A proof of the multiplicity 1 conjecture for min-max minimal surfaces in arbitrary codimension

On the Morse index of critical points in the viscosity method

“…Therefore, the main Theorem 1.1 applies to the viscosity method for minimal surfaces. Combining the recent resolution of the multiplicity one conjecture proved in this setting by Pigati and Rivière [21] with the previous result of Rivière [27], one can obtain the lower semi-continuity of the index. Willmore surfaces [14,16,26].…”

Morse Index Bound for Minimal Torus

“…The parametric approach produces immersed (rather than embedded) minimal surfaces with possible branch points for arbitrary codimension. The Morse index bound of such immersed minimal surfaces is shown in [PR20]. We like to remark here that the Morse index in [PR20] is for the area functional, while the Morse index in Theorem 1.1 is for the energy functional.…”

“…The Morse index bound of such immersed minimal surfaces is shown in [PR20]. We like to remark here that the Morse index in [PR20] is for the area functional, while the Morse index in Theorem 1.1 is for the energy functional. For a conformal harmonic map, its Morse index for area is bounded by its Morse index for energy plus a constant depending on the genus and branch points [EM08].…”

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).…”