An index theory for paths that are solutions of a class of strongly indefinite variational problems

We generalize the Morse index theorem of [12,15] and we apply the new result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of semi-Riemannian manifolds. More specifically, we consider semi-Riemannian manifolds admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for . In particular we obtain Morse relations for stationary semi-Riemannian manifolds (see [7]) and for the G?del-type manifolds (see [3]). Received: 4 April 2001 / Accepted: 27 September 2001 / Published online: 23 May 2002 The authors are partially sponsored by CNPq (Brazil) Proc. N. 301410/95 and N. 300254/01-6. Parts of this work were done during the visit of the two authors to the IMPA, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, in January and February 2001. The authors wish to express their gratitude to all Faculty and Staff of the IMPA for their kind hospitality.     

Homoclinic solutions of an infinite-dimensional Hamiltonian system

We consider the system which is an unbounded Hamiltonian system in . We assume that the constant function is a stationary solution, and that H and V are periodic in the t and x variables. We present a variational formulation in order to obtain homoclinic solutions z=(u,v) satisfying as . It is allowed that V changes sign and that has essential spectrum below (and above) 0. We also treat the case of a bounded domain instead of with Dirichlet boundary conditions. Received: 21 March 2001; in final form: 11 June 2001 / Published online: 1 February 2002     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002     

Stability for hypersurfaces of constant mean curvature with free boundary

The partitioning problem for a smooth convex bodyB ³ consists in to study, among surfaces which divideB in two pieces of prescribed volume, those which are critical points of the area functional.We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case thatB is a Euclidean ball we obtain stronger results.Antonio Ros is partially supported by DGICYT grant PB91-0731 and Enaldo Vergasta is partially supported by CNPq grant 202326/91-8.     

A new proof of regularity of weak solutions of the <Emphasis Type="Italic">H</Emphasis>-surface equation

Abstract. We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions of the H-surface system are locally H?lder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in , this time assuming in addition that decays at infinity like . Received: 10 May 2001 / Accepted: 7 June 2001 / Published online: 18 January 2002 The author gratefully acknowledges the generous support of Alexander von Humboldt Foundation, and the hospitality of Mathematisches Institut der Universit?t Bonn, where this research has been carried out. In particular, many thanks are due to Professor Stefan Hildebrandt.     

Ruled surfaces of Weingarten type in Minkowski 3-space

In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K_II, H, H_II}, where K is the Gaussian curvature, K_II is the second Gaussian curvature, H is the mean curvature, and H_II is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK_II + bH + cH_II + dK is constant along each ruling for some constants a, b, c, d with a² + b² + c² ≠ 0.     

Local and global existence criteria for capillary surfaces in wedges

This paper addresses a conjecture of Concus and Finn [Capillary Wedges Revisited, SIAM J. Math. Anal., in press] on conditions for local existence of solutions of the zero-gravity capillarity equation at a boundary protruding corner pointP of prescribed opening 2. Geometrically, surfaces of constant mean curvatureH are sought as graphs which meet vertical walls over the boundary in prescribed angles, which are locally constant except for a possible jump discontinuity atP. The conjecture is settled more or less completely in the affirmative, depending on whetherH is to be prescribed. The proof proceeds through a global existence theorem for moon domains, which seems of independent interest.     

Complete space-like hypersurfaces with constant scalar curvature

Let M ⁿ be a complete space-like hypersurface with constant normalized scalar curvature R in the de Sitter space S ^{n + 1} ₁ and denote . We prove that if the norm square of the second fundamental form of M ⁿ satisfies , then either and M ⁿ is a totally umbilical hypersurface; or , and, up to rigid motion, M ⁿ is a hyperbolic cylinder . Received: 8 February 2001 / Revised version: 27 April 2001     

The asymptotic Plateau problem in Gromov hyperbolic manifolds

We solve the asymptotic Plateau problem in every Gromov hyperbolic Hadamard manifold (X,g) with bounded geometry. That is, we prove existence of complete (possibly singular) k-dimensional area minimizing surfaces in X with prescribed boundary data at infinity, for a large class of admissible limit sets and for all . The result also holds with respect to any riemannian metric on X which is lipschitz equivalent to g. Received: 23 January 2001 / Accepted: 25 October 2001 Published online: 28 February 2002     

Perturbed closed geodesics are periodic orbits: Index and transversality

We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in M. The potential $V\in C^\infty(M\times S^1,\mathbb{R})$ serves as perturbation and we show that both functionals are Morse for generic V. In this case we prove that the Morse index of a critical point x of equals minus its Conley-Zehnder index when viewed as a critical point of and if is trivial. Otherwise a correction term +1 appears. Received: 21 May 2001; in final form: 10 October 2001 / Published online: 4 April 2002     

On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

Existence result for minimal hypersurfaces¶with a prescribed finite number of planar ends

Paralleling what has been done for minimal surfaces in ℝ³, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝ ^{n +1} which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature. Received: 1 July 1999 / Revised version: 31 May 2000     

Existence of constant mean curvature graphs in hyperbolic space

We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible. Received July 18, 1997 / Accepted April 24, 1998     

Asymptotic behaviour of families of real curves

We consider the family of fibres of a polynomial function f on a smooth noncompact algebraic real surface and we characterise the regular fibres of f which are atypical due to their asymptotic behaviour at infinity. We compare to the similar problem in the complex case. Received: 5 May 1998 / Revised version: 20 March 1999     

Boundary regularity for minimizing currents with prescribed mean curvature

We prove complete boundary regularity for energy minimizing integer multiplicity rectifiablen currents in ⁿ⁺¹ of prescribed mean curvatureH with boundaryB= represented by an oriented smooth submanifold of dimensionn – 1 in sun+1. We also give applications to the Plateau problem for surfaces with prescribed mean curvature.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations

We derive the total energy decay and boundedness for the solutions to the initial boundary value problem for the wave equation in an exterior domain : with , where and a(x) is a nonnegative function which is positive near some part of the boundary and near infinity. We apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f(u) like . We note that no geometrical condition is imposed on the boundary . Received: 16 June 1999; in final form: 13 March 2000 / Published online: 4 May 2001     

Existence Results for Nonlinear Schrödinger Equations With Electromagnetic Fields

 We consider the nonlinear Schr?dinger equation

Actions and irreducible representations of the mapping class group

Let G be a countable discrete group. Call two subgroups and of G commensurable if has finite index in both and . We say that an action of G on a discrete set X has noncommensurable stabilizers if the stabilizers of any two distinct points of X are not commensurable. We prove in this paper that the action of the map ping class group on the complex of curves has noncommensurable stabilizers. Following a method due to Burger and de la Harpe, this action leads to constructions of irreducible unitary representations of the mapping class group. Received: 26 July 1999 / Revised version: 14 May 2001 / Published online: 19 October 2001