Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: minimization

Ikoma, Norihisa; Malchiodi, Andrea; Mondino, Andrea

doi:10.1112/plms.12047

Cited by 9 publications

(23 citation statements)

References 45 publications

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“…As in our first paper [12] the proof relies on a Lyapunov-Schmidt reduction (encoding the variational structure of the problem, see [1,2] and the book [3]). Using such techniques, together with the stability property of Clifford tori proved by Weiner [40] (see also the related gap-energy result [26]), we reduce the problem of finding area-constrained Willmore tori to a finite dimensional variational problem.…”

Section: Outline Of the Strategymentioning

confidence: 99%

“…under composition of the immersion with isometries, homotheties and inversions with respect to spheres), so the theory of Willmore surfaces can be seen as a natural merging between conformal invariance and minimal surface theory. This was indeed the motivation of Blaschke and Thomsen in the 1920-'30 to introduce such an energy, rediscovered by Willmore [41] in the 60's and thoroughly studied in the last twenty years by a number of authors [5,6,15,21,22,33,35,36,37] (for more details see the introduction of our first paper [12]). Here let us just recall that the minimum of W among all immersed surfaces in R 3 is achieved by the round sphere [41], the minimum among immersed surfaces of strictly positive genus is achieved by the Clifford torus and its Möbius deformations (the existence of a smooth minimum among genus one surfaces was proved by Simon [36], the characterization of the minimum was the long standing Willmore conjecture recently proved by Marques-Neves [22]), and for every positive genus the infimum is achieved by a smooth immersion (the proof of Bauer-Kuwert [5] is built on top of Simon's work [36] and some geometric ideas of Kusner [13]; see also the different approach by Rivière [33,34]) but it is a challenging open problem to characterize such immersion.…”

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confidence: 96%

“…The goal of the present (and the previous [12]) work is to construct smooth embedded Willmore tori with small area constraint in Riemannian 3-manifolds, under some curvature/topological condition but without any symmetry assumption. More precisely we obtain the following main result.…”

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confidence: 99%

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Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory

Ikoma¹,

Malchiodi²,

Mondino³

2017

American Journal of Mathematics

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This is the second part of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Möbius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory. To this aim we establish new geometric expansions of the derivative of the Willmore functional on small Clifford tori (in geodesic normal coordinates) which degenerate to small geodesic spheres with a small handle under the action of the Möbius group. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori which are stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.

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Section: Outline Of the Strategymentioning

confidence: 99%

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confidence: 96%

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Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory

Ikoma¹,

Malchiodi²,

Mondino³

2017

American Journal of Mathematics

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“…Under the area constraint condition, the existence of Willmore type spheres and their properties have been investigated by Lamm-Metzger [17,18], Lamm-Metzger-Schulze [19], and the third author in collaboration with Laurain [21]. The existence of area-constrained Willmore tori of small size have been recently addressed by the authors of this work in [10,11].…”

Section: Introductionmentioning

confidence: 99%

Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

Ikoma

Malchiodi

Mondino

2018

International Mathematics Research Notices

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Let (M, g) be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if P 0 ∈ M is a non-degenerate critical point of the scalar curvature, then a neighborhood of P 0 is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than 32π, moreover it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the first multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass.

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“…Furthermore, this functional is also appears in biology, under the name of Helfrich energy, and it is used to describe the behaviour of some lipid bilayer cell membranes. For further details and references, we suggest to see [18,14,15].…”

Section: Introductionmentioning

confidence: 99%

Clifford Tori and the singularly perturbed Cahn–Hilliard equation

Rizzi

2017

Journal of Differential Equations

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In this paper we construct entire solutions u ε to the Cahn-Hilliard equation −ε 2 ∆(−ε 2 ∆u + W ′ (u)) + W ′′ (u)(−ε 2 ∆u + W ′ (u)) = 0, under the volume constraint R 3 (1 − u ε )dx = 4 √ 2π 2 , whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/ √ 2 embedded in R 3 , as ε → 0. What is crucial is that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov-Schmidt reduction and on careful geometric expansions of the laplacian.Keywords: Lyapunov-Schmidt reduction; Cahn-Hilliard equation; Willmore surface; Clifford Torus. IntroductionThe Allen-Cahn equationarises in several physical contexts, such as the study of the stable configurations of two different fluids confined in a bounded container Ω. If u(x) is the density of one of the two fluids at a point x ∈ Ω and the energy per unit volume is given by a function W of u, it looks reasonable to obtain stable configurations by minimizing the energy functionalamong all distributions fulfilling the volume constraintIf, for instance, W (u) = (1 − u 2 ) 2 , and m ∈ (−1, 1), any piecewise constant function taking only the values ±1 and satisfying (2) is a minimizer, irrespectively of the shape of the interface. Therefore this model is unsatisfactory, since it is very far from the reasonable physical assumption that the interfaces are area minimizers, so one replaces the energy byWe can see that there is a competition between the potential energy, that forces u to be close to ±1, and the gradient term that penalizes the phase transition. By minimizing this functional, we are looking for the physical interfaces in which the phase transition can occur. The minimizers u ε of E ε are solutions to the Euler Lagrange equation, that is (1). In order to see if the interfaces are actually minimal surfaces, it is interesting to study the asymptotic behaviour of the level sets {u ε = c} as the parameter ε → 0. It is useful to exploit the variational structure of the problem. It was shown by Modica and 2 Mortola that the energy E ε , seen as a functional on L 1 (Ω) and extended to be +∞ when the integrand is not an L 1 function, Γ−converges to the functional E(u) = cP er Ω ({u = 1}) if u = ±1 a.e. in Ω +∞ otherwise in L 1 (Ω) in the strong topology of L 1 (Ω) (see [23]), where c > 0 is a suitable constant. Moreover, Modica showed that, if u ε are minimizers of F ε under the volume constraintfor some m ∈ (−1, 1), then there exists a sequence ε k → 0 such that u ε k converges to some function u in L 1 (Ω) (see proposition 3 of [22]). Furthermore, Theorem 1 of [22] asserts that u = ±1 a. e. in Ω, and the set E = {x ∈ Ω : u(x) = 1} is actually a perimeter minimizer between all the subsets F ⊂ Ω satisfying the volume constraintFurther results about the relation between the minimizers of E ε and the minimizers of the perimeter can be found in [22] and in [7], where Choksi and Sternberg also described the relation between phase transition theory and the ...

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Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: minimization

Cited by 9 publications

References 45 publications

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory

Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

Clifford Tori and the singularly perturbed Cahn–Hilliard equation

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