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Richard Schoen - One of the best experts on this subject based on the ideXlab platform.
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the first steklov Eigenvalue conformal geometry and minimal surfaces
Advances in Mathematics, 2011Co-Authors: Ailana Fraser, Richard SchoenAbstract:We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first Nonzero Eigenvalue σ1 of the Dirichlet-to-Neumann map (Steklov Eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ1L(∂Σ)⩽2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ1(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.
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the first steklov Eigenvalue conformal geometry and minimal surfaces
arXiv: Differential Geometry, 2009Co-Authors: Ailana Fraser, Richard SchoenAbstract:We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first Nonzero Eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov Eigenvalue). For surfaces Sigma with genus gamma and k boundary components we obtain the upper bound sigma_1L(\partial \Sigma) \leq 2(2gamma+k)\pi. We attempt to find the best constant in this inequality for annular surfaces (gamma=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that $\sigma_1(\sig)L(\p\Sigma)$ is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. We prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least \pi.
Ailana Fraser - One of the best experts on this subject based on the ideXlab platform.
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the first steklov Eigenvalue conformal geometry and minimal surfaces
Advances in Mathematics, 2011Co-Authors: Ailana Fraser, Richard SchoenAbstract:We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first Nonzero Eigenvalue σ1 of the Dirichlet-to-Neumann map (Steklov Eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ1L(∂Σ)⩽2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ1(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.
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the first steklov Eigenvalue conformal geometry and minimal surfaces
arXiv: Differential Geometry, 2009Co-Authors: Ailana Fraser, Richard SchoenAbstract:We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first Nonzero Eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov Eigenvalue). For surfaces Sigma with genus gamma and k boundary components we obtain the upper bound sigma_1L(\partial \Sigma) \leq 2(2gamma+k)\pi. We attempt to find the best constant in this inequality for annular surfaces (gamma=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that $\sigma_1(\sig)L(\p\Sigma)$ is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. We prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least \pi.
Jose F Escobar - One of the best experts on this subject based on the ideXlab platform.
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an isoperimetric inequality and the first steklov Eigenvalue
Journal of Functional Analysis, 1999Co-Authors: Jose F EscobarAbstract:Let (Mn, g) be a compact Riemannian manifold with boundary. In this paper we give upper and lower estimates for the first Nonzero Steklov EigenvalueΔϕ=0inM,∂ϕ∂η=ν1ϕon∂M,where ν1 is a positive real number. The estimate from below is for a star-shaped domain on a manifold whose Ricci curvature is bounded from below. The upper estimate is for a convex manifold with nonnegative Ricci curvature and is given in terms of the first Nonzero Eigenvalue for the Laplacian on the boundary. We prove a comparison theorem for simply connected domains in a simply connected manifold. We exhibit annuli domains for which the comparison theorem fails to be true. In (J. F. Escobar, J. Funct. Anal.60 (1997), 544–556) we introduced the isoperimetric constant I(M) defined asI(M)=infΩ⊂MVol(Σ)min{Vol(Ω1), Vol(Ω2)}, where Ω1=Ω∩∂M is a nonempty domain with boundary in the manifold ∂M, Ω2=∂M−Ω1, and Σ=∂Ω∩int(M), where int(M) is the interior of M. We proved a Cheeger's type inequality for ν1 using the constant I(M). In this paper we give upper and lower estimates for the constant I in terms of isoperimetric constants of the boundary of M.
Sharief Deshmukh - 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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First Nonzero Eigenvalue of a minimal hypersurface in the unit sphere
Annali di Matematica Pura ed Applicata, 2011Co-Authors: Sharief DeshmukhAbstract:In this paper, it is shown that the first Nonzero Eigenvalue λ1 of the Laplacian operator on a compact immersed minimal hypersurface M in the unit sphere S n+1 satisfies one of the following $$ (i)\lambda _{1}=n, \quad (ii)\lambda _{1} \leq (1+k_{0})n, \quad (iii)\lambda _{1}\geq n+\frac{n}{2}(nk_{0}-(n-1))$$ where k 0 is the infimum of the sectional curvatures of M. It is also shown that a compact immersed minimal hypersurface of the unit sphere S n+1 with λ1 = n is either isometric to the unit sphere S n or else k 0 < n −1(n−1).
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Spherical submanifolds in a Euclidean space
Monatshefte für Mathematik, 2007Co-Authors: Haila Alodan, Sharief DeshmukhAbstract:In this paper, we are interested in extending the study of spherical curves in R ^3 to the submanifolds in the Euclidean space R ^ n + p . More precisely, we are interested in obtaining conditions under which an n -dimensional compact submanifold M of a Euclidean space R ^ n + p lies on the hypersphere S ^ n + p −1( c ) (standardly imbedded sphere in R ^ n + p of constant curvature c ). As a by-product we also get an estimate on the first Nonzero Eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n -dimensional sphere S ^ n ( c ) (cf. Theorem 4.1).
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Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature
The Journal of Geometric Analysis, 2005Co-Authors: Sharief Deshmukh, Afifah Al-eidAbstract:Let (M, g) be an n-dimensional compact and connected Riemannian manifold of constant scalar curvature. If the sectional curvatures of M are bounded below by a constant α > 0, and the Ricci curvature satisfies Ric < (n − 1)αδ, δ ≥ 1, then it is shown that either M is isometric to the n-sphere S^n(α) or else each Nonzero Eigenvalue λ of the Laplacian acting on the smooth functions of M satisfies the following: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae83UdW% 2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaad6gacqaHXoqy% caGGOaGae8hTdqMaeyOeI0IaaGOmaiaacMcacqWF7oaBcqWFRaWkcq% WFYaGmcaWGUbGaeqySde2aaWbaaSqabeaacaaIYaaaaOGae8hTdqMa% e8hkaGIae8xmaeJae83kaSIae8hkaGccbiGaa4NBaiab-jHiTiab-f% daXiab-LcaPiab-r7aKjab-LcaPiab-5da+iab-bdaWaaa!556A! $$\lambda ^2 + 3n\alpha (\delta - 2)\lambda + 2n\alpha ^2 \delta (1 + (n - 1)\delta ) > 0$$ .
David Rideout - One of the best experts on this subject based on the ideXlab platform.
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Oriented matroidscombinatorial structures underlying loop quantum gravity
Classical and Quantum Gravity, 2010Co-Authors: Johannes Brunnemann, David RideoutAbstract:We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator [ 1 ] in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence vertices of a graph in 3-dimensional Riemannian space, and can be represented by sign strings containing relative orientations of embedded edges. We demonstrate that these signature factors are a special representation of the general mathematical concept of an oriented matroid [ 2, 3 ]. Moreover, we show that oriented matroids can also be used to describe the topology (connectedness) of directed graphs. Hence the mathematical methods developed for oriented matroids can be applied to the difficult combinatorics of embedded graphs underlying the construction of LQG. As a first application we revisit the analysis of [ 4, 5 ], and find that enumeration of all possible sign configurations used there is equivalent to enumerating all realizable oriented matroids of rank 3 [ 2, 3 ], and thus can be greatly simplified. We find that for 7-valent vertices having no coplanar triples of edge tangents, the smallest Nonzero Eigenvalue of the volume spectrum does not grow as one increases the maximum spin jmax at the vertex, for any orientation of the edge tangents. This indicates that, in contrast to the area operator, considering large jmax does not necessarily imply large volume Eigenvalues. In addition we give an outlook to possible starting points for rewriting the combinatorics of LQG in terms of oriented matroids.
