The Regularity of Parametrized Integer Stationary Varifolds in Two Dimensions

Abstract

We establish an optimal regularity result for parametrized two‐dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant.We provide some applications of this regularity result, especially in the calculus of variations for the area functional. © 2020 Wiley Periodicals LLC

The viscosity method for min-max free boundary minimal surfaces

“…The fact that Φ| Σ\∂Σ is a branched minimal immersion then follows as discussed in the last step of the proof of [31,Theorem 5.7]. As already mentioned, the interior regularity was already established in [31]. Here we show again how it can be obtained when N = 1-a fact proved in [38] and used in [31]-presenting a slightly simplified proof which covers also the boundary regularity.…”

“…Later, in [32], the authors show that actually N ≡ 1 in the variational setting, by exploiting the results from [37,31]. This fact allows to obtain an upper bound on the Morse index of the limit minimal immersion in terms of the number of min-max parameters.…”

“…A crucial feature is that stationarity can be localized with respect to the domain. This allows to obtain the full regularity for the limit object: the regularity theory was started in [38], where N ≡ 1 is assumed, and carried out in full generality in [31], where parametrized stationary varifolds are axiomatically studied.…”

Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture

“…In [35] the author introduced a PDE strategy for producing minmax minimal surfaces based on a relaxation procedure of the area that he called viscosity method. After a series of works [35,36,38,28,29] partly in collaboration with Alessandro Pigati and also after using a work by Alexis Michelat [19] the following result has been finally obtained Theorem IV.2. [29] Let (N n , g) be an arbitrary closed and smooth riemannian manifold, let Σ be a smooth closed surface and A be an admissible homological family of M(Σ) of dimension d such that…”

“…This relaxation is preceded by a small "viscosity" parameter that is sent to zero once the minmax critical points have been obtained for the regularized Lagrangian by the Palais-Smale theory. This approach, also called viscosity method, which is very much based on the analysis of Partial differential Equations, has been successfully implemented for the area of immersions of surfaces in [35,23,36,38,19,28,29,27,30] and the authors obtain the realization of any non trivial minmax problem by a possibly branched smooth minimal immersion satisfying various properties (Morse index bound, genus bound, free-boundary property, Lagrangian property...etc).…”