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Morse index and bifurcation of p-geodesics on semi Riemannian manifolds

Monica Musso, Jacobo Pejsachowicz, Alessandro Portaluri (2007)

ESAIM: Control, Optimisation and Calculus of Variations

Given a one-parameter family { g λ : λ ∈ [ a , b ] } of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials { V λ : λ ∈ [ a , b ] } and a family { σ λ : λ ∈ [ a , b ] } of trajectories connecting two points of the mechanical system defined by ( g λ , V λ ) , we show that there are trajectories bifurcating from the trivial branch σ λ if the generalized Morse indices μ ( σ a ) and μ ( σ b ) are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate...

Motion by curvature of planar networks

Carlo Mantegazza, Matteo Novaga, Vincenzo Maria Tortorelli (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two–dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which...

Multiplicity results for the prescribed scalar curvature on low spheres

Mohamed Ben Ayed, Mohameden Ould Ahmedou (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar curvature on standard spheres 𝕊 3 , 𝕊 4 . Under generic conditions we establish someMorse Inequalities at Infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of its critical points at Infinityto the difference of topology between the level sets of the associated Euler-Lagrange functional. As a by-product of our arguments we derive a new existence...

Multivector fields and connections. Applications to field theories.

Arturo Echeverría-Enríquez, Miguel Carlos Muñoz-Lecanda, Narciso Román-Roy (2002)

RACSAM

Se estudia la integrabilidad de campos multivectoriales en variedades diferenciables y la relación entre algunos tipos de campos multivectoriales en un fibrado de jets y conexiones en dicho fibrado. Como caso particular se relacionan los campos multivectoriales integrables y las conexiones cuyas secciones integrales son holonómicas. Como aplicación de todo ello, estos resultados permiten escribir las ecuaciones de campo de las teorías clásicas de campos de primer orden en varias formas equivalentes....

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