Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement:

Statement. There exists a length decreasing homotopy $\varphi_t:S^1\to M$, $t\in [0,1]$, such that the $\varphi_1(S^1)$ is locally length minimizing curve (i.e. can not be slightly perturbed to a shorter curve.) or a point.

Remark. In the case when $M$ has no boundary, of course, $\varphi_1(S^1)$ should be a closed geodesic.

Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement:

Statement. There exists a length decreasing homotopy $\varphi_t:S^1\to M$, $t\in [0,1]$, such that the $\varphi_1(S^1)$ is locally length minimizing curve (i.e. can not be slightly perturbed to a shorter curve.) or a point.

$\varphi$ is length decreasing if for any $t_1<t_2$ the length of $\varphi_{t_2}(S^1)$ is smaller than the length $\varphi_{t_1}(S^1)$.

Remark. In the case when $M$ has no boundary, of course, $\varphi_1(S^1)$ should be a closed geodesic.

Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement:

Statement. There exists a length decreasing homotopy $\varphi_t:S^1\to M$, $t\in [0,1]$, such that the curve $\varphi_1(S^1)$ is locally length minimizing (i.e. can not be slightly perturbed to a shorter curve.)

Remark. In the case when $M$ has no boundary, of course, $\varphi_1(S^1)$ should be a closed geodesic.

Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement:

Statement. There exists a length decreasing homotopy $\varphi_t:S^1\to M$, $t\in [0,1]$, such that the $\varphi_1(S^1)$ is locally length minimizing curve (i.e. can not be slightly perturbed to a shorter curve.) or a point.

Remark. In the case when $M$ has no boundary, of course, $\varphi_1(S^1)$ should be a closed geodesic.