Noether’s theorem and the Willmore functional

Bernard, Yann

doi:10.1515/acv-2014-0033

Cited by 22 publications

(48 citation statements)

References 59 publications

(97 reference statements)

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“…Let Σ be a closed Riemann surface and M m be a closed sub-manifold M m ⊂ R Q . A map Φ ∈ W 1,2 (Σ, M m ) together with an L ∞ bounded integer multiplicity N x is called " integer target harmonic" if for almost every 2 domain Ω ⊂ Σ and any smooth function F supported in the complement of an open neighborhood of Φ(∂Ω) we have…”

Section: Introductionmentioning

confidence: 99%

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

Rivière

2017

Publ.math.IHES

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: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 ...

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Section: Introductionmentioning

confidence: 99%

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

Rivière

2017

Publ.math.IHES

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“…Our proof is based on the regularity theory for Willmore surfaces developed by Rivière [Riv08]. It relies on conservation laws discovered by Rivière [Riv08] in the context of the Willmore energy and adjusted by Bernard [Ber16] for the Canham-Helfrich energy. We get the following final result.…”

Section: Theorem Supposementioning

confidence: 99%

“…An important step in Riviere's regularity theory is the discovery of hidden conservation laws for weak Willmore immersions. In the framework of Canham-Helfich immersions, the corresponding hidden conservation laws were discovered by Bernard [Ber16].…”

Section: Theorem (Weak Closure and Lower Semi-continuity Of Bubble Trmentioning

confidence: 99%

Existence and Regularity of Spheres Minimising the Canham–Helfrich Energy

Mondino

Scharrer

2020

Arch Rational Mech Anal

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We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the 2-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main results we establish the lower semicontinuity of the Canham-Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions.

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“…En particulier le théorème de Noether, voir [28], nous assure de l'existence de lois de conservation associées à chaque transformation 11 , voir [2]. Contrairement aux fonctionnelles conformément invariantes, ici il s'agit d'une invariance au but, et pas à la source, ce qui donne des lois de conservation globales et non-plus locales.…”

Section: Xii-17unclassified

Analyse des problèmes conformément invariants

Laurain¹

2017

Séminaire Laurent Schwartz — EDP Et Applications

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Noether’s theorem and the Willmore functional

Cited by 22 publications

References 59 publications

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

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

Existence and Regularity of Spheres Minimising the Canham–Helfrich Energy

Analyse des problèmes conformément invariants

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