Topic
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.
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84 citations
TL;DR: In this paper, the authors study the Asharp-interface model for phase transitions, which incorporates the interaction of the phase boundaries with the walls of a container, and prove that if the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero.
Abstract: We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.
84 citations
TL;DR: In this article, the authors investigated the stability of cosmological scaling solutions describing a barotropic fluid with p = (-1) and a non-interacting scalar field with an exponential potential V() = V0e.
Abstract: We investigate the stability of cosmological scaling solutions describing a barotropic fluid with p = (-1) and a non-interacting scalar field with an exponential potential V() = V0e-. We study homogeneous and isotropic spacetimes with non-zero spatial curvature and find three possible asymptotic future attractors in an ever-expanding universe. One is the zero-curvature power-law inflation solution where 1 ( (2/3),2 3). We find that this matter scaling solution is unstable to curvature perturbations for > (2/3). The third possible future asymptotic attractor is a solution with negative spatial curvature where the scalar field energy density remains proportional to the curvature with 2/2 ( > (2/3),2 > 2). We find that solutions with 0 are never late-time attractors.
83 citations
Journal Article•
TL;DR: In this paper, it was shown that the Laplacian on an asymptotically flat manifold is an isomorphism between certain weighted Sobolev spaces, which is used to find a necessary and sufficient condition for a metric with vanishing scalar curvature to be conformally equivalent to one with zero curvature.
Abstract: It is shown that the Laplacian on an asymptotically flat manifold is an isomorphism between certain weighted Sobolev spaces. This is used to find a necessary and sufficient condition for an asymptotically flat metric to be conformally equivalent to one with vanishing scalar curvature. This in turn is used to give an example of a metric which cannot be conformally deformed within the class of asymptotically flat metrics to one with zero scalar curvature.
83 citations