Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat three-manifolds as a limit of solutions natural mountain pass problem with diverging boundaries? In this work, we give evidence that might be true by analyzing related aspects the case exact Riemannian Schwarzschild manifold. More precisely, observe simplest surface space has Morse index one. We prove also relati...

Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and area a free-boundary surface Riemannian manifold with boundary, show that is controlled from above topology surface. Combining these results Fraser-Li, conclude in convex domain Euclidean three-space, bounded linear function its genus number boundary components. We also prove bounds for submanifolds h...

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces uniform area and Morse index upper bound is in sense smoothly graphical convergence away from finitely many points. We show limit sequence such always inherits non-trivial Jacobi field when it has multiplicity one. In forthcoming paper, will construc...