We revisit the classical Poincaré inequality on closed surfaces, and prove its natural analogue for quadratic differentials. In stark contrast to the classical case, our inequality does not degenerate when we work on hyperbolic surfaces that themselves are degenerating, and this fact turns out to be essential for applications to the Teichmüller harmonic map flow.

A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A boundary-value problem for this example is formulated and is shown to possess weak solutions. MSC2000: 35M10, 58J99.

Hyperbolic plasmonic metamaterials provide numerous opportunities for designing unusual linear and nonlinear optical properties. We show that the modal overlap of fundamental and second-harmonic light in an anisotropic plasmonic metamaterial slab results in the broadband enhancement of radiated second-harmonic intensity by up to 2 orders of magnitudes for TMand TE-polarized fundamental light co...

In the present work the classical problem of harmonic oscillator in the hyperbolic space H 2 : z 2 0+z 2 1−z 2−z 3 = R has been completely solved in framework of Hamilton–Jacobi equation. We have shown that the harmonic oscillator on H 2 , as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the...

The Poisson boundary of a group G with a probability measure μ on it is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded μ-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimat...

1. Introduction. In [Ch] we gave some large sieve type inequalities involving elements of harmonic analysis in Riemann surfaces and compact Riemannian manifolds. In this paper we present some of their applications. Our results are related to the hyperbolic circle problem, which is a generalization of the classical circle problem. The latter can be formulated as counting the images of a point in...

This paper gives some examples of hypersurfaces φt(M ) evolving in time with speed determined by functions of the normal curvatures in an (n+ 1)-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean curvature. The examples converge to a totally geodesic submanifold of any dimension from 1 to n, and include cases which exist for infinite time. Convergence to a point was...

The harmonic in distribution systems becomes an important problem due to an increase in nonlinear loads. This paper presents a new approach based on a graph algorithm for optimum placement of passive harmonic filters in a multi-bus system, which suffers from harmonic current sources. The objective of this paper is to minimize the network loss, the cost of the filter and the total harmonic disto...

This note presents elementary geometric descriptions of several simple families of harmonic functions on the upper sheet of the unit hyperboloid in Minkowski three-space. As is briefly discussed here, these calculations grew out of an earlier attempt to construct Poincaré series on punctured surfaces using Minkowski geometry. Introduction The material in this note grew out of an attempt (discus...