Related papers: A gap theorem for positive Einstein metrics on the…
It is shown in the paper "Variational Properties of the Gauss-Bonnet Curvatures" of M.L. Labbi, that metrics with constant 2k-Gauss-Bonnet curvature on a closed n-dimensional manifold, 1<2k<n, are critical points for a certain Hilbert type…
It is known that if gauge conditions have Gribov zero modes, then topological symmetry is broken. In this paper we apply it to topological gravity in dimension $n \geq 3$. Our choice of the gauge condition for conformal invariance is…
In Newtonian theory, gravity inside a constant density static sphere is independent of spacetime dimension. Interestingly this general result is also carried over to Einsteinian as well as higher order Einstein-Gauss-Bonnet (Lovelock)…
In this paper, we study vacuum static spaces with positive isotropic curvature. We prove that if $(M^n, g, f)$, $n \ge 4$, is a compact vacuum static space with positive isotropic curvature, then up to finite cover, $M$ is isometric to a…
The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M. (To be precise, one only considers those constant-scalar-curvature…
The classification of compact homogeneous spaces of the form $M=G/K$, where $G$ is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are $4$ infinite families and $3$ isolated spaces…
We prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space $W^{2, n/2}_{\mathrm{loc}}(M)$. We argue that this is the largest class of metrics with scalar curvature a positive a.c. measure for which the positive…
We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for…
In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function,…
The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(\Theta)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we…
For non-trivial solutions to the zero mode equation on a closed spin manifold \[D \varphi=iA\cdot \varphi,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$…
In this paper we provide a sharp characterization of the smooth four-dimensional sphere. The assumptions of the theorem are conformally invariant, and can be reduced to an L^2 inequality of the Weyl tensor and positivity of the Yamabe…
We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor…
4-dimensional homogeneous isotropic cosmological models obtained from solutions of vacuum 5-dimensional Einstein equations are considered. It is assumed, that the G(55)-component of the 5-d metric simulates matter in the comoving frame of…
We consider the existence and stability of the Einstein static universe under the Generalized Uncertainty Principle (GUP) effects. We show that this solution in the presence of perfect fluid with a minimal length is cyclically stable around…
We consider Einstein gravity with positive cosmological constant coupled to matter in an asymptotically de Sitter universe with sphere boundary at timelike infinity. In this setting we show that, to one-loop order and at late time, the norm…
On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant…
We consider a Yang-Mills type gauge theory of gravity based on the conformal group SO(4,2) coupled to a conformally invariant real scalar field. The goal is to generate fundamental dimensional constants via spontaneous breakdown of the…
For warped products with harmonic curvature, nonconstant warping functions $\phi$, and compact two-dimensional bases $(M,h)$, we establish a dichotomy: either the Gaussian curvature $K$ of the metric $g=\phi^{-2}h$ is constant and negative,…
We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant $c \geqslant 0$. Our first main result builds on previous results by Yang and establishes the existence of solutions to the…