I'm studying minimal surfaces. In a variety of sources people seem to be using the following two results which I can't prove.
1. Let $(M,g)$ be a compact $n$-dimensional riemannian manifold and $\pi:(\tilde{M},\tilde{g}) \to (M,g)$ a covering map where $\tilde{g}$ is the riemannian metric in $\tilde{M}$ given by the pullback of $g$ by $\pi$, i.e., $\tilde{g}=\pi^*g$.
Question: If $f:(\Sigma,h) \to (\tilde{M},\tilde{g})$ is a minimal isometric immersion where $(\Sigma,h)$ is a $(n-1)$-dimensional riemannian manifold with $h=f^*g$, then $\pi\circ f: \Sigma \to M$ is a minimal isometric immersion?
2. Let $(M,g)$ be a $n$-dimensional riemannian manifold and $f: (\Sigma,h) \to (M,g)$ an minimal isometric immersion with $(\Sigma,h)$ a $(n-1)$-dimensional riemannian manifold and $h=f^*g$.
Question: If $\pi : \tilde{\Sigma} \to \Sigma$ is a covering map such that $\tilde{\Sigma}$ is endowed with the metric $\tilde{g}=\pi^*g$ then $f\circ \pi : \tilde{\Sigma}\to M$ is a minimal isometric immersion?
Any help or reference would be highly valuable.
Thank you.