We adapt the viscosity method introduced by Rivière to the free boundary case. Namely, given a compact oriented surface S, possibly with boundary, a closed ambient Riemannian manifold M and a closed embedded submanifold N in M, we study the asymptotic behavior of (almost) critical maps for a functional on immersions from S to M with the constraint that the boundary of S is mapped to N, as a scaling parameter goes to zero, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection F of compact subsets of the space of smooth immersions, assuming F to be stable under isotopies of this space, we show that the corresponding min-max value is the sum of the areas of finitely many branched free boundary minimal immersions, whose (connected) domains can be different from S but cannot have a more complicated topology. Contrary to other min-max frameworks, the present one applies in an arbitrary codimension. We adopt a point of view which exploits extensively the diffeomorphism invariance of the energy and, along the way, we simplify and streamline several arguments from the initial work of Rivière. Some parts generalize to closed higher-dimensional domains, for which we get an integral stationary varifold in the limit.
The viscosity method for min-max free boundary minimal surfaces
Pigati, Alessandro
2022
Abstract
We adapt the viscosity method introduced by Rivière to the free boundary case. Namely, given a compact oriented surface S, possibly with boundary, a closed ambient Riemannian manifold M and a closed embedded submanifold N in M, we study the asymptotic behavior of (almost) critical maps for a functional on immersions from S to M with the constraint that the boundary of S is mapped to N, as a scaling parameter goes to zero, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection F of compact subsets of the space of smooth immersions, assuming F to be stable under isotopies of this space, we show that the corresponding min-max value is the sum of the areas of finitely many branched free boundary minimal immersions, whose (connected) domains can be different from S but cannot have a more complicated topology. Contrary to other min-max frameworks, the present one applies in an arbitrary codimension. We adopt a point of view which exploits extensively the diffeomorphism invariance of the energy and, along the way, we simplify and streamline several arguments from the initial work of Rivière. Some parts generalize to closed higher-dimensional domains, for which we get an integral stationary varifold in the limit.
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