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.

Asymptotic behavior of solutions to the σk-Yamabe equation near isolated singularities

Abstract: σ k -Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In (J. Funct. Anal. 233: 380–425, 2006) YanYan Li proved that an admissible solution with an isolated singularity at 0∈ℝ n to the σ k -Yamabe equation is asymptotically radially symmetric. In this work we prove that such a solution is asymptotic to a radial solution to the same equation on ℝ n ∖{0}. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al., we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and σ k curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.

4D gravity localized in non Z_2-symmetric thick branes

Abstract: We present a comparative analysis of localization of 4D gravity on a non Z_2-symmetric scalar thick brane in both a 5-dimensional Riemannian space time and a pure geometric Weyl integrable manifold. This work was mainly motivated by the hypothesis which claims that Weyl geometries mimic quantum behaviour classically. We start by obtaining a classical 4-dimensional Poincare invariant thick brane solution which does not respect Z_2-symmetry along the (non-)compact extra dimension. The scalar energy density of our field configuration represents several series of thick branes with positive and negative energy densities centered at y_0. The only qualitative difference we have encountered when comparing both frames is that the scalar curvature of the Riemannian manifold turns out to be singular for the found solution, whereas its Weylian counterpart presents a regular behaviour. By studying the transverse traceless modes of the fluctuations of the classical backgrounds, we recast their equations into a Schroedinger's equation form with a volcano potential of finite bottom (in both frames). By solving the Schroedinger equation for the massless zero mode m^2=0 we obtain a single bound state which represents a stable 4-dimensional graviton in both frames. We also get a continuum gapless spectrum of KK states with positive m^2>0 that are suppressed at y_0, turning into continuum plane wave modes as "y" approaches spatial infinity. We show that for the considered solution to our setup, the potential is always bounded and cannot adopt the form of a well with infinite walls; thus, we do not get a discrete spectrum of KK states, and we conclude that the claim that Weylian structures mimic, classically, quantum behaviour does not constitute a generic feature of these geometric manifolds.

Deforming hypersurfaces of the sphere by their mean curvature

Abstract: is satisfied. In [6] we studied hypersurfaces moving along their mean curvature vector in a general Riemannian manifold N" + 1. It was shown that all hypersurfaces Mo satisfying a suitable convexity condition will contract to a single point in finite time during this evolution. Here we want to show that in a spherical spaceform some convergence results can be obtained without assuming convexity for the initial hypersurface Mo. In particular, we will see that some hypersurfaces do not contract during this flow, but straighten out and become totally geodesic, i.e. in case N" + 1 = S" + 1 they converge to a "big S" ". To be precise, let g = {gij} and A = {hi j} be the induced metric and the second fundamental form on M and denote by H = giih~j, [A 12= h~Jh~j the mean curvature and the squared norm of the second fundamental form respectively.