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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TL;DR: The vapor-liquid coexistence line for the Lennard-Jones fluid is constructed using the fact that the correlation length, and therefore the thermodynamic scalar curvature, must be equal in the two coexisting phases using the modified Benedict-Webb-Rubin equation of state.
Abstract: The behavior of thermodynamic response functions and the thermodynamic scalar curvature in the supercritical region have been studied for a Lennard-Jones fluid based on a revised modified Benedict-Webb-Rubin equation of state. Response function extrema are sometimes used to estimate the Widom line, which is characterized by the maxima of the correlation lengths. We calculated the Widom line for the Lennard-Jones fluid without using any response function extrema. Since the volume of the correlation length is proportional to the Riemannian thermodynamic scalar curvature, the locus of the Widom line follows the slope of maximum curvature. We show that the slope of the Widom line follows the slope of the isobaric heat capacity maximum only in the close vicinity of the critical point and that, therefore, the use of response function extrema in this context is problematic. Furthermore, we constructed the vapor-liquid coexistence line for the Lennard-Jones fluid using the fact that the correlation length, and therefore the thermodynamic scalar curvature, must be equal in the two coexisting phases. We compared the resulting phase envelope with those from simulation data where multiple histogram reweighting was used and found striking agreement between the two methods.
95 citations
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TL;DR: Soit (M,g) une variete de Riemann compacte de dimension n≥3 et soit Γ une sous-variete lisse fermee of dimension d. Alors il y a metrique conforme complete ĝ sur M^=M/Γ a courbure scalaire negative constante si et seulement si d>(n-2)/2
Abstract: Soit (M,g) une variete de Riemann compacte de dimension n≥3 et soit Γ une sous-variete lisse fermee de dimension d. Alors il y a une metrique conforme complete ĝ sur M^=M/Γ a courbure scalaire negative constante si et seulement si d>(n-2)/2
95 citations
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TL;DR: In this article, the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry K-means was studied.
Abstract: This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kaehler metrics.
95 citations
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01 Oct 2001
TL;DR: The Fermi Lectures of the Scuola Normale Superiore, Pisa, Italy, in June 2001 as mentioned in this paper focused on the noncollapsing situation of Riemannian manifolds.
Abstract: These notes are based on the Fermi Lectures delivered at the Scuola Normale Superiore, Pisa, in June 2001. The principal aim of the lectures was to present the structure theory developed by Toby Colding and myself, for metric spaces which are Gromov-Hausdorff limits of sequences of Riemannian manifolds which satisfy a uniform lower bound of Ricci curvature. The emphasis in the lectures was on the "non-collapsing" situation. A particularly interesting case is that in which the manifolds in question are Einstein (or Kahler-Einstein). Thus, the theory provides information on the manner in which Einstein metrics can degenerate.
95 citations
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15 Sep 2011TL;DR: In this paper, the existence of closed conformal and nonparallel vector fields on complete Riemannian hypersurfaces with nonpositive Ricci curvature has been examined and a generalization of theorems Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres has been obtained.
Abstract: In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.
95 citations