The Experts below are selected from a list of 56796 Experts worldwide ranked by ideXlab platform
Gabriel Eduard Vilcu - One of the best experts on this subject based on the ideXlab platform.
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Hypersurfaces of a Sasakian Manifold
Mathematics, 2020Co-Authors: Haila Alodan, Sharief Deshmukh, Nasser Bin Turki, Gabriel Eduard VilcuAbstract:We extend the study of orientable Hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the Hypersurface, namely the tangential component of ξ to Hypersurface, and it also gives a smooth function ρ on the Hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable Hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the Hypersurface, showing that if the equality holds then the Hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ∇ ρ on a compact orientable Hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the Hypersurface is isometric to a sphere. Finally, we study compact orientable Hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the Hypersurface. In particular, we show that this upper bound is attained if and only if the Hypersurface is isometric to a sphere, provided that the Ricci curvature of the Hypersurface along ∇ ρ has a certain lower bound.
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a geometric perspective on the generalized cobb douglas production functions
Applied Mathematics Letters, 2011Co-Authors: Gabriel Eduard VilcuAbstract:Abstract In this work we obtain an interesting link between some fundamental notions in the theory of production functions and the differential geometry of Hypersurfaces in Euclidean spaces. In fact we establish that a generalized Cobb–Douglas production function has decreasing/increasing return to scale if and only if the corresponding Hypersurface has positive/negative Gaussian curvature. Moreover, this production function has constant return to scale if and only if the corresponding Hypersurface is developable.
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A geometric perspective on the generalized Cobb–Douglas production functions
Applied Mathematics Letters, 2011Co-Authors: Gabriel Eduard VilcuAbstract:Abstract In this work we obtain an interesting link between some fundamental notions in the theory of production functions and the differential geometry of Hypersurfaces in Euclidean spaces. In fact we establish that a generalized Cobb–Douglas production function has decreasing/increasing return to scale if and only if the corresponding Hypersurface has positive/negative Gaussian curvature. Moreover, this production function has constant return to scale if and only if the corresponding Hypersurface is developable.
Miguel Sanchez - One of the best experts on this subject based on the ideXlab platform.
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further results on the smoothability of cauchy Hypersurfaces and cauchy time functions
Letters in Mathematical Physics, 2006Co-Authors: Antonio N Bernal, Miguel SanchezAbstract:Recently, folk questions on the smoothability of Cauchy Hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy Hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy Hypersurface S, a Cauchy temporal function \(\mathcal{T}\) (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy Hypersurfaces as levels) with \(S= \mathcal{T}^{-1}(0)\) is constructed – thus, the spacetime splits orthogonally as \(\mathbb{R} \times S\) in a canonical way. Even more, accurate versions of this last result are obtained if the Cauchy Hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).
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further results on the smoothability of cauchy Hypersurfaces and cauchy time functions
arXiv: General Relativity and Quantum Cosmology, 2005Co-Authors: Antonio N Bernal, Miguel SanchezAbstract:Recently, folk questions on the smoothability of Cauchy Hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy Hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy Hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy Hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R \times S$ in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy Hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).
Jie Qing - One of the best experts on this subject based on the ideXlab platform.
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Möbius curvature, Laguerre curvature and Dupin Hypersurface
Advances in Mathematics, 2017Co-Authors: Jie Qing, Changping WangAbstract:Abstract In this paper we show that a Dupin Hypersurface with constant Mobius curvatures is Mobius equivalent to either an isoparametric Hypersurface in the sphere or a cone over an isoparametric Hypersurface in a sphere. We also show that a Dupin Hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric Hypersurface. These results solve the major issues related to the conjectures of Cecil et al. on the classification of Dupin Hypersurfaces.
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Hypersurfaces in hyperbolic space with support function
Advances in Mathematics, 2015Co-Authors: Vincent Bonini, Jose M Espinar, Jie QingAbstract:Abstract Based on [19] , we develop a global correspondence between immersed Hypersurfaces ϕ : M n → H n + 1 satisfying an exterior horosphere condition, also called here horospherically concave Hypersurfaces, and complete conformal metrics e 2 ρ g S n on domains Ω in the boundary S n at infinity of H n + 1 , where ρ is the horospherical support function, ∂ ∞ ϕ ( M n ) = ∂ Ω , and Ω is the image of the Gauss map G : M n → S n . To do so we first establish results on when the Gauss map G : M n → S n is injective. We also discuss when an immersed horospherically concave Hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed Hypersurfaces in H n + 1 and conformal metrics on domains in S n . Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave Hypersurface in H n + 1 of constant mean curvature.
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Mobius and Laguerre geometry of Dupin Hypersurfaces
arXiv: Differential Geometry, 2015Co-Authors: Jie Qing, Changping WangAbstract:In this paper we show that a Dupin Hypersurface with constant Mobius curvatures is Mobius equivalent to either an isoparametric Hypersurface in the sphere or a cone over an isoparametric Hypersurface in a sphere. We also show that a Dupin Hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hy- persurface. These results solve the major issues related to the conjectures of Cecil et al on the classification of Dupin Hypersurfaces.
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Hypersurfaces in hyperbolic space with support function
arXiv: Differential Geometry, 2012Co-Authors: Vincent Bonini, Jose M Espinar, Jie QingAbstract:In this paper we develop a global correspondence between immersed horospherically convex Hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex Hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles of elliptic problems of both Weingarten Hypersurfaces and complete conformal metrics and relations between them. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically convex Hypersurface of constant mean curvature in hyperbolic space.
Antonio N Bernal - One of the best experts on this subject based on the ideXlab platform.
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further results on the smoothability of cauchy Hypersurfaces and cauchy time functions
Letters in Mathematical Physics, 2006Co-Authors: Antonio N Bernal, Miguel SanchezAbstract:Recently, folk questions on the smoothability of Cauchy Hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy Hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy Hypersurface S, a Cauchy temporal function \(\mathcal{T}\) (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy Hypersurfaces as levels) with \(S= \mathcal{T}^{-1}(0)\) is constructed – thus, the spacetime splits orthogonally as \(\mathbb{R} \times S\) in a canonical way. Even more, accurate versions of this last result are obtained if the Cauchy Hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).
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further results on the smoothability of cauchy Hypersurfaces and cauchy time functions
arXiv: General Relativity and Quantum Cosmology, 2005Co-Authors: Antonio N Bernal, Miguel SanchezAbstract:Recently, folk questions on the smoothability of Cauchy Hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy Hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy Hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy Hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R \times S$ in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy Hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).
Young Jin Suh - One of the best experts on this subject based on the ideXlab platform.
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Contact Hypersurfaces in Kaehler manifolds
arXiv: Differential Geometry, 2013Co-Authors: Jurgen Berndt, Young Jin SuhAbstract:A contact Hypersurface in a Kaehler manifold is a real Hypersurface for which the induced almost contact metric structure determines a contact structure. We carry out a systematic study of contact Hypersurfaces in Kaehler manifolds. We then apply these general results to obtain classifications of contact Hypersurfaces with constant mean curvature in the complex quadric SO(n+2)/SO(n)SO(2) and its noncompact dual space SO(n,2)/SO(n)SO(2) for n > 2.
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real Hypersurfaces with isometric reeb flow in complex quadrics
International Journal of Mathematics, 2013Co-Authors: Jurgen Berndt, Young Jin SuhAbstract:We classify real Hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such Hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real Hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kahler manifolds which do not admit a real Hypersurface with isometric Reeb flow.
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real Hypersurfaces with isometric reeb flow in complex quadrics
arXiv: Differential Geometry, 2013Co-Authors: Jurgen Berndt, Young Jin SuhAbstract:We classify real Hypersurfaces with isometric Reeb flow in the complex quadrics Q^m for m > 2. We show that m is even, say m = 2k, and any such Hypersurface is an open part of a tube around a k-dimensional complex projective space CP^k which is embedded canonically in Q^{2k} as a totally geodesic complex submanifold. As a consequence we get the non-existence of real Hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics.
