arXiv subjects
Publications and source records attributed to M. L. Bialy.
We consider a sequences of symplectic twist maps corresponding to Frenkel Kontorova variational functional. For such a sequence without conjugate points we construct an invariant Lagrangian subbundle. Using the ideas by E.Hopf from Riemannian geometry we show that absence of conjugate points is a rare property for this class of maps.
We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a ``twisted'' version of the recent generalisation of Hopf's rigidity result obtained by Burago and Ivanov.
We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally minimising. However, in dimension two the situation is different. We show that for radially symmetric potentials there always exist solutions which are not locally minimal unless the potential vanishes identically. We discuss the relations of this with the so-called Hopf rigidity phenomenon.