a minimal ruled surface. We also prove a perturbation result that implies that there are non-trivial examples of higher constant scalar…
We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the…
We give examples of smooth surfaces with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kahler classes that do not admit any constant scalar curvature Kahler metrics. We also compare…
Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically…
We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.
We prove that the existence of constant scalar curvature K\"ahler metrics with cone singularities along a divisor implies log $K$-polystability and $G$-uniform log $K$-stability, where $G$ is the automorphism group which preserves the…
We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.
We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains…
We prove the existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds provided there are barriers.
In this paper we study spherically symmetric metrics on a symmetric space in $\mathbb{R}^n$ with scalar and constant flag curvature and we also obtain families of this type of metrics. Many explicit examples are provided for Douglas metrics…
We study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions, for instance the Timelike Convergence Condition (TCC). We characterize stability when the target…