# American Institute of Mathematical Sciences

July  1997, 3(3): 439-450. doi: 10.3934/dcds.1997.3.439

## Periodic orbits on Riemannian manifolds with convex boundary

 1 Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 BARI, Italy

Received  November 1996 Published  April 1997

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.

Citation: Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439
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