arXiv research

Classifies smooth metric measure spaces with two weighted Einstein representatives.

Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.

Study classifies Einstein spaces and warped products in weighted geometry.

Survey on Kähler-Einstein and weighted solitons on Fano manifolds.

Proves conditions for weighted Hermite-Einstein metrics on vector bundles.

In this paper we prove a number of triviality results for Einstein warped products and quasi-Einstein manifolds using different techniques and under assumptions of various nature. In particular we obtain and exploit gradient estimates for solutions of weighted Poisson-type equations and adaptations to the weighted sett…

An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.

Classifies solutions to vacuum weighted Einstein equations on pr-waves.

Study on Einstein solitons with bounds and asymptotic behavior.

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.

The article characterizes gradient ρ-Einstein solitons under specific conditions.

We propose a definition of the weighted $σ_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $σ_k$-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when $k=1,2$ or the smooth metric measure space is lo…

We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.

Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.

The paper proves conditions for Einstein solitons to split into line and manifold.

New invariants help solve existence of weighted cscK metrics.

New conformally invariant forms help identify Einstein metrics.

The paper studies a new vacuum field equation and its solutions.

The paper studies weighted Ricci curvatures and characterizes Randers metrics.

Existence and uniqueness of discrete Einstein metrics on trees proven.

Extremal Kahler metrics and Sasaki-Einstein metrics characterized via coercive energy.

We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to superg…

We compute the hybrid limit (in the sense of Boucksom-Jonsson) of the family of Kähler-Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial valuations, determined by the natural algebro-geometric limit of the family. We also make s…

A new definition of canonical conformal differential operators $P_k$ ($k=1,2,...)$, with leading term a $k^{\rm th}$ power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction …

Study geometric and analytical properties of $ρ$-Einstein solitons.

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted …

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

The study shows ends of shrinking gradient $ρ$-Einstein solitons are non-parabolic.

We introduce the coupled Ricci-Calabi functional and the coupled H-functional which measure how far from a coupled Kähler-Einstein metric in the sense of Hultgren-Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give …

The paper studies the curvature behavior near the boundary of certain domains.

In this paper, the radiation field is defined for solutions to Einstein vacuum equations which are close to Minkowski space-time with spacial dimension $n\geq 4$. The regularity properties and asymptotic behavior of those Einstein vacuum solutions are established at the same time. In particular, the map from Cauchy int…

The paper studies how certain solitons on Fano manifolds extend to nearby deformations.

The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of Kähler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the…

We compute all 2-covariant tensors naturally constructed from a semiriemannian metric which are divergence-free and have weight greater than -2. As a consequence, it follows a characterization of the Einstein tensor as the only, up to a constant factor, 2-covariant tensor naturally constructed from a semiriemannian met…

The paper presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds.

Negative Sasakian manifolds, where the first Chern class of the contact subbundle is a torsion class, can be viewed as Seifert-$S^1$ bundles where the base orbifold has an ample orbifold canonical class. We use this framework to settle completely an open problem formulated by C.Boyer and K.Galicki which asks whether or…

In this paper, we prove that the set of solutions of constraint equations for coupled Einstein and scalar fields in classical general relativity possesses Hilbert manifold structure. We follow the work of R. Bartnik [2] and use weighted Sobolev spaces and Implicit Function Theorem to prove our results.

A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs …

Paper explores new Kähler metrics from old, aiming to solve YTD conjecture.

In this article, we give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted pr…

In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymp…

Positive-curvature metrics on trees identified for specific configurations.

The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-li…

We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain …

In the class of metrics of a generic conformal structure there exists a distinguishing metric. This was noticed by Albert Einstein in a lesser-known paper of 1921 (Berl. Ber., 1921, pp. 261-264). We explore this finding from a geometrical point of view. Then, we obtain a family of scalar conformal invariants of weight …

Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.