The study of real Einstein submanifolds in Kähler geometry.
problem Understanding real Einstein submanifolds in Kähler geometry.
method Provided a necessary and sufficient condition for anti-holomorphic automorphisms to determine real Einstein submanifolds.
result A condition for determining real Einstein submanifolds in compact Kähler-Einstein manifolds.
The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.
problem Understanding projective and Carrollian geometries at infinity for Ricci flat Einstein manifolds.
method Developed a new type of Cartan geometry based on non-effective homogeneous models for projective geometry.
result Carrollian geometries are determined by the projective compactification data of Ricci flat Einstein manifolds.
Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.
problem Defining metrics and Einstein tensors on Riemannian manifolds.
method Defines bilinear functionals of vector fields and differential forms, generalizing to non-commutative geometry.
result Proves the vanishing of the Einstein functional for the conformally rescaled geometry of the noncommutative two-torus.
The study of special metrics in various parabolic geometries.
problem Finding special metrics in parabolic geometries.
method Investigating invariant conditions for special metrics in different parabolic geometries.
result Characterization of special metrics in hypersurface CR and contact Legendrean cases.
The paper studies Einstein-type manifolds with structural conditions.
problem Investigating geometric structures on Riemannian manifolds.
method Unified approach to various geometric structures and curvature conditions.
result Rigidity results for Einstein-type manifolds under specific curvature conditions.
Linearized Einstein equations simplified via Calabi operator.
problem Linearizing Einstein equations for cosmological applications.
method Using the Calabi operator from projective differential geometry.
result Linearized Einstein equations simplified.
The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.
problem Understanding the connection between Kähler-Einstein metrics and algebraic geometry.
method Exploring metric limits and rescalings of Kähler-Einstein metrics in relation to moduli spaces and singularities.
result Proposes tentative conjectural pictures connecting Kähler-Einstein metrics and algebraic geometry.
We study eta-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem for Sasakian geometry to prove the existence of eta-Einstein structures on many different compact manifolds, including exotic spheres. We also relate…
Classifies a specific type of Lie groups related to Einstein geometry.
problem Classifying Einstein Lorentzian 3-nilpotent Lie groups with 1-dimensional nondegenerate center.
method Complete classification through mathematical analysis.
result A full classification of the specified Lie groups.
The paper proves rigidity results for Einstein manifolds with specific geometric constraints.
problem Understanding the rigidity of Einstein manifolds under bounded covering geometry.
method Analyzing Einstein manifolds with bounded covering geometry to prove rigidity results.
result Compact Einstein manifolds with specific geometric properties are isometric to space forms.
This article is a summary of some of the author's work on Sasaki-Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki-Einstein geometry, in low dimensions, to superconformal field theory; properties of the latter are therefore reflected in the former, and vi…
Study on 4D Einstein manifolds with Kähler conformal geometry.
problem Exploring 4D Poincaré-Einstein manifolds with Kähler metrics.
method Formulated a Dirichlet boundary value problem and established existence and uniqueness theory.
result Existence and uniqueness of new Poincaré-Einstein metrics.
We discuss Sasakian-Einstein geometry under a quasi-regularity assumption. It is shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it. Furthermore, necessary and sufficient conditions are given for the `product' of two Sasakian-Einstein manifolds to be a smooth Sasaki…
Geometries and dual field theories linked by AdS/CFT.
problem Understanding the AdS/CFT correspondence.
method Geometric extremization principles informed by physical considerations.
result Key role of Sasaki-Einstein and GK geometry.
We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely …
Einstein's non-symmetric geometry uses Bochner's technique to prove decomposition and vanishing results.
problem Analyzing Einstein's non-symmetric geometry with Bochner's technique.
method Defining concepts, proving decomposition formula, and showing vanishing results.
result Vanishing results about the null space of Bochner and Hodge type Laplacians.
Study on Einstein deformations of negative Kähler Einstein metrics.
problem Understanding Einstein deformations of Kähler Einstein metrics.
method Relate second order Einstein deformation theory to complex geometry, gauge normalise, and use Taylor expansion.
result Taylor expansion to order two of an Einstein deformation is determined by h12 and the divergence of the Kodaira-Spencer bracket. Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.
problem Characterizing compact complex manifolds with holomorphic GL(2)-geometry.
method Analyzing Kähler-Einstein and Fano manifolds, using GL(2) and SL(2) geometries.
result Only compact Kähler-Einstein manifolds with holomorphic GL(2)-geometry are covered by compact complex tori, three dimensional quadric, or three dimensional Lie ball.
Proves finite step termination of Kähler-Einstein metric singularity formation.
problem Singularity formation of Kähler-Einstein metrics.
method Finite step termination of bubble trees for singularity formation.
result Finite step termination of Kähler-Einstein metric singularity formation proved in non-collapsing situation.
We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifold…
This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution σ of the adjoint bundle $\mc{A}$ gives rise, given certain algebraic conditions, to a unique preferred affine connection ∇…
In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of n+1-dimensional, n≥3, spatially compact spacetimes which generalizes the k=−1 Friedmann--Robertson--Walker vacuum spacetime. Our results demonstrate causal geodes…
Normalizes pseudo-Einstein contact forms for easier analysis.
problem Understanding pseudo-Einstein contact forms.
method Constructing intrinsic CR normal coordinates using parabolic normal coordinates.
result Normal form for pseudo-Einstein contact forms.
We show that every K-contact Einstein manifold is Sasakian-Einstein and discuss several corollaries of this result.
New bound on partition function proves Kähler-Einstein stability.
problem Proving Kähler-Einstein metrics on complex manifolds.
method Quantitative bound on partition function, connecting probabilistic and quantization approaches.
result Direct analytic proof of Kähler-Einstein stability for uniformly Gibbs stable manifolds.
We consider local geometry of sub-pseudo-Riemannian structures on contact manifolds. We construct fundamental invariants of the structures and show that the structures give rise to Einstein-Weyl geometries in dimension 3, provided that certain additional conditions are satisfied.
New rigidity results for quasi-Einstein metrics with non-zero divergence-free vector fields.
problem Classifying quasi-Einstein metrics with specific vector field properties.
method Analyzing quasi-Einstein metrics on closed manifolds and near-horizon geometries of extreme black holes.
result These metrics always admit a one-parameter group of isometries generated by the divergence-free vector field.
Study on quasi-Einstein manifolds with boundary estimates and inequalities.
problem Understanding the geometry of compact quasi-Einstein manifolds with boundary.
method Sharp boundary estimates and characterization theorems for quasi-Einstein manifolds.
result New geometric inequalities and boundary estimates for quasi-Einstein manifolds.
The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
problem Understanding the singular sets of degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
method Developing quantitative differentiation theory, stratification, Minkowski estimates, and ε-regularity results.
result Uniform Hausdorff measure estimates for the singular sets of degenerate/singular elliptic operators.
Research connects probabilistic and variational approaches to Kahler-Einstein metrics.
problem Constructing Kahler-Einstein metrics on complex projective varieties.
method Combines probabilistic construction and variational methods.
result Non-Archimedean geometry of X emerges from probabilistic framework.
We review our study of Sasakian geometry as an agent for proving the existence of Einstein metrics on odd dimensional manifolds. Particular emphasis is given to the Sasakian structures occuring on links of isolated hypersurface singularities.
Paper extends curvature estimates to new tensor types.
problem Mean curvature and volume comparison estimates for integral generalized quasi-Einstein tensors.
method Extends existing comparison results to new tensor types.
result Global diameter estimates derived from comparison results.
Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
Introduces pqc structures, generalizing para 3-Sasakian geometry.
problem Generalizing and studying para 3-Sasakian geometry.
method Defines pqc structures, derives a distinguished linear connection, and presents structure equations.
result Para 3-Sasakian spaces are pqc manifolds and pqc-Einstein.
The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on d…
Extends construction of Kähler-Einstein metrics to noncompact manifolds.
problem Construct complete Kähler-Einstein metrics on noncompact manifolds.
method Iterative construction using Berndtsson's method.
result Induces semipositively curved metric on relative canonical bundle.
The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry of the foliations …
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. In this paper, we introduce the trans-para-Sasakian manifolds and we study their geometry. These manifolds are an analogue of the trans-Sasakian manifolds in the Riemannian geometry. We shall investigate many curvature properties of these manifolds and we shall give many conditions under which the manifolds are either …
Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
problem Characterize compact quasi-Einstein metrics with constant scalar curvature.
method Connection to Sasakian geometry and circle bundles over Einstein metrics.
result Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
We briefly review a few aspects of the development of differential geometry which may be considered as being influenced by Einstein's general relativity. We focus on how Einstein's quest for a complete geometrization of matter and electromagnetism gave rise to an enormous amount of theoretical work both on physics and …
Study on generalized quasi-Einstein structures in contact geometry.
problem Characterizing and understanding generalized quasi-Einstein structures in contact geometry.
method Investigation of properties, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifolds.
result Normal metric contact pair manifolds with generalized quasi-constant curvature are generalized quasi-Einstein manifolds.
Study Einstein warped products with Einstein base and fiber.
problem Characterize Einstein warped products with Einstein base and fiber.
method Investigate necessary and sufficient conditions for a warped product to be Einstein.
result Explicitly determine the warping function when the base is hyperbolic space.
On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [KW]. We expand on the recent work of Demailly and Kollár [DK] and Johnson and Kollá…
This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nea…
New findings on static near horizon geometries and quasi-Einstein manifolds, including rigidity results for negative cosmological constant.
problem Rigidity of quasi-Einstein manifolds under different cosmological constant conditions.
method Analysis of quasi-Einstein equations on closed manifolds, focusing on static vacuum solutions and their properties.
result For negative cosmological constant, rigidity holds under specific conditions on the 1-form \(X\), including incompressibility, constant norm, and nontrivial cohomology.
All local solutions of the two dimensional Einstein-Weyl equations are found, and related to the compact examples which I obtained in "Moebius structures and two dimensional Einstein-Weyl geometry" J. reine angew. Math. 504 (1998).