We establish a min-max theory for constructing minimal disks with free boundary in
any closed Riemannian manifold. The main result is an effective version of
the partial Morse theory for minimal disks with free boundary established
by Fraser. Our theory also includes as a special case the min-max theory
for the Plateau problem of minimal disks, which can be used to generalize
the famous work by Morse–Tompkins and Shiffman on minimal surfaces in
to the
Riemannian setting.
More precisely, we generalize, to the free boundary setting, the min-max construction
of minimal surfaces using harmonic replacement introduced by Colding–Minicozzi. As
a key ingredient to this construction, we show an energy convexity for weakly
harmonic maps with mixed Dirichlet and free boundaries from the half unit
–disk
in into
any closed Riemannian manifold, which in particular yields the uniqueness of such
weakly harmonic maps. This is a free boundary analogue of the energy convexity and
uniqueness for weakly harmonic maps with Dirichlet boundary on the unit
–disk
proved by Colding and Minicozzi.
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