We show that Einstein equations are compatible with the presence of massive point particle idealization and find the corresponding two parameter family of solutions. They are complete defined by the bare mechanical mass M > 0 and the Keplerian mass m > 0 (m < M) of the point source of gravity. The global analytical properties of these solutions in the complex plane define a unique preferable ra...

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,\,g,\,u)$ of dimension $n\geq 4$ with boundary $\partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, the standard hemisphere $\Bbb {S}^n_+,$ or $g=dt^{2}+\psi ^{2}(t)g_{L}$ $u=u(t),$ where $g_{L}$ Einstein Ricci curvature. A similar classification result obtained by assum...

We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton’s equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometr...