It's well known that $C^1$ minimal surfaces (surfaces that are locally area minimzing) in $\mathbb{R}^n$ are automatically smooth, and one can prove this result solving the Dirichlet problem of MSE. However, in Riemannian manifolds, I can't find a reference how to prove $C^1$ minimal surfaces are smooth, and some papers just say "it's well known that minimal surfaces are smooth". Can anyone give me some references?

It's well known that $C^1$ minimal surfaces (surfaces that are locally area minimzing) in $\mathbb{R}^n$ are automatically smooth, and one can prove this result by solving the Dirichlet problem of the minimal surface equation over convex domains. However, in Riemannian manifolds, I can't find a reference how to prove $C^1$ minimal surfaces are smooth, and some papers just say "it's well known that minimal surfaces are smooth". Can anyone give me some references?