The Experts below are selected from a list of 103365 Experts worldwide ranked by ideXlab platform
Simon Brendle - One of the best experts on this subject based on the ideXlab platform.
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the isoperimetric inequality for a minimal submanifold in Euclidean Space
Journal of the American Mathematical Society, 2021Co-Authors: Simon BrendleAbstract:We prove a Sobolev inequality which holds on submanifolds in Euclidean Space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean Space of codimension at most 2.
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the logarithmic sobolev inequality for a submanifold in Euclidean Space
Communications on Pure and Applied Mathematics, 2020Co-Authors: Simon BrendleAbstract:We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean Space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.
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the isoperimetric inequality for a minimal submanifold in Euclidean Space
arXiv: Differential Geometry, 2019Co-Authors: Simon BrendleAbstract:We prove an isoperimetric inequality which holds for minimal submanifolds in Euclidean Space of arbitrary dimension and codimension. Our estimate is sharp if the codimension is at most $2$.
Rafael Lopez - One of the best experts on this subject based on the ideXlab platform.
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invariant surfaces in Euclidean Space with a log linear density
Advances in Mathematics, 2018Co-Authors: Rafael LopezAbstract:Abstract A λ-translating soliton with density vector v → is a surface in Euclidean Space whose mean curvature H satisfies 2 H = 2 λ + 〈 N , v → 〉 , where N is the Gauss map. We classify all λ-translating solitons that are invariant by a one-parameter group of translations and a one-parameter group of rotations.
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translation surfaces in Euclidean Space with constant gaussian curvature
arXiv: Differential Geometry, 2018Co-Authors: Thomas Hasanis, Rafael LopezAbstract:We prove that the only surfaces in $3$-dimensional Euclidean Space $\R^3$ with constant Gaussian curvature $K$ and constructed by the sum of two Space curves are cylindrical surfaces, in particular, $K=0$.
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invariant surfaces in Euclidean Space with a log linear density
arXiv: Differential Geometry, 2018Co-Authors: Rafael LopezAbstract:A $\lambda$-translating soliton with density vector $\vec{v}$ is a surface in Euclidean Space whose mean curvature $H$ satisfies $2H=2\lambda+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map. We classify all $\lambda$-translating solitons that are invariant by a one-parameter group of translations and a one-parameter group of rotations.
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minimal translation surfaces in Euclidean Space
Journal of Geometric Analysis, 2017Co-Authors: Rafael Lopez, Oscar PerdomoAbstract:A translation surface in Euclidean Space is a surface that is the sum of two regular curves \(\alpha \) and \(\beta \). In this paper we characterize all minimal translation surfaces. In the case that \(\alpha \) and \(\beta \) are non-planar curves, we prove that the curvature \(\kappa \) and the torsion \(\tau \) of both curves must satisfy the equation \(\kappa ^2 \tau = C\) where C is constant. We show that, up to a rigid motion and a dilation in the Euclidean Space and, up to reparametrizations of the curves generating the surfaces, all minimal translation surfaces are described by two real parameters \(a,b\in \mathbb {R}\) where the surface is of the form \(\phi (s,t)=\beta _{a,b}(s)+\beta _{a,b}(t)\).
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minimal surfaces in Euclidean Space with a log linear density
arXiv: Differential Geometry, 2014Co-Authors: Rafael LopezAbstract:We study surfaces in Euclidean Space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(\alpha,\beta,\gamma)$ and the surface must be rotational. We also classify all minimal surfaces of translation type.
R L Jaffe - One of the best experts on this subject based on the ideXlab platform.
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quantum mechanics on manifolds embedded in Euclidean Space
Annals of Physics, 2003Co-Authors: Philip Schuster, R L JaffeAbstract:Abstract Quantum particles confined to surfaces in higher-dimensional Spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when confinement is implemented by forces that act normal to the surface. We review this confining potential formalism applied to the confinement of a particle to an arbitrary manifold embedded in a higher-dimensional Euclidean Space. We devote special attention to the geometrically induced gauge potential that appears in the effective Hamiltonian for motion on the surface. We emphasize that the gauge potential is only present when the Space of states describing the degrees of freedom normal to the surface is degenerate. We also distinguish between the effects of the intrinsic and extrinsic geometry on the effective Hamiltonian and provide simple expressions for the induced-scalar potential. We discuss examples including the case of a three-dimensional manifold embedded in a five-dimensional Euclidean Space.
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quantum mechanics on manifolds embedded in Euclidean Space
arXiv: High Energy Physics - Theory, 2003Co-Authors: Philip Schuster, R L JaffeAbstract:Quantum particles confined to surfaces in higher dimensional Spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when confinement is implemented by forces that act normal to the surface. We review this confining potential formalism applied to the confinement of a particle to an arbitrary manifold embedded in a higher dimensional Euclidean Space. We devote special attention to the geometrically induced gauge potential that appears in the effective Hamiltonian for motion on the surface. We emphasize that the gauge potential is only present when the Space of states describing the degrees of freedom normal to the surface is degenerate. We also distinguish between the effects of the intrinsic and extrinsic geometry on the effective Hamiltonian and provide simple expressions for the induced scalar potential. We discuss examples including the case of a 3-dimensional manifold embedded in a 5-dimensional Euclidean Space.
Ni Xiang - One of the best experts on this subject based on the ideXlab platform.
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a class of hessian quotient equations in Euclidean Space
Journal of Differential Equations, 2020Co-Authors: Xiaojuan Chen, Ni XiangAbstract:Abstract In this paper, we consider a class of Hessian quotient equations in Euclidean Space. Under some sufficient conditions, we obtain an existence result using standard degree theory based on a priori estimates for solutions to Hessian quotient equations.
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a class of hessian quotient equations in Euclidean Space
arXiv: Analysis of PDEs, 2020Co-Authors: Xiaojuan Chen, Ni XiangAbstract:In this paper, we consider a Class of Hessian quotient equations in Euclidean Space. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the Hessian quotient equations.
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A Class of Hessian quotient equations in Euclidean Space
2020Co-Authors: Chen Xiaojuan, Tu Qiang, Ni XiangAbstract:In this paper, we consider a Class of Hessian quotient equations in Euclidean Space. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the Hessian quotient equations.Comment: 23 page
Adam Timar - One of the best experts on this subject based on the ideXlab platform.
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a nonamenable factor of a Euclidean Space
Annals of Probability, 2021Co-Authors: Adam TimarAbstract:Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean Space Rd, d≥3, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in Rd as an isometry-invariant random partition of Rd to bounded polyhedra, and also as an isometry-invariant random partition of Rd to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID’s.
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a nonamenable factor of a Euclidean Space
arXiv: Probability, 2017Co-Authors: Adam TimarAbstract:Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean Space $\mathbb{R}^d$, $d\geq 3$, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in $\mathbb{R}^d$ as an isometry-invariant random partition of $\mathbb{R}^d$ to bounded polyhedra, and also as an isometry-invariant random partition of $\mathbb{R}^d$ to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of iid's.
