A note on the shrinking sector problem for surfaces of variable negative curvature

Given the universal cover ? for a compact surface V of variable negative curvature and a point x? ₀ ∈ ?, we consider the set of directions \({\widetilde v_0} \in {S_{\widetilde {{x_0}}}}\widetilde V\) for which a narrow sector in the direction ?, and chosen to have unit area, contains exactly k points from the orbit of the covering group. We can consider the size of the set of such ? in terms of the induced measure on \({S_{{{\widetilde x}_0}}}\widetilde V\) by any Gibbs measure for the geodesic flow. We show that for each k the size of such sets converges as the sector grows narrower and describe these limiting values. The proof involves recasting a similar result by Marklof and Vinogradov, for the particular case of surfaces of constant curvature and the volume measure, by using the strong mixing property for the geodesic flow, relative to the Gibbs measure.     

Solitons of curvature

An intristic geometry of surfaces is discussed. In geodesic coordinates the Gauss equation is reduced to the Schrödinger equation where the Gaussian curvature plays the role of a potential. The use of this fact provides an infinite set of explicit expressions for the curvature and metric of a surface. A special case is governed by the KdV equation for the Gaussian curvature. We consider the integrable dynamics of curvature via the KdV equation, higher KdV equations and (2+1)-dimensional integrable equations with breaking solitons.     

Vertical sectional curvature and K-contactness

We prove that a contact form with riemannian characteristic flow is K-contact. We also present a purely riemannian hypothesis which implies the existence of a K-contact form with a prescribed unit Killing vector field as characteristic vector field. Our hypothesis is weaker than that previously presented by Hatakeyama, Ogawa and Tanno.     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

The Webster scalar curvature problem on the three dimensional CR manifolds

In this paper, we prove some existence results for the Webster scalar curvature problem on the three dimensional CR compact manifolds locally conformally CR equivalent to the unit sphere S³ of C². Our methods are based on the techniques related to the theory of critical points at infinity.     

The compact category and multiple periodic solutions of Hamiltonian systems on symmetric starshaped energy surfaces

Dieter Puppe zum 60. Geburtstag gewidmet     

Local invariants of singular surfaces in an almost complex four-manifold

In this paper we define two local invariants, the local self-intersection index and the Maslov index, for singular surfaces in an almost complex four-manifold and prove formulae involving these invariants, which generalize formulae of Lai and Givental.Dedicated to Professor Masahisa Adachi on his 60th birthday.     

Prescribed Gauss curvature problem on singular surfaces

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface \(\Sigma \) admitting conical singularities of orders \(\alpha _i\)’s at points \(p_i\)’s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min–max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity \(\chi (\Sigma )+\sum _i \alpha _i\) approaches a positive even integer, where \(\chi (\Sigma )\) is the Euler characteristic of the surface \(\Sigma \).     

Isoscattering deformations for complete manifolds of negative curvature

We construct continuous families of nonisometric metrics on simply connected manifolds of dimension n ≥ 9which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.     

The isoperimetric problem in complete surfaces of nonnegative curvature

We show that in a complete plane with nonnegative curvature there is a perimeter minimizing set of any given area. This set is a disc whose boundary is a closed embedded curve with constant geodesic curvature.     

The prescribed curvature problem in dimension four

Necessary and sufficient conditions for a g-valued differential 2-form on a 4-dimensional manifold to be, locally, a curvature form, are given. The dimension four is exceptional for the problem of prescribed curvature as, in this dimension, Bianchi's identities can be eliminated for a large class of Lie algebras, including semisimple algebras. Hence, the curvature forms are characterized as the solutions to a second-order partial differential system, which is proved to be formally integrable.     

The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

     

The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space

The purpose of this paper is to study some uniqueness, existence and regularity properties of the Dirichlet problem at infinity for proper harmonic maps from the hyperbolic m-space to the open unit n-ball with a specific incomplete metric. When m=n=2, harmonic solutions of this Dirichlet problem yield complete constant mean curvature surfaces in the hyperbolic 3-space. Received: 25 January 2001 / Accepted: 23 February 2001 / Published online: 25 June 2001     

The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, N ≥ 2, we show that positive solutions to the problem

The problem of the principal chiral field with the parameters depending on free arguments and its integration

A method to determine the solutions for the equation of the principal chiral field with the parameters depending on independent arguments, is worked out for arbitrary semisimple algebras. Each solution depends on (N-r)/2 arbitrary functions of independent arguments ₁ and ₂. Moreover, the number of derivatives of the arbitrary functions appearing in the solution distinguishes them, gathering them into series.     

Simultaneous metric uniformization of foliations by Riemann surfaces

We consider a two-dimensional linear foliation on torus of arbitrary dimension. For any smooth family of complex structures on the leaves we prove existence of smooth family of uniformizing (conformal complete flat) metrics on the leaves. We extend this result to linear foliations on and families of complex structures with bounded derivatives C ³-close to the standard complex structure. We prove that the analogous statement for arbitrary C two-dimensional foliation on compact manifold is wrong in general, even for suspensions over in dimension 3 the uniformizing metric can be nondifferentiable at some points; in dimension 4 the uniformizing metric of each noncompact leaf can be unbounded.