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Periodic orbits on Riemannian manifolds with convex boundary

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  • We look for $T$-periodic solutions on a convex Riemannian manifold $\mathcal{M}$ of the differential equation

    $D_s\dot x(s) + \nabla V_x(x(s),s) = 0$

    where $D_s\dot x(s)$ is the covariant derivative of $\dot x(s)$, $V$ is a $\mathcal{C}^2$ real function on $\mathcal{M}\times \mathbf{R}$, $T$-periodic in $s$. The manifold is allowed to be noncompact and to have boundary, so the action integral associated to the equation does not satisfy the Palais-Smale compactness condition. We overcome this problem under a assumption on the sectional curvature of $\mathcal{M}$ which allows to control the Morse index of the critical points of $f$ at "infinity". If $\mathcal{M}$ has a "rich" topology it is proved that there exist infinitely many periodic solutions.

    Mathematics Subject Classification: 58E05, 58F, 70H35.


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