The Experts below are selected from a list of 114 Experts worldwide ranked by ideXlab platform
Fabiano G B Brito - One of the best experts on this subject based on the ideXlab platform.
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degree of the gauss map and curvature integrals for closed hypersurfaces
Results in Mathematics, 2018Co-Authors: Fabiano G B Brito, Icaro GoncalvesAbstract:Given a Unit Vector Field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of invariants which combines the second fundamental form of the hypersurface and the covariant derivative of the Vector Field. We show how these invariants can be used as obstructions to the existence of codimension one foliations with prescribed geometric properties.
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a topological lower bound for the energy of a Unit Vector Field on a closed euclidean hypersurface
arXiv: Differential Geometry, 2017Co-Authors: Fabiano G B Brito, Icaro Goncalves, Adriana V. NicoliAbstract:For a Unit Vector Field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\geq 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the Unit sphere $\mathbb{S}^{2n+1}$, immersed with degree one, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals $\mathcal{B}_k$ on a compact Riemannian manifold $M^{m}$, $1\leq k\leq m$, and show that, when the underlying manifold is a closed hypersurface, these functionals possess similar properties regarding the degree of the immersion. In addition, we prove that Hopf flows minimize $\mathcal{B}_n$ on $\mathbb{S}^{2n+1}$.
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a topological lower bound for the energy of a Unit Vector Field on a closed hypersurface of the euclidean space the 3 dimensional case
arXiv: Differential Geometry, 2016Co-Authors: Fabiano G B Brito, Andre O Gomes, Adriana V. NicoliAbstract:In this short note we prove that the degree of the Gauss map {\nu} of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.
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degree of the gauss map and curvature integrals for closed hypersurfaces
arXiv: Differential Geometry, 2016Co-Authors: Fabiano G B Brito, Icaro GoncalvesAbstract:Given a Unit Vector Field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of topological invariants which combines the second fundamental form of the hypersurface and the Vector Field itself. We show how these invariants can be used as obstructions to certain codimension one foliations.
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on the volume of Unit Vector Fields on spaces of constant sectional curvature
Commentarii Mathematici Helvetici, 2004Co-Authors: Fabiano G B Brito, Pablo M Chacon, A M NaveiraAbstract:A Unit Vector Field X on a Riemannian manifold determines a submanifold in the Unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel Fields are the trivial minima.
Arghir Zarnescu - One of the best experts on this subject based on the ideXlab platform.
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orientability and energy minimization in liquid crystal models
Archive for Rational Mechanics and Analysis, 2011Co-Authors: J M Ball, Arghir ZarnescuAbstract:Uniaxial nematic liquid crystals are modelled in the Oseen–Frank theory through a Unit Vector Field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to −n. This symmetry is preserved in the constrained Landau–de Gennes theory that works with the tensor \({Q=s \left(n\otimes n-\frac{1}{3} Id\right)}\). We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class W1,2 the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains with holes and various boundary conditions, for the simplest form of the energy functional, we completely characterise the instances in which the predictions of the constrained Landau–de Gennes theory differ from those of the Oseen–Frank theory.
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orientability and energy minimization in liquid crystal models
arXiv: Analysis of PDEs, 2010Co-Authors: J M Ball, Arghir ZarnescuAbstract:Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a Unit Vector Field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory.
Alexander Yampolsky - One of the best experts on this subject based on the ideXlab platform.
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On the mean curvature of a Unit Vector Field.
arXiv: Differential Geometry, 2005Co-Authors: Alexander YampolskyAbstract:We present an explicit formula for the mean curvature of a Unit Vector Field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal Unit Vector Fields. We also give an example of a Vector Field of constant mean curvature on the Lobachevsky (n + 1) space.
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On the intrinsic geometry of a Unit Vector Field
arXiv: Differential Geometry, 2005Co-Authors: Alexander YampolskyAbstract:We study the geometrical properties of a Unit Vector Field on a Riemannian 2-manifold, considering the Field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature K, we give a description of the totally geodesic Unit Vector Fields for K=0 and K=1 and prove a non-existence result for K not equal to 0 and 1. We also found a family of Vector Fields on the hyperbolic 2-plane L^2 of curvature -c^2 which generate foliations on Unit tangent bundle over L^2 with leaves of constant intrinsic curvature -c^2 and of constant extrinsic curvature -c^2/4.
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on the intrinsic geometry of a Unit Vector Field
Commentationes Mathematicae Universitatis Carolinae, 2002Co-Authors: Alexander YampolskyAbstract:We study the geometrical properties of a Unit Vector Field on a Riemannian 2-manifold, considering the Field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature K, we give a description of the totally geodesic Unit Vector Fields for K = 0 and K = 1 and prove a non-existence result for K 6 0,1. We also found a family �! of Vector Fields on the hyperbolic 2-plane L 2 of curvature −c 2 which generate foliations on T1L 2 with leaves of constant intrinsic curvature −c 2 and of constant extrinsic curvature − c 2 4 .
Adriana V. Nicoli - One of the best experts on this subject based on the ideXlab platform.
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Loxodromic Unit Vector Field on punctured spheres.
arXiv: Differential Geometry, 2019Co-Authors: Jackeline Conrado, Adriana V. Nicoli, Giovanni N. NunesAbstract:In these short notes we characterize the loxodromic Unit Vector Fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincar\'e indexes around their singularities.
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a topological lower bound for the energy of a Unit Vector Field on a closed euclidean hypersurface
arXiv: Differential Geometry, 2017Co-Authors: Fabiano G B Brito, Icaro Goncalves, Adriana V. NicoliAbstract:For a Unit Vector Field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\geq 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the Unit sphere $\mathbb{S}^{2n+1}$, immersed with degree one, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals $\mathcal{B}_k$ on a compact Riemannian manifold $M^{m}$, $1\leq k\leq m$, and show that, when the underlying manifold is a closed hypersurface, these functionals possess similar properties regarding the degree of the immersion. In addition, we prove that Hopf flows minimize $\mathcal{B}_n$ on $\mathbb{S}^{2n+1}$.
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a topological lower bound for the energy of a Unit Vector Field on a closed hypersurface of the euclidean space the 3 dimensional case
arXiv: Differential Geometry, 2016Co-Authors: Fabiano G B Brito, Andre O Gomes, Adriana V. NicoliAbstract:In this short note we prove that the degree of the Gauss map {\nu} of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.
J M Ball - One of the best experts on this subject based on the ideXlab platform.
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orientability and energy minimization in liquid crystal models
Archive for Rational Mechanics and Analysis, 2011Co-Authors: J M Ball, Arghir ZarnescuAbstract:Uniaxial nematic liquid crystals are modelled in the Oseen–Frank theory through a Unit Vector Field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to −n. This symmetry is preserved in the constrained Landau–de Gennes theory that works with the tensor \({Q=s \left(n\otimes n-\frac{1}{3} Id\right)}\). We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class W1,2 the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains with holes and various boundary conditions, for the simplest form of the energy functional, we completely characterise the instances in which the predictions of the constrained Landau–de Gennes theory differ from those of the Oseen–Frank theory.
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orientability and energy minimization in liquid crystal models
arXiv: Analysis of PDEs, 2010Co-Authors: J M Ball, Arghir ZarnescuAbstract:Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a Unit Vector Field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory.
