Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics

Abstract: Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive hermitian line bundle (L,h) → M over a compact Kahler manifold: the expected distribution K crit (z) of critical points dlog |s(z)|h = 0 of a Gaussian random holomorphic section s ∈ H 0 (M,L) with respect to h. It is a measure on M whose total mass is the average number N crit h of critical points of a random holomorphic section. We are interested in the metric dependence of N crit h , especially metrics h which minimize N crit h . We concentrate on the asymptotic minimization problem for the sequence of tensor powers (L N ,h N ) → M of the line bundle and their critical point densities K critN (z). We prove that K critN (z) has a complete asymptotic expansion in N whose coefficients are curvature invariants of h. The first two terms in the expansion of N crit hN are topological invariants of (L,M). The third term is a topological invariant plus a constantm 2 (depending only on the dimension m of M) times the Calabi functional R M � 2 dV olh, whereis the scalar curvature of the curvature form of h. We give an integral formula form 2 and show, by a computer assisted calculation, thatm 2 > 0 for m ≤ 3, hence that N crit hN is asymptotically minimized by the Calabi extremal metric (when one exists). We conjecture thatm 2 > 0 in all dimensions, i.e. that the Calabi extremal metric is always the asymptotic minimizer.

Spherically symmetric black holes in $f(R)$ gravity: Is geometric scalar hair supported ?

Abstract: gravity. We focus then on the analysis and quest of non-trivial (i.e. hairy) asymptotically at (AF) BH solutions in static and spherically symmetric (SSS) spacetimes in vacuum having the property that the Ricci scalar does not vanish identically in the domain of outer communication. To do so, we provide and enforce the regularity conditions at the horizon in order to prevent the presence of singular solutions there. Specically, we consider several classes of f(R) models like those proposed recently for explaining the accelerated expansion in the universe and which have been thoroughly tested in several physical scenarios. Finally, we report analytical and numerical evidence about the absence of geometric hair in AFSSSBH solutions in those f(R) models. First, we submit the models to the available no-hair theorems, and in the cases where the theorems apply, the absence of hair is demonstrated analytically. In the cases where the theorems do not apply, we resort to a numerical analysis due to the complexity of the non-linear dierential equations. Within that aim, a code to solve the equations numerically was built and tested using well know exact solutions. In a future investigation we plan to analyze the problem of hair in De Sitter and Anti-De Sitter backgrounds.

Isotropic transport process on a Riemannian manifold

Abstract: We construct a canonical Markov process on the tangent bundle of a complete Riemannian manifold, which generalizes the isotropic scattering transport process on Euclidean space. By inserting a small parameter it is proved that the transition semigroup converges to the Brownian motion semigroup provided that the latter preserves the class C0. The special case of a manifold of negative curvature is considered as an illustration.