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.

Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces

Abstract: In this paper, using the Schauder fixed point theorem, we prove existence results of radial solutions for Dirichlet problems in the unit ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces.

Wormholes in viable $f(R)$ modified theories of gravity and Weak Energy Condition

Abstract: In this work wormholes in viable $$f(R)$$ gravity models are analyzed. We are interested in exact solutions for stress-energy tensor components depending on different shape and redshift functions. Several solutions of gravitational equations for different $$f(R)$$ models are examined. The solutions found imply no need for exotic material, while this need is implied in the standard general theory of relativity. A simple expression for weak energy condition (WEC) violation near the throat is derived and analyzed. High curvature regime is also discussed, as well as the question of the highest possible values of the Ricci scalar for which the WEC is not violated near the throat, and corresponding functions are calculated for several models. The approach here differs from the one that has been common since no additional assumptions to simplify the equations have been made, and the functions in $$f(R)$$ models are not considered to be arbitrary functions, but rather a feature of the theory that has to be evaluated on the basis of consistency with observations for the Solar System and cosmological evolution. Therefore in this work we show that the existence of wormholes without exotic matter is not only possible in simple arbitrary $$f(R)$$ models, but also in models that are in accordance with empirical data.

Hypersurfaces in IRn+1 with prescribed gaussian curvature and related equations of monge—ampére type

Abstract: (1984). Hypersurfaces in IRn+1 with prescribed gaussian curvature and related equations of monge—ampere type. Communications in Partial Differential Equations: Vol. 9, No. 8, pp. 807-838.

On the volume functional of compact manifolds with boundary with constant scalar curvature

Abstract: We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.