Reference on these two affirmations on Differential Geometry

Question

I'm reading the paper "A gap theorem for free boundary minimal surfaces in the three-ball".

In what follows $\Sigma$ is a minimal compact free boundary surface in the unit ball $B^3$ contained in $\mathbb{R}^3$.

On page 5 we have two affirmations, here they are:

1 - "In particular, $\partial \Sigma$ is strictly convex in $\Sigma$. This implies that for all $p,q \in \Sigma$ there exists a minimising geodesic in $\Sigma$ joining $p$ to $q$."

2 - "Given $[\alpha] \in \pi_1(\Sigma,p)$, let us assume that $[\alpha]$ is a non trivial homotopy class. Since $\partial \Sigma$ is strictly convex we can find a geodesic loop $\gamma:[0,1] \to \Sigma$, $\gamma(0)=\gamma(1) = p$, such that $\gamma \in [\alpha]$."

I read the paper and these are the only two things I don't understand let alone know how to prove.

Where can I find a reference for these two facts?

Thank you.

Answer 1

1. Since $\Sigma$ ia orthogonal to $\partial B$, the shape operator of $\partial\Sigma$ in $\Sigma$ is the restriction of the shape operator of $\partial B$ in $B$. In particular it is positive definite and therefore $\partial\Sigma$ is locally convex. So a minimizing geodesic between the interior point $p$ to $q$ in $\Sigma$ can not touch the boundary.

2. Minimal surfaces have non-positive curvature and their universal cover is $\mathrm{CAT}[0]$. Hence the statement follows.