The classical Sobolev and Escobar inequalities are embedded into the same oneparameter family of sharp trace-Sobolev inequalities on half-spaces. Equality cases are characterized for each inequality in this family by tweaking a well-known mass transportation argument. The case of the Dirichlet energy corresponds to a family of variational problems on conformally flat metrics, whose absolute min...

For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmon’s ...

Abstract We consider a magnetic Laplacian −∆A = (id + A)?(id + A) on a noncompact hyperbolic surface M with finite area. A is a real one-form and the magnetic field dA is constant in each cusp. When the harmonic component of A satifies some quantified condition, the spectrum of −∆A is discrete. In this case we prove that the counting function of the eigenvalues of −∆A satisfies the classical We...

In this paper we examine the Laplacian on the product of two asymptotically hyperbolic spaces from the point of view of geometric scattering theory. In particular, we describe the asymptotic behavior of the resolvent applied to Schwartz functions and that of the resolvent kernel itself. We use these results to find the Martin boundary of the product space. This behaves (nearly) as expected when...

In his paper Mc], whose object was to show that the spectrum of the Laplacian of a Riemannian surface S determines the surface up to nitely many possibilities , Henry McKean proved the following result, which he called the \Riemann hypothesis for Riemann surfaces": Theorem (McKean Mc]). If S is a hyperbolic Riemann surface, then the rst eigenvalue 1 (S) of the Laplacian on S satisses 1 (S) 14: ...

The Hecke L-function Hj(s) attached to the jth Maass form for the full modular group is estimated in the mean square over a spectral interval for s = 1 2 + it. As a corollary, we obtain the estimate Hj( 1 2 + it) ¿ t1/3+ε for t À κ j , where 1/4 + κj is the respective jth eigenvalue of the hyperbolic Laplacian. This extends a result due to T. Meurman.

Context. An approach to the problem of estimating several physical properties heliospheric current sheet (HCS) local structure from a single spacecraft observation point view was published in 2003. This method, HYperbolic TAngent ROtation (HYTARO), is based on following two main assumptions: (a) that HCS locally flat, and (b) its magnetic topology described by modified Harris field.

the aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. in [1], ungar and chen showed that the algebra of the group sl(2,c) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the lorentz group and its underlying hyperbolic geometry. they defined the chen addition and then chen model of hyperbolic geometry. in this paper,...