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William Beckner - One of the best experts on this subject based on the ideXlab platform.
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sobolev inequalities the poisson semigroup and analysis on the Sphere Sn
Proceedings of the National Academy of Sciences of the United States of America, 1992Co-Authors: William BecknerAbstract:Abstract Hypercontractive estimates are obtained for the Poisson semigroup on Lp(Sn). Such estimates are determined by a sharp logarithmic Sobolev inequality that relates the entropy of a function on Sn to its smoothness.
Qingming Cheng - One of the best experts on this subject based on the ideXlab platform.
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hypersurfaces in a unit Sphere Sn 1 1 with constant scalar curvature
Journal of The London Mathematical Society-second Series, 2001Co-Authors: Qingming ChengAbstract:The paper considers n -dimensional hypersurfaces with constant scalar curvature of a unit Sphere S n −1 (1). The hypersurface S k ( c 1 )× S n − k ( c 2 ) in a unit Sphere S n +1 (1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit Sphere S n +1 (1) which are not congruent to each other in it. In particular, it is proved that if M is an n -dimensional ( n > 3) complete locally conformally flat hypersurface with constant scalar curvature n ( n −1) r in a unit Sphere S n +1 (1), then r > 1−2/ n , and (1) when r ≠ ( n −2)/( n −1), if then M is isometric to S 1 (√1− c 2 )× S n −1 ( c ), where S is the squared norm of the second fundamental form of M ; (2) there are no complete hypersurfaces in S n +1 (1) with constant scalar curvature n ( n −1) r and with two distinct principal curvatures, one of which is simple, such that r = ( n −2)/( n −1) and
Xiong Changwei - One of the best experts on this subject based on the ideXlab platform.
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Inequalities of Alexandrov-Fenchel type for convex hypersurfaces in hyperbolic space and in the Sphere
University of California, 2018Co-Authors: Wei Yong, Xiong ChangweiAbstract:By applying the unit normal flow to well-known inequalities in hyperbolic space Hn+1 and in the Sphere Sn+1, we derive some new inequalities of Alexandrov-Fenchel type for closed convex hypersurfaces in these spaces. We also use the inverse mean curvature flow in the Sphere to prove an optimal Sobolev-type inequality for closed convex hypersurfaces in the Sphere
Amacha Inas - One of the best experts on this subject based on the ideXlab platform.
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Yamabe flow with prescribed scalar curvature
2017Co-Authors: Amacha InasAbstract:Cette thèse est consacrée à l'étude d'une famille des flots géométriques associés au problème de la courbure scalaire prescrite sur une variété riemannienne compacte. Plus précisément, si on désigne par (M,g0) une variété riemannienne compacte de dimension n≥3, et si F∈C∞ (M) est une fonction donnée, le problème de la courbure scalaire prescrite consiste à trouver une métrique g conforme à g0 telle que F soit sa courbure scalaire. Ce problème est équivalent à la résolution de l'EDP suivante :-4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Où R0 est la courbure scalaire de la métrique initiale g0 et ∆ est le laplacien associé à g0. Il s'agit d'une équation elliptique non-linéaire dont la difficulté principale provient du terme u((n+2)/(n-2 )). Hormis le cas de la sphère standard Sn , tous les travaux consacrés à l'étude de l'équation (E) sont basés sur la méthode variationnelle. Dans cette thèse, on développe une autre approche basée sur l'étude d'une famille de flots géométriques qui permet, entre autres, de résoudre l'équation (E). La question dépend bien entendu de la métrique initiale g0 et en particulier du signe de sa courbure scalaire R0. Les flots introduits sont des flots de gradient associés à deux fonctionnelles distinctes dépendant du signe de R0. La première partie de cette thèse est consacrée au cas R00. Dans les deux cas, on démontre l'existence globale du flot et on étudie son comportement asymptotique à l'infini.This thesis is devoted to the study of a family of geometric flows associated with the prescribed scalar curvature problem. More precisely, if we denote by (M,g0) a compact riemannian manifold with dimension n≥3, and if F∈C∞ (M) is a given function, the prescribed scalar curvature problem consists of finding a conformal metric g to g0 such that F is its scalar curvature. This problem is equivalent to the resolution of the following PDE : -4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Where R0 is the scalar curvature of the initial metric g0 and ∆ is the laplacian associated with g0.It is a nonlinear elliptic equation, whose the main difficulty comes from the term u((n+2)/(n-2 )). Apart from the case of the standard Sphere Sn all the works that study the equation (E) are based on the variational method. In this thesis, we develop another approach based on the study of a family of geometric flows which allows to solve equation (E).The flows introduced are gradient flows associated with two distinct functional functions depending on the sign of R0.The first part of this thesis is devoted to the case R00. In both cases, our aim is to proof the global existence of the flow and study its asymptotic behavior at infinity
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Flot de Yamabe avec courbure scalaire prescrite
HAL CCSD, 2017Co-Authors: Amacha InasAbstract:This thesis is devoted to the study of a family of geometric flows associated with the prescribed scalar curvature problem. More precisely, if we denote by (M,g0) a compact riemannian manifold with dimension n≥3, and if F∈C∞ (M) is a given function, the prescribed scalar curvature problem consists of finding a conformal metric g to g0 such that F is its scalar curvature. This problem is equivalent to the resolution of the following PDE : -4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Where R0 is the scalar curvature of the initial metric g0 and ∆ is the laplacian associated with g0.It is a nonlinear elliptic equation, whose the main difficulty comes from the term u((n+2)/(n-2 )). Apart from the case of the standard Sphere Sn all the works that study the equation (E) are based on the variational method. In this thesis, we develop another approach based on the study of a family of geometric flows which allows to solve equation (E).The flows introduced are gradient flows associated with two distinct functional functions depending on the sign of R0.The first part of this thesis is devoted to the case R00. In both cases, our aim is to proof the global existence of the flow and study its asymptotic behavior at infinity.Cette thèse est consacrée à l'étude d'une famille des flots géométriques associés au problème de la courbure scalaire prescrite sur une variété riemannienne compacte. Plus précisément, si on désigne par (M,g0) une variété riemannienne compacte de dimension n≥3, et si F∈C∞ (M) est une fonction donnée, le problème de la courbure scalaire prescrite consiste à trouver une métrique g conforme à g0 telle que F soit sa courbure scalaire. Ce problème est équivalent à la résolution de l'EDP suivante :-4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Où R0 est la courbure scalaire de la métrique initiale g0 et ∆ est le laplacien associé à g0. Il s'agit d'une équation elliptique non-linéaire dont la difficulté principale provient du terme u((n+2)/(n-2 )). Hormis le cas de la sphère standard Sn , tous les travaux consacrés à l'étude de l'équation (E) sont basés sur la méthode variationnelle. Dans cette thèse, on développe une autre approche basée sur l'étude d'une famille de flots géométriques qui permet, entre autres, de résoudre l'équation (E). La question dépend bien entendu de la métrique initiale g0 et en particulier du signe de sa courbure scalaire R0. Les flots introduits sont des flots de gradient associés à deux fonctionnelles distinctes dépendant du signe de R0. La première partie de cette thèse est consacrée au cas R00. Dans les deux cas, on démontre l'existence globale du flot et on étudie son comportement asymptotique à l'infini
Wei Yong - One of the best experts on this subject based on the ideXlab platform.
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Inequalities of Alexandrov-Fenchel type for convex hypersurfaces in hyperbolic space and in the Sphere
University of California, 2018Co-Authors: Wei Yong, Xiong ChangweiAbstract:By applying the unit normal flow to well-known inequalities in hyperbolic space Hn+1 and in the Sphere Sn+1, we derive some new inequalities of Alexandrov-Fenchel type for closed convex hypersurfaces in these spaces. We also use the inverse mean curvature flow in the Sphere to prove an optimal Sobolev-type inequality for closed convex hypersurfaces in the Sphere
