A quadratic Bolza-type problem in a Riemannian manifold

A classical nonlinear equation $D_s\dot x + \nabla_x V(x,s) = 0$ on a complete Riemannian manifold $\m$ is considered. The existence of solutions connecting any two points $x_0 , x_1 \in \m$ is studied, i.e., for $T > 0$ the critical points of the functional $J_T(x) = {1\over 2} \int_0^T \langle\dot x,\dot x\rangle ds -\int_0^T V(x,s) ds$ with $x(0)=x_0, x(T)=x_1$. When the potential $V$ has a subquadratic growth with respect to $x$, $J_T$ admits a minimum critical point for any $T>0$ (infinitely many critical points if the topology of $\m$ is not trivial). When $V$ has an at most quadratic growth, i.e., $V(x,s) \leq \lambda d^2(x,\bar x) + k$, this property does not hold, but an optimal arrival time $T(\lambda) > 0$ exists such that, if $0 < T<T(\lambda)$, any pair of points in $\m$ can be joined by a critical point of the corresponding functional. For the existence and multiplicity results, variational methods and Ljusternik-Schnirelman theory are used. The optimal value $T(\lambda) = \pi/\sqrt{2\lambda}$ is fulfilled by the harmonic oscillator. These ideas work for other related problems.