The well known local version of the maximum principle for minimal hypersurfaces asserts that if two minimal hypersurfaces $ M_1 $ and $ M_2 $ of $ R^n $ has a common point $ x_0 \in M_1 \cap M_2 $ where $ M_1 $ lies (locally) on one side of $ M_2 $ then $ M_1 = M_2 $ in a neighbourod of $ x_0 $. This result can be found for example in 'A course in minimal surfaces' of Colding Minicozzi.
Now i want to prove that if one of them is complete, for example $ M_1 $, then $ M_2 \subset M_1 $.
It is quite obvious to proceed in this way: if $ f_i:M_i \rightarrow R^n $, $ i=1,2 $, are the immersion maps, for the local version of the maximum principle there exist $ U_i \subset M_i $ such that $ f_i :U_i \rightarrow R^n $ is an embedding and $ f_1(U_1)=f_2(U_2) $. Now it is easy to check that $f_1^{-1} \circ f_2 : U_2 \rightarrow U_1 $ is an isometry. Now the most obviuos thing to do is an application of the unique continuation theorem since $ f_1 $ and $ f_2 $ are harmonic maps on the manifolds (unique continuation is suggested in Colding Minicozzi, but they do not give a proof). My problem is that i don't know how i can apply the unique continuation for elliptic operators? In fact $ f_1 $ and $ f_2 $ are defined on two hypersurfaces of $ R^n $ that a priori can be distinct. Thank you