The viscosity method for min-max free boundary minimal surfaces

Abstract

We adapt the viscosity method introduced by Rivière in [37] to the free boundary case. Namely, given a compact oriented surface Σ, possibly with boundary, a closed ambient Riemannian manifold (M m , g) and a closed embedded submanifold N n ⊂ M, we study the asymptotic behavior of (almost) critical maps Φ for the functionalon immersions Φ : Σ → M with the constraint Φ(∂Σ) ⊆ N , as σ → 0, 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 (Σ, ∂Σ) → (M, N ), assuming F to be stable under isotopies of this space we show that the min-max valueis the sum of the areas of finitely many branched minimal immersions Φ (i) : Σ (i) → M with ∂ν Φ (i) ⊥ T N along ∂Σ (i) , whose (connected) domains Σ (i) can be different from Σ but cannot have a more complicated topology.We adopt a point of view which exploits extensively the diffeomorphism invariance of Eσ and, along the way, we simplify several arguments from the original work [37]. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.

Multiplicity one for min-max theory in compact manifolds with boundary and its applications

“…We have seen immense existence results concerning free boundary minimal hypersurfaces in recent years, c.f. [16], [15], [40], [30], [28], [32], [20], [54], [45], [8]. In this article, we prove the free boundary version of the Multiplicity One Conjecture in min-max theory.…”

Shape optimization for combinations of Steklov eigenvalues on Riemannian surfaces

“…Other Techniques follow Algrem Pitt's min-max theory on varifolds [Li15], [GLWZ21]. Other ones are somewhat intermediate by the so-called viscosity method [Pi20]. In order to build examples with more elaborate topology, many authors focused on the particular case of target balls.…”