Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension

We study the fractional Plateau problem for $s$-minimal sets with prescribed exterior datum. For fixed exterior data, minimizers need not be unique, and singularities may occur beyond the critical dimension. We first prove a generic uniqueness theorem: along any strictly increasing family of exterior data, nonuniqueness occurs for at most countably many parameters. We then show that one can make arbitrarily small perturbations for which the interior regularity theory improves by one dimension.