I'm looking for an existence theorem of minimal surface. In a paper by L. Andersson and J Metzger, 'The Area of Horizons and the Trapped Region', they used Jang's equaiton to prove existence of stable MOTS (a generalization of stable minimal surface) when the outer boundary of a compact region is outer non-trapped (generalization of mean-convex) while the inner boundary is outer-trapped (generalization of mean-concave).
There must be a similar theorem for minimal surfaces since MOTS reduces to minimal surfaces for time-symmetric Cauchy slice. Could anyone point to me a reference on such existence theorems of minimal surfaces? Thanks in advance!
More generally, I would appreciate any references on minimal surfaces in complete 3-dim Riemannian manifold, satisfying condition like conformal flat.