The Experts below are selected from a list of 56934 Experts worldwide ranked by ideXlab platform
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).
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).
Francesco Russo - One of the best experts on this subject based on the ideXlab platform.
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On cubic Hypersurfaces with vanishing hessian
Journal of Pure and Applied Algebra, 2015Co-Authors: Rodrigo Gondim, Francesco RussoAbstract:We prove that for N≤6 an irreducible cubic hypersurface with vanishing hessian in PN is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of Special Perazzo Cubic Hypersurfaces, which, a posteriori, exhaust the class of cubic Hypersurfaces with vanishing hessian, not cones, for N≤6. Finally we show by pertinent examples the technical difficulties arising for N≥7.
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On cubic Hypersurfaces with vanishing hessian
arXiv: Algebraic Geometry, 2013Co-Authors: Rodrigo Gondim, Francesco RussoAbstract:If $X = V(f) \subset \mathbb P^N$ is a reduced complex hypersurface, the hessian of $f$ (or by abusing the terminology the hessian of $X$) is the determinant of the matrix of the second derivatives of the form $f$, that is the determinant of the hessian matrix of $f$. Hypersurfaces with vanishing hessian were studied systematically for the first time in the fundamental paper [GN], where Gordan and M. Noether analyze Hesse's claims in [Hesse1, Hesse2] according to which these Hypersurfaces are necessarily cones. Of course cones have vanishing hessian. Clearly the claim is true if deg(X)=2 so that the first relevant case for the problem is that of cubic Hypersurfaces. One immediately sees that $V(x_0x_3^2 + x_1x_3x_4 + x_2x_4^2)\subset \mathbb P^4$ is a cubic hypersurface with vanishing hessian but not a cone (for example one could check that the first partial derivatives of the equation are linearly independent). As firstly pointed out in [GN], the claim is true for $N\leq 3$ and in general false for every $N\geq 4$. Here we prove that for $N\leq 6$ an irreducible cubic hypersurface with vanishing hessian in $\mathbb P^N$ is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of {\it Special Perazzo Cubic Hypersurfaces}, which, a posteriori, exhaust the class of cubic Hypersurfaces with vanishing hessian, not cones, for $N\leq 6$. Finally we show by pertinent examples the technical difficulties arising for $N\geq 7$.
Tommaso De Fernex - One of the best experts on this subject based on the ideXlab platform.
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Birational geometry of singular Fano Hypersurfaces
arXiv: Algebraic Geometry, 2012Co-Authors: Tommaso De FernexAbstract:We establish birational superrigidity for a large class of singular projective Fano Hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) Hypersurfaces with semi-homogeneous singularities of multiplicity roughly bounded by half of the dimension of the hypersurface, (2) Hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) Hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree close enough to the expected degree.
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Birational rigidity of singular Fano Hypersurfaces
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 2012Co-Authors: Tommaso De FernexAbstract:We establish birational superrigidity for a large class of singular projective Fano Hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) Hypersurfaces with semi-homogeneous singularities of multiplicity asymptotically bounded by twice the square root of the dimension of the hypersurface, (2) Hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) Hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree sufficiently close to the expected degree.
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.
