### Some existence results for dynamical systems on non-complete Riemannian manifolds

#### Abstract

Let $\mathcal M^*$ be a non-complete Riemannian manifold with bound-ed topological

boundary and $V: \mathcal M \to \mathbb R$ a $C^2$ potential function subquadratic

at infinity.

In this paper we look for curves $x: [0,T]\to\mathcal M$ having prescribed period $T$ or joining two

fixed points of $\mathcal M$, satisfying the system

$$

D_t (\dot x(t))=-\nabla_R V(x(t)),

$$

where $D_t(\dot x(t))$ is the covariant derivative of $\dot x$ along the direction

of $\dot x$ and $\nabla_R V$ the Riemannian gradient of $V$.

We assume that $V(x) \to -\infty$ if $d(x,\partial\mathcal M)\to 0$ and, in the periodic case,

suitable hypotheses on the sectional curvature of $\mathcal M$ at infinity.

We use variational methods in addition with a penalization technique and Morse

index estimates.

boundary and $V: \mathcal M \to \mathbb R$ a $C^2$ potential function subquadratic

at infinity.

In this paper we look for curves $x: [0,T]\to\mathcal M$ having prescribed period $T$ or joining two

fixed points of $\mathcal M$, satisfying the system

$$

D_t (\dot x(t))=-\nabla_R V(x(t)),

$$

where $D_t(\dot x(t))$ is the covariant derivative of $\dot x$ along the direction

of $\dot x$ and $\nabla_R V$ the Riemannian gradient of $V$.

We assume that $V(x) \to -\infty$ if $d(x,\partial\mathcal M)\to 0$ and, in the periodic case,

suitable hypotheses on the sectional curvature of $\mathcal M$ at infinity.

We use variational methods in addition with a penalization technique and Morse

index estimates.

#### Keywords

Variational methods; Riemannian manifold; Morse index; sectional curvature

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