Related papers: On the Bartnik conjecture for the static vacuum Ei…
It is shown that gravity can be incorporated into the Standard Model (SM) in a way solving the hierarchy problem. For this, the SM effective action in flat spacetime is adapted to curved spacetime via not only the general covariance but…
The stability properties of the Einstein Static solution of General Relativity are altered when corrective terms arising from modification of the underlying gravitational theory appear in the cosmological equations. In this paper the…
On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge3$, we consider the stationary Schr\"odinger equation $\Delta_gu+h_0u=\left|u\right|^{2^*-2}u$, where $\Delta_g:=-\text{div}_g\nabla$, $h_0\in C^1\left(M\right)$…
We consider a $(7 + k)$-dimensional Einstein-Gauss-Bonnet model with the cosmological $\Lambda$-term. A cosmological model with three factor spaces of dimensions $3$, $3$ and $k$, $k > 2$ is considered. Exact stable solutions with three…
We present the exact solution of Einstein's equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below $z=0$. This solution depends essentially on two constants: the density…
In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $\gamma$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex…
We construct low regularity solutions of the vacuum Einstein constraint equations on compact manifolds. On 3-manifolds we obtain solutions with metrics in $H^s$ where $s>3/2$. The constant mean curvature (CMC) conformal method leads to a…
A number of recent observations have suggested that the Einstein's theory of general relativity may not be the ultimate theory of gravity. The f(R) gravity model with R being the scalar curvature turns out to be one of the best bet to…
We have investigated a cosmological model with variable speed of light (c), gravitational constant (G) and cosmological constant (Lambda). The model is shown to solve the horizon, flatness and monopole problems of the early universe. We…
We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_\lambda:M\to\mathbb R$ is the viscosity solution of the discounted equation $$ \lambda…
In this note we prove the following result: There is a positive constant $\epsilon(n,\Lambda)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $\Lambda$, diameter…
The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be…
The positive mass theorem states that the total mass of a complete asymptotically flat manifold with non-negative scalar curvature is non-negative; moreover, the total mass equals zero if and only if the manifold is isometric to the…
We consider the homogeneous space $M=H\times H/\Delta K$, where $H/K$ is an irreducible symmetric space and $\Delta K$ denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of $H\times H$-invariant…
Einstein equations with $T_{\mu\nu} = k_\mu k_\nu + \ell_\mu \ell_\nu$ where $k, \ell$ are null are considered with spherical symmetry and staticity. The solution has naked singularity and is not asymptotically flat. However, it may be…
We study the H^n-Yamabe constants of Riemannian products (H^n \times M^m, g_h^n +g), where (M,g) is a compact Riemannian manifold of constant scalar curvature and g_h^n is the hyperbolic metric on H^n. Numerical calculations can be carried…
We study existence and regularity properties of stable positive solutions to the nonvariational problem - Delta u - b(x)|nabla u|^2 = lambda g(u) in a bounded smooth domain. In the case where b is constant, by means of a Hopf-Cole…
We consider solutions of the Einstein equations with cosmological constant $\Lambda\neq 0$ admitting conformal compactification with smooth scri $\mathscr{I^+}$. Metrics are written in the Bondi-Sachs coordinates and expanded into inverse…
It is shown that, $(a \Lambda^2 + b |H|^2)R$ in a spacetime of curvature $R$ is a natural ultraviolet $(U\!V)$ completion of $(a \Lambda^4 + b \Lambda^2 |H|^2)$ in the flat-spacetime Standard Model $(S\!M)$ with Higgs field $H$, $U\!V$…
The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using…