A viscosity method in the min-max theory of minimal surfaces

Abstract

:We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface Σ into a given closed manifold, we add to the area Lagrangian a term equal to the L q norm of the second fundamental form of the immersion times a "viscosity" parameter. This relaxation of the area functional satisfies the Palais-Smale condition for q > 2. This permits to construct critical points of the relaxed Lagrangian using classical min-max arguments such as the mountain pass lemma. The goal of this work is to describe the passage to the limit when the "viscosity" parameter tends to zero. Under some natural entropy condition, we establish a varifold convergence of these critical points towards a parametrized integer stationary varifold realizing the min-max value. It is proved in [36] that parametrized integer stationary varifold are given by smooth maps exclusively. As a consequence we conclude that every surface area minmax is realized by a smooth possibly branched minimal immersion.Math. Class. 49Q05, 53A10, 49Q15, 58E12, 58E20 I IntroductionThe study of minimal surfaces, critical points of the area, has stimulated the development of entire fields in analysis and in geometry. The calculus of variations is one of them. The origin of the field is very much linked to the question of proving the existence of minimal 2-dimensional discs bounding a given curve in the euclidian 3-dimensional space and minimizing the area. This question, known as Plateau Problem, has been posed since the XVIIIth century by Joseph-Louis Lagrange, the founder of the Calculus of Variation after Leonhard Euler. This question has been ultimately solved independently by Jesse Douglas and Tibor Radó around 1930. In brief the main strategy of the proofs was to minimize the Dirichlet energy instead of the area, which is lacking coercivity properties, the two lagrangians being identical on conformal maps. After these proofs, successful attempts have been made to solve the Plateau Problem in much more general frameworks. This has been in particular at the origin of the field of Geometric Measure Theory during the 50's, where the notions of rectifiable current which were proved to be the ad-hoc objects for the minimization process of the area (or the mass in general) in the most general setting.The search of absolute or even local minimizers is of course the first step in the study of the variations of a given lagrangians but is far from being exhaustive while studying the whole set of critical points. In many problems there is even no minimizer at all, this is for instance the case of closed surfaces in simply connected manifolds with also trivial two dimensional homotopy groups. This problem is already present in the 1-dimensional counter-part of minimal surfaces, the study of closed geodesics. For instance in a sub-manifold of R 3 diffeomorphic to S 2 there is obviously no closed geodesic minimizing the length. In order to construct closed geodesics in such manifold, Birkhoff around ...

Min-max minimal disks with free boundary in Riemannian manifolds

“…A similar strong convergence property was first proven by Colding-Minicozzi in [6] for the min-max construction of minimal spheres, and it played an essential role in their proof of the finite time extinction for certain 3-dimensional Ricci flow. Similar property was also obtained by the last author for the min-max construction of closed minimal surfaces of higher genus [55,57], and by Riviére for min-max construction of closed minimal surfaces via viscosity method [41]. To the authors' knowledge, our work is the first occasion to obtain such a strong property in the context of free boundary problems.…”

Lower Semi-continuity of the Index in the Viscosity Method for Minimal Surfaces

“…in Gauge Theory while studying the Hilbert Bundle structure of the quotient of H s connections by the Gauge group for s > n/2 and away from reducible connections (see [5]). ✷ Once this Hilbert Bundle structure will be established we shall be considering the following application to the viscosity method for the area functional introduced by the author in [13].…”

“…where σ > 0. The work [13] has been devoted to the asymptotic analysis of sequences of critical points of A σ k , with uniformly bounded A σ k energy and satisfying Struwe's entropy condition…”

“…Since we are assuming N ∞ ≡ 1 a.e. on S ∞ and, following the proof of the main theorem of [13], we can extract a subsequence that we keep denoting Φ k such that we have a bubble tree strong W 1,2 convergence of Φ k towards a minimal (possibly branched) immersion Ψ ∞ of a surface S ∞ . More precisely, if one denotes {S j ∞ } j∈J to be the connected components of S ∞ , for every j ∈ J there exists N j points {a j,l } l=1···N j (containing in particular the possible branched points of Ψ ∞ and a converging sequence of constant scalar curvature metrics h j k of volume one and for any δ > 0 a sequence of conformal embeddings…”

The regularity of conformal target harmonic maps

“…In [5] the author developed a viscosity method in order to produce closed minimal 2 dimensional surfaces into any arbitrary closed oriented sub-manifolds N n of any euclidian spaces R m by min-max type arguments. The method consists in adding to the area of an immersion Φ of a surface Σ into N n a more coercive term such as the L 2p norm of the second fundamental form preceded by a small parameter σ 2 :…”