The existence and uniqueness of standard static splitting

New techniques are introduced to study the existence and uniqueness of the static standard splitting of a static spacetime, and as a consequence, several results are provided in the field of geometric analysis. In particular, we obtain a rigidity result for compact spacelike hypersurfaces with a boundary, we solve a Dirichlet problem for the non-linear elliptic maximal hypersurface equation in a standard static spacetime, and finally we provide an upper bound for the first nontrivial eigenvalue of the Laplacian operator for a compact spacelike surface in certain standard static spacetimes.