Showing papers in "Annals of Global Analysis and Geometry in 1992"
TL;DR: In this paper, the authors considered the Agmon-Miranda maximum principle for strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point and gave necessary and sufficient conditions for the validity of this principle.
Abstract: In this paper the Agmon-Miranda maximum principle for solutions of strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point is considered. Necessary and sufficient conditions for the validity of this principle are given both for smooth solutions of the equation Lu = 0 in G and for the generalized solution of the problem Lu = 0 in G, D
u = gk on ∂G (k = 0,...,m-1). It will be shown that for every elliptic operator L of order 2m > 2 there exists such a cone in ℝn (n≥4) that the Agmon-Miranda maximum principle fails in this cone.
31 citations
TL;DR: In this paper, the existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2.
Abstract: Existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2. Application to prescribed curvature problems: scalar curvature in a quasi-isometry class (including a contribution to the Lichnerowicz-York equation of General Relativity); Ricci curvature in a weighted Kahler class (with a related result in equiaffine geometry). A new asymptonic behaviour is allowed throughout, called partial decay, which requires its own maximum principle.
28 citations
TL;DR: In this article, the Laplacian of a harmonic mapping from a Riemannian 3-manifold to a 2-manivold is computed and a maximum principle is applied to derive criteria when a harmonic map must be a harmonic morphism.
Abstract: Let ϕ be a harmonic mapping from a Riemannian 3-manifold to a Riemannian 2-manifold. A smooth function on M is associated to ϕ, derived from the eigenvalues of the first fundamental form, the vanishing of which is equivalent to ϕ being a harmonic morphism. The Laplacian of this function is computed and a maximum principle applied to derive criteria when a harmonic map must be a harmonic morphism.
18 citations
TL;DR: In this paper, upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds were derived for spheres of constant curvature, and the first eigenvalues were sharp for spheres with constant curvatures.
Abstract: We derive upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds. In the case of the classical Dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature.
16 citations
TL;DR: In this article, the smooth structure of principal lines near a closed principal cycle was determined in terms of integral expressions evaluated on the cycle, involving the curvatures and their derivatives, and the smoothness of the principal line near the closed one.
Abstract: The smooth structure of the net of principal lines near a closed one (principal cycle) is determined in terms of integral expressions evaluated on the cycle, involving the curvatures and their derivatives
16 citations
TL;DR: In this paper, the Navier-Stokes equation is viewed as a perturbation of a geodesic flow on the group of volume preserving diffeomorphisms on a compact Riemannian manifold.
Abstract: Following Ebin and Marsden the Navier-Stokes equation is viewed as a perturbation of a geodesic flow on the group of volume preserving diffeomorphisms on a compact Riemannian manifold. Existence and uniqueness of bounded solutions for all position time is shown by taking a higher order diffusion term.
8 citations
TL;DR: In this paper, it was shown that the question of whether there are 2-type Willmore surfaces in E3 has a negative answer when the surface is the image under stereographic projection of a minimal surface in S3.
Abstract: In this paper we deal with the following particular case of a weaker conjecture by B. Y. Chen: Are there 2-type Willmore surfaces in E3? In particular we prove that the above question has a negative answer when the surface is the image under stereographic projection of a minimal surface in S3.
3 citations
TL;DR: In this paper, the authors give examples of spaces with 3-torsion in the homology admitting Z3-tight polyhedral immersions into euclidean space.
Abstract: We give examples of spaces with 3-torsion in the homology admitting Z3-tight polyhedral immersions into euclidean space. Polyhedral tubes are used to construct embedded hypersurfaces of this kind.
2 citations
TL;DR: In this paper, it was shown that the restriction of a Verma module over a semi-simple Lie algebra to a subalgebra of Levi type can be decomposed into a direct sum of standard indecomposables in the category O.
Abstract: We observe that the restriction of a Verma module over a semi-simple Lie algebra to a subalgebra of Levi type may be viewed as a projective functor. By simple arguments we prove that this restriction can be decomposed into a direct sum of standard indecomposables in the category O. For the restriction problem from sl(n+1) to gl(n) we describe the complete answer. We study the properties of the modules with Verma flag also and prove that any module with Verma flag is a submodule of some projective.
1 citations
TL;DR: In this paper, a comparison theorem between the Mean Exit Time function defined on a Kahler manifold and the mean exit time from a geodesic ball of the complex projective space ℂℙ n (λ) is presented.
Abstract: Let M be a Kahler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let ω be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary ∂ω. The main result of this paper is a comparison theorem between the Mean Exit Time function defined on ω and the Mean Exit Time from a geodesic ball of the complex projective space ℂℙ
n
(λ) which involves a characterization of the geodesic balls among the domain ω. In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of ω, using the Index Lemma for Submanifolds.
1 citations
TL;DR: In this article, the authors define a smooth real function Λx(t) : (1 + ⋋�i(t)), where the eigenvalues of AAT are the eigrams of the non-singular linear map dϕ2t, restricted to v at the point ϕx-t ∈ Mn.
Abstract: Let ϕt be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold Mn, whose orbits are geodesics. Then the (n-1)-plane field normal to v, ⊥v, is invariant under dϕt and, for each x ∈ M, we define a smooth real function Λx(t) : (1 + ⋋i(t)), where the ⋋i(t) are the eigenvalues of AAT, A being the matrix (with respect to orthonormal bases) of the non-singular linear map dϕ2t, restricted to ⊥v at the point ϕx-t ∈ Mn.
TL;DR: In this paper, the existence of Riemannian metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold is studied.
Abstract: We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.
TL;DR: In this paper, the dimension of the space of harmonic spinors on hyperelliptic Riemann surfaces for all spin structures is calculated and the maximal possible number of linearly independent spinors is achieved.
Abstract: We calculate the dimension of the space of harmonic spinors on hyperelliptic Riemann surfaces for all spin structures. Furthermore, we present non-hype relliptic examples of genus 4 and 6 on which the maximal possible number of linearly independent harmonic spinors is achieved.
TL;DR: In this paper, a complete geodesic for flat pseudo-Riemannian homogeneous manifolds with non-trivial holonomy is given, and a complete homogeneous manifold with a nilpotent transitive group of isometries is given.
Abstract: Flat pseudo-Riemannian manifolds with a nilpotent transitive group of isometries are shown to be complete. Also flat pseudo-Riemannian homogeneous manifolds with non-trivial holonomy are shown to contain a complete geodesic.
TL;DR: In this paper, the vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in the sense that they derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields.
Abstract: The vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in this paper. From Banach Lie groups associated with the MKdV hierarchy of differential equations, we derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields. In the same way, we obtain homogeneous Banach manifolds of solutions to the sine-Gordon equation. The scattering and inverse scattering maps in this set-up are also discussed.
TL;DR: In this article, the authors show that pseudodifferential operators with symbols in the Grushin classes can be characterized by uniform ellipticity of the symbol, which leads to an index theorem for the Fredholm operators in these classes.
Abstract: The pseudodifferential operators with symbols in the Grushin classes \~S
inf0
supρ,δ
, 0 ≤ δ < ρ ≤ 1, of slowly varying symbols are shown to form spectrally invariant unital Frecher-*-algebras (Ψ*-algebras) in L(L
2(R
n
)) and in L(H
γ
st
) for weighted Sobolev spaces H
infγ
sup
defined via a weight d function γ. In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and γ. The characterization of the Fredholm property by uniform ellipticity leads to an index theorem for the Fredholm operators in these classes, extending results of Fedosov and Hormander.
TL;DR: In this article, it was shown that γ0 and γ1 are isotopic as usual Legendrian knots with nonintersecting front faces, and the same invariants rot and link.
Abstract: Let γ0 and γ1 be Legendrian knots which are isotopic as usual knots, and which have the same obvious invariants rot and link. It seems to be an open question whether γ0 and γ1 are isotopic as Legendrian knots. In the paper we give a positive answer to this question for the (rather restricted) class of Legendrian knots with nonintersecting fronts.
TL;DR: For a Riemannian foliation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0x
Abstract: For a Riemannian foliation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\] on a compact manifold M with a bundle-like metric, the de Rham complex of M is C∞-splitted as the direct sum of the basic complex and its orthogonal complement. Then the basic component % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciGae8NUdS% 2aaSbaaSqaaiaadkgaaeqaaaaa!38B9!\[\kappa _b \] of the mean curvature form of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\] is closed and defines a class ξ (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\]) in the basic cohomology that is invariant under any change of the bundle-like metric. Moreover, any element in ξ(% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\]) can be realized as the basic component of the mean curvature of some bundle-like metric.
TL;DR: In this article, the spectral properties of a Riemannian foliation F on a manifold (M, g) are determined by the two types of spectral invariants, namely the Jacobi operator and the Laplacian.
Abstract: Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants?