The Hessian and Jacobi morphisms for higher order calculus of variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler-Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.     

Lightlike rays in stationary spacetimes with critical growth

The aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary Lorentzian manifolds whose metric coefficients have an optimal growth.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.     

A remark on convex functions andp-harmonic maps

We prove that the constant maps are the onlyp-harmonic maps for anyp 2 from an arbitrary compact Riemannian manifold into a complete Riemannian manifold which admits a strictly convex function.     

Multiple solutions for a class of nonlinear Schrödinger equations

In this paper, we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the Schrödinger equation

     

Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms

Based on new information concerning strongly indefinite functionals without Palais-Smale conditions, we study existence and multiplicity of solutions of the Schrödinger equation

     

Morse Theory for Geodesics in Conical Manifolds

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. In a previous paper [15] we defined these manifolds as submanifolds of with a finite number of conical singularities. To formulate a good Morse theory we use an appropriate definition of geodesic, introduced in the cited work. The main theorem of this paper (see Theorem 3.6, section 3) proofs that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful. In section 5 we compare our theory with the weak slope approach existing in literature. Some examples are also provided.     

Transversality for equivariant exact 1-forms and gauge theory on 3-manifolds

An equivariant jet transversality framework is developed for the study of critical sets of invariant functions on G manifolds. Techniques are developed to extend transversality results to the infinite dimensional Fredholm setting. As an application, the generic structure of the SU(4) perturbed flat moduli space of an integral homology three-sphere is described, as well as the generic structure of the parameterized moduli space for a path of perturbations. A similar analysis of the U(3) moduli space for rational homology three-spheres is also carried out.     

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

On bifurcation of solutions of the Yamabe problem in product manifolds

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds.     

The explicit construction of Finsler metrics with special curvature properties

The S-curvature is one of most important non-Riemannian quantities in Finsler geometry. It delicately related to Riemannian quantities. This note gives an explicit construction of 3-parameter family of non-locally projectively flat Finsler metrics of non-constant isotropic S-curvature. The necessary and sufficient condition that these Finsler metrics are of constant flag curvature is given.     

The Euler-Lagrange Method for Biharmonic Curves

In this article we consider the Euler-Lagrange method associated to a suitable bilagrangian to study biharmonic curves of a Riemannian manifold. We apply this method to characterize biharmonic curves of the three-dimensional Lie group Sol. We also classify, using a geometric method, the biharmonic curves of the three-dimensional Cartan-Vranceanu manifolds. Dedicated to the memory of Professor Aldo Cossu Work partially supported by GNSAGA (Italy); the third author was supported by a CNR-NATO fellowship (Italy), and by the Grant At, 191/2006, CNCSIS (Romania).     

Equilibrium maps between metric spaces

We show the existence of harmonic mappings with values in possibly singular and not necessarily locally compact complete metric length spaces of nonpositive curvature in the sense of Alexandrov. As a technical tool, we show that any bounded sequence in such a space has a subsequence whose mean values converge. We also give a general definition of harmonic maps between metric spaces based on mean value properties and-convergence.     

Bounded Palais-Smale sequences for functionals with a mountain pass geometry

We prove the existence of bounded Palais-Smale sequences for abstract functionals with a mountain pass geometry under hypotheses weaker than those commonly used in the literature. This is obtained via a generalization of a generic result of Jeanjean, combined with a rescaling argument. Applications to the existence of nontrivial solutions to semilinear elliptic problems are given. Received: 17 November 2005     

Global inversion and covering maps on length spaces

In order to obtain global inversion theorems for mappings between length metric spaces, we investigate sufficient conditions for a local homeomorphism to be a covering map in this context. We also provide an estimate of the domain of invertibility of a local homeomorphism around a point, in terms of a kind of lower scalar derivative. As a consequence, we obtain an invertibility result using an analog of the Hadamard integral condition in the frame of length spaces. Some applications are given to the case of local diffeomorphisms between Banach-Finsler manifolds. Finally, we derive a global inversion theorem for mappings between stratified groups.     

Constructions of harmonic polynomial maps between spheres

The complexity of _q -eigenmaps, i.e. homogeneous degreeq harmonic polynomial mapsf:S ^m S ⁿ, increases fast with the degreeq and the source dimensionm. Here we introduce a variety of methods of manufacturing new eigenmaps out of old ones. They include degree and source dimension raising operators. As a byproduct, we get estimates on the possible range dimensions of full eigenmaps and obtain a geometric insight of the harmonic product of ₂-eigenmaps.     

On the non-Riemannian quantity H of a Finsler metric

One of fundamental problems in Finsler geometry is to establish some delicate equations between Riemannian invariants and non-Riemannian invariants. Inspired by results due to Akbar-Zadeh etc., this note establishes a new fundamental equation between non-Riemannian quantity H and Riemannian quantities on a Finsler manifold. As its application, we show that all R-quadratic Finsler metrics have vanishing non-Riemannian invariant H generalizing result previously only known in the case of Randers metric.     

A simple proof of removable singularities for coupled fermion fields

This paper presents a simple proof of a removable singularity theorem for coupled fermion fields on compact four-dimensional manifolds. New methods are employed and the hypotheses here are weak.     

Variational properties of the Gauss–Bonnet curvatures

The Gauss–Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss–Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss–Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss–Bonnet curvature.