The Allen-Cahn equation on closed manifolds

Gaspar, Pedro Dinis; Guaraco, Marco A. M.

doi:10.48550/arxiv.1608.06575

Cited by 8 publications

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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%

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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confidence: 65%

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

Rivière

2017

Publ.math.IHES

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“…One is led then to the problem of describing the limit interfaces which arise from this strategy. For some results along these lines see, for instance, [24,10], for finite index solutions on surfaces, and [2] for least area limit interfaces.…”

Section: Introductionmentioning

confidence: 99%

The Second Inner Variation of Energy and the Morse Index of Limit Interfaces

Gaspar

2019

J Geom Anal

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In this article we study the second variation of the energy functional associated to the Allen-Cahn equation on closed manifolds. Extending well known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of N. Le [8,9] and F. Hiesmayr [4].

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“…As in the proof of Theorem 19, by (13) we have for any minimal surface in (Int(X), g hyp ) intersecting Γ:…”

Section: Yau's Conjecture For Finite Volume Hyperbolic 3-manifoldsmentioning

confidence: 96%

“…The search for minimal hypersurfaces in compact manifolds has enjoyed significant progress recently, thanks to the development of various min-max methods, such as the systematic extension of Almgren-Pitts' min-max theory [37] led by Marques and Neves [27,29,28,30], the Allen-Cahn approach [18,13,5], or others [39,23,8,48,38,36]. One central motivation was the following conjecture of S.-T. Yau:…”

Section: Introductionmentioning

confidence: 99%

A dichotomy for minimal hypersurfaces in manifolds thick at infinity

Song¹

2019

Preprint

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Let (M n+1 , g) be a complete (n + 1)-dimensional Riemannian manifold with 2 ≤ n ≤ 6. Our main theorem generalizes the solution of Yau's conjecture for minimal surfaces and builds on a result of Gromov. Suppose that (M, g) is thick at infinity, i.e. any connected finite volume complete minimal hypersurface is closed. Then the following dichotomy holds for the space of closed hypersurfaces in M : either there are infinitely many saddle points of the n-volume functional, or there is none.Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface if (M, g) has finite volume, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result of Irie-Marques-Neves when (M, g) is shrinking to zero at infinity.

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The Allen-Cahn equation on closed manifolds

Cited by 8 publications

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

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

The Second Inner Variation of Energy and the Morse Index of Limit Interfaces

A dichotomy for minimal hypersurfaces in manifolds thick at infinity

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