Related papers: Energy properness and Sasakian-Einstein metrics

The question of whether a Sasakian metric can admit an additional compatible (K-)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold…

Using Hilbert's criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to…

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $\nu$-entropy. We then investigate some properties the energy shares with these constants,…

We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of…

In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the…

Using the Sasakian join construction with homology 3-spheres, we give a countably infinite number of examples of Sasakian manifolds with perfect fundamental group in all odd dimensions greater than 1. These have extremal Sasaki metrics with…

We identify a set of "energy" functionals on the space of metrics in a given Kaehler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast…

We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski-Demianski metrics one obtains a family of local toric Kahler-Einstein metrics. These can be used to construct local Sasaki-Einstein metrics in five…

For any configuration of a static plane-symmetric distribution of matter along space-time, there are coordinates where the metric can be put explicitly as a functional of the energy density and pressures. It satisfies Einstein equations as…

General relativity and its extensions including torsion identify stress energy momentum as being proportional to the Einstein tensor, thus ensuring both symmetry and conservation. Here we visualize stress energy and momentum by identifying…

We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it…

The potential failure of energy equality for a solution $u$ of the Euler or Navier-Stokes equations can be quantified using a so-called `energy measure': the weak-$*$ limit of the measures $|u(t)|^2\,\mbox{d}x$ as $t$ approaches the first…

On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of K\"ahler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with…

We show that there are isometrically nonequivalent Robertson-Walker metrics which have the same set of geodesics. While one of these metrics satisfies the Einstein equations of pure dust without a cosmological constant, all the other…

We consider the familiar junction conditions described by Israel for thin timelike walls in Einstein-Hilbert gravity. One such condition requires the induced metric to be continuous across the wall. Now, there are many spacetimes with…

A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the…

Deformations of the Reeb flow of a Sasakian manifold as transversely K\"ahler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence…

In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe…

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^\infty$. It is well known that the original form of the inequality with the sharp exponent (proved…

We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such…