arXiv research

Einstein metrics on products are shown to be warped.

Paper proves uniqueness of Einstein metrics on balls.

An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincare-Lovelock metrics, also have Fefferman-Graham…

Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.

In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology…

New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.

Study of conformally compact metrics and Lovelock tensors in even dimensions.

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles…

In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \ci…

Constructs conformal metrics with negative curvature on manifolds with boundary.

Compact metrics found with specific curvature properties on 3D surfaces.

We define an invariant for compact spin manifolds $X$ of dimension $4k$ equipped with a metric $h$ of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of $h$ to be the conformal infinity of a conformally compact Einstein metric on $X$.

In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive …

Study on filling 3D metrics with 4D Poincaré-Einstein structures.

Derives stress-energy identities in Liouville theory on compact surfaces.

We show that C^2 conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.

A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of con…

We discuss a number of topics in the area of conformally compact Einstein metrics, mostly centered around the global existence question of finding such metrics with an arbitrarily prescribed conformal infinity. The paper is partly a survey of this area but also presents new results and a number of open problems.

Extended Vaisman theorem to compact spaces with singularities.

Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.

Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.

Conformal vector fields on lcK manifolds are shown to be Killing or holomorphic.

New tensors help determine if metrics are related to Poincaré-Einstein ones.

This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence.

Canonical metrics and conformal invariants are presented for closed oriented even-dimensional manifolds with non-degenerate conformal structures and in particular for compact Riemann surfaces.

The article classifies curvature functions on compact manifolds with boundaries.

Paper examines stability of minimizing metrics on manifolds with boundary.

We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has no $C^2$ conformal compactification.

We obtain an example of a compact locally conformal symplectic nilmanifold which admits no locally conformal Kähler metrics. This gives a new positive answer to a question raised by L. Ornea and M. Verbitsky.

The main result of this paper is that the space of conformally compact Einstein metrics on a given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We also prove full boundary regularity for such metrics in dimension 4, and a loca…

Study finds conditions for metrics on curved spaces.

Conformal vector fields on LCP manifolds are orthogonal and Killing.

We study locally conformal calibrated $G_2$-structures whose underlying Riemannian metric is Einstein, showing that in the compact case the scalar curvature cannot be positive. As a consequence, a compact homogeneous $7$-manifold cannot admit an invariant Einstein locally conformal calibrated $G_2$-structure unless the…

New proof of instability for certain Einstein metrics.

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remai…

The paper studies eigenvalues of a special Laplacian system on compact manifolds.

We examine here the space of conformally compact metrics $g$ on the interior of a compact manifold with boundary which have the property that the $k^{th}$ elementary symmetric function of the Schouten tensor $A_g$ is constant. When $k=1$ this is equivalent to the familiar Yamabe problem, and the corresponding metrics a…

We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

Study on curvature functions for compact manifolds with boundary.

The study finds points on surfaces where a tensor is conformal to a metric.

Study locally conformally balanced metrics on specific Lie algebras.

We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the correspon…

New scalars measure failure of CC metrics to solve singular Yamabe problem.

This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $Σ$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $Σ\backslash \{{\rm singularities}\}$, called the {\it developing …