Min-max for phase transitions and the existence of embedded minimal hypersurfaces

Guaraco, Marco A. M.

doi:10.48550/arxiv.1505.06698

Cited by 6 publications

(26 citation statements)

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“…Genus bounds were proven recently by Ketover [15] (see also De Lellis-Pellandini [12]), as conjectured by Pitts and Rubinstein [26]. We note that another variant of min-max theory, based on phase transition partial differential equations, has been recently proposed by Guaraco [14].…”

Section: Introductionmentioning

confidence: 59%

“…Because x ∈ Y → M(Φ(x)) is continuous, we get from property (iv) above that for every y ∈ X(k i ) 0 , M(φ i (y)) ≤ M(Φ(y)) + η i with η i → 0 as i → ∞. If we apply Lemma 4.1 of [20] with S = Φ(X), property [(ii)] above implies (14) sup{F(φ i (x), Φ(x)) : x ∈ X(k i ) 0 } → 0 as i → ∞.…”

Section: Appendix a Various Interpolation Resultsmentioning

confidence: 96%

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Morse index and multiplicity of min-max minimal hypersurfaces

Marques

Neves

2016

Cambridge Journal of Mathematics

107

188

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The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the minmax minimal hypersurface.We advance the theory further and prove the first general Morse index bounds for minimal hypersurfaces produced by it. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.

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Section: Introductionmentioning

confidence: 59%

Section: Appendix a Various Interpolation Resultsmentioning

confidence: 96%

Morse index and multiplicity of min-max minimal hypersurfaces

Marques

Neves

2016

Cambridge Journal of Mathematics

107

188

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“…This general existence result has to be put in perspective with the previous min-max existence results partly discussed above either in GMT (see [37], [48], [7], [27], [28]• • • ) in harmonic map theory (see [8], [9], [57], [58]) or using level set-PDE approaches (see [19], [56], [15], [14], [49], [50]). Combined with the main regularity results in [42] and [36] theorem I.2 implies in particular all known results for the realization of arbitrary minmax by minimal surfaces.…”

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confidence: 65%

“…One difficulty at this stage is that we can not remove the absolute values inside the upper integral of (III.139). If we would be able to do so, we would be proving the strong convergence for ∇ Φ k towards ∇ Φ ∞ and the lemma would be proven 15 . The rest of the argument consists in proving that the limiting un-oriented varifold associated to the current ( Φ k ) * [B r (x)] is going to be equal, asymptotically as r goes to zero, to an integer times Ξ α * T x Σ.…”

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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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“…Inspired by Guaraco's work on the Allen-Cahn min-max [12] and the well known connection between Ginzburg-Landau functionals and the codimension two area functional (see, e.g., [8], [11], [15] for some of the major results in this line), we began to investigate in [19] the energy concentration of these min-max solutions in the limit ǫ → 0, with an eye to providing a p.d.e.-based alternative to Almgren's min-max construction ( [1], [16]) of stationary integral varifolds in codimension two.…”

Section: Introductionmentioning

confidence: 99%

Energy Concentration for Min-Max Solutions of the Ginzburg-Landau Equations on Manifolds with $b_1(M)\neq 0$

Stern

2017

Preprint

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We establish a new estimate for the Ginzburg-Landau energies) 2 of complex-valued maps u on a compact, oriented manifold M with b1(M ) = 0, obtained by decomposing the harmonic component hu of the one-form ju := u 1 du 2 − u 2 du 1 into an integral and fractional part. We employ this estimate to show that, for critical points uǫ of Eǫ arising from the two-parameter minmax construction considered by the author in previous work, a nontrivial portion of the energy must concentrate on a stationary, rectifiable (n − 2)-varifold as ǫ → 0.

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Min-max for phase transitions and the existence of embedded minimal hypersurfaces

Cited by 6 publications

References 0 publications

Morse index and multiplicity of min-max minimal hypersurfaces

Morse index and multiplicity of min-max minimal hypersurfaces

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

Energy Concentration for Min-Max Solutions of the Ginzburg-Landau Equations on Manifolds with $b_1(M)\neq 0$

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