Noncompact Riemannian Research Articles

Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator H h , depending on the semiclassical parameter h > 0 , on a noncompact Riemannian manifold M such that H 1 ( M , R ) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ 0 ( H h ) of the spectrum of the operator H h in L 2 ( M ) . Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator H h in the region close to λ 0 ( H h ) , as h → 0 . In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.

“Positive Solutions of Quasilinear Elliptic Inequalities on Noncompact Riemannian Manifolds”

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality − Δu ≥ u q, q > 1, on spherically symmetric noncompact Riemannian manifolds.

Existence of Infinitely Many Distinct Solutions to the Quasirelativistic Hartree-Fock Equations

We establish existence of infinitely many distinct solutions to the semilinear elliptic Hartree-Fock equations forN-electron Coulomb systems with quasirelativistic kinetic energy−α−2Δxn+α−4−α−2for thenthelectron. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total chargeZtotofKnuclei is greater thanN−1and thatZtotis smaller than a critical chargeZc. The proofs are based on a new application of the Fang-Ghoussoub critical point approach to multiple solutions on a noncompact Riemannian manifold, in combination with density operator techniques.

Estimates for functions of the Laplacian on manifolds with bounded geometry

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hormander type conditions of appropriate order at infinity, then the operator m(L) extends to an operator of weak type 1. This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M, but without any assumption on the decay of the heat kernel.

Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds

In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation $$u_t=\Delta u+au\, {\rm log}\, u+bu\quad {\rm in}\,M$$ , where a, b are real constants, M is a complete noncompact Riemannian manifold. As a corollary, we give a local gradient estimate for the corresponding elliptic equation: $$\Delta u+au\,{\rm log}\, u+bu=0\quad {\rm in}\,M$$ , which improves and extends the result of Ma (J Funct Anal 241:374–382, 2006) and get a bound for the positive solution to this elliptic equation.

Noncompact Riemannian spaces with the holonomy group spin(7) and 3-Sasakian manifolds

We complete the study of the existence of Riemannian metrics with Spin(7) holonomy that smoothly resolve standard cone metrics on noncompact manifolds and orbifolds related to 7-dimensional 3-Sasakian spaces.

Hölder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity

AbstractWe consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K ⊂ M so that the outward normal exponential map off the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well-defined Hölder structure independent of K. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

The heat flow of harmonic maps from noncompact manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As a corollary, we show that if the tension field is in L p , the heat flow exists globally and uniquely which converges at infinity under an additional condition.

Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SL n ( R ) , SU ∗ ( 2 n ) and Sp ( n , R ) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO ∗ ( 2 n ) , SO ( p , q ) , SU ( p , q ) and Sp ( p , q ) . Applying a duality principle we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO ( n ) , SU ( n ) and Sp ( n ) equipped with semi-Riemannian metrics.

Periodic magnetic Schrödinger operators: Spectral gaps and tunneling effect

A periodic Schrödinger operator on a noncompact Riemannian manifold M such that H 1(M, ℝ) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions on the magnetic field, the existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit is established. The proofs are based on the study of the tunneling effect in the corresponding quantum system.

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let ( M , g ) (M,g) be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation \[ ∂ u ∂ t = Δ u + a u log ⁡ u + b u \frac {\partial u}{\partial t}=\Delta u+au\log u+bu \] on M × [ 0 , + ∞ ) M\times [0,+\infty ) , where a a , b b are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).

Lie groups locally isomorphic to generalized Heisenberg groups

We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group N N , there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to N N and a union of certain quotients of noncompact Riemannian symmetric spaces.

One end theorem and application to stable minimal hypersurfaces

In this paper, we prove a one end theorem for complete noncompact Riemannian manifolds and apply it to complete noncompact stable minimal hypersurfaces.

G 2-holonomy metrics connected with a 3-Sasakian manifold

We construct complete noncompact Riemannian metrics with G 2-holonomy on noncompact orbifolds that are ℝ3-bundles with the twistor space as a spherical fiber.

Hölder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity

We consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K of M so that the outward normal exponential map off of the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well defined H\older structure independent of K. The H\older constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

Local two-radii theorem in symmetric spaces

Various classes of functions on a non-compact rank-one Riemannian symmetric space X with vanishing integrals over all balls of fixed radius are studied. A description in the form of a series in hypergeometric functions is obtained for such classes and a uniqueness theorem is proved. This makes it possible to establish the local two-radii theorem in X in a definitive form.Bibliography: 45 titles.

Semiclassical asymptotics and gaps in the spectra of periodic Schrödinger operators with magnetic wells

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrödinger operator with magnetic wells on a noncompact Riemannian manifold $M$ such that $H^1(M, \mathbb {R})=0$, equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries, has an arbitrarily large number of spectral gaps in the semi-classical limit.

A Paley-Wiener theorem for the inverse Fourier transform on some homogeneous spaces

We formulate and prove a version of Paley-Wiener theorem for the inverse Fourier transforms on noncompact Riemannian symmetric spaces and Heisenberg groups. The main ingredient in the proof is the Gutzmer’s formula.

Gutzmer’s formula and Poisson integrals on the Heisenberg group

In 1978 M. Lassalle obtained an analogue of the Laurent series for holomorphic functions on the complexification of a compact symmetric space and proved a Plancherel type formula for such functions. In 2002 J. Faraut established such a formula, which he calls Gutzmer?s formula, for all noncompact Riemannian symmetric spaces. This was immediately put into use by B. Krotz, G. Olafsson and R. Stanton to characterise the image of the heat kernel transform. In this article we prove an analogue of Gutzmer?s formula for the Heisenberg motion group and use it to characterise Poisson integrals associated to the sublaplacian. We also use the Gutzmer?s formula to study twisted Bergman spaces.

A priori upper bounds of solutions satisfying a certain differential inequality on complete manifolds

In this article we study a priori upper bounds of subsolutions satisfying a certain differential inequality (*) below on a non-compact complete Riemannian manifold $(M,g)$ without any Ricci curvature condition. Our method depends on a volume estimate of open subsets where those solutions satisfy a certain strong subharmonicity. Several applications in conformal deformation of metrics and value distribution of harmonic maps are given.