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Register a new userOn the existence of closed magnetic geodesics via symplectic reduction
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Let $(M,g)$ be a closed Riemannian manifold and $sigma$ be a closed 2-form on $M$ representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair $(g,sigma)$ can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle $T^*E$ of a suitable $S^1$-bundle $E$ over $M$ or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of $E$. We then study the relation between the stability property of energy hypersurfaces in $(T^*M,dpwedge dq+pi^*sigma)$ and of the corresponding codimension 2 coisotropic submanifolds in $(T^*E,dpwedge dq)$ arising via symplectic reduction. Finally, we reprove the main result of [9] in this setting.
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We show that on every compact Riemannian 2-orbifold there exist infinitely many closed geodesics of positive length.
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
Let $(M,g)$ be a closed Riemannian manifold, $L: TMrightarrow mathbb R$ be a Tonelli Lagrangian. Given two closed submanifolds $Q_0$ and $Q_1$ of $M$ and a real number $k$, we study the existence of Euler-Lagrange orbits with energy $k$ connecting $Q_0$ to $Q_1$ and satisfying the conormal boundary conditions. We introduce the Ma~ne critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies. We also provide counterexamples showing that all the results are sharp.
We show that, on a complete and possibly non-compact Riemannian manifold of dimension at least 2 without close conjugate points at infinity, the existence of a closed geodesic with local homology in maximal degree and maximal index growth under iteration forces the existence of infinitely many closed geodesics. For closed manifolds, this was a theorem due to Hingston.
We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.
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