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

Ikoma, Norihisa; Malchiodi, Andrea; Mondino, Andrea

doi:10.1353/ajm.2017.0033

Cited by 5 publications

(4 citation statements)

References 41 publications

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“…Under the areaconstraint condition, the existence and the geometric properties of Willmore-type spheres have been investigated by Lamm-Metzger-Schulze [23], Lamm-Metzger [21,22], Laurain and the second author [24] and Eichmair-Körber [10]. Area-constrained Willmore tori of small area have been recently constructed by Ikoma, Malchiodi and the second author [16,17]. All the aforementioned results are perturbative in nature, i.e.…”

Section: Introductionmentioning

confidence: 99%

Quantization of the Willmore Energy in Riemannian Manifolds

Michelat¹,

Mondino²

2021

Preprint

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We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area are uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay in a compact domain of the moduli space.

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Section: Introductionmentioning

confidence: 99%

Quantization of the Willmore Energy in Riemannian Manifolds

Michelat¹,

Mondino²

2021

Preprint

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“…The goal of the present (and the subsequent ) work is to construct smooth embedded Willmore tori with small area constraint in Riemannian three‐manifolds, under some curvature condition but without any symmetry assumption. Let us recall that area‐constrained Willmore surfaces satisfy the equation

{normalΔ}_{\overset{g}{true}} H + H {| \overset{˚}{A} |}^{2} + H Ric false(n, n false) = λ H,

for some

λ \in R

playing the role of Lagrange multiplier.…”

Section: Introductionmentioning

confidence: 99%

“…This is the first of a series of two papers where the construction of embedded area-constrained Willmore tori in Riemannian three-manifolds is performed under curvature conditions: here via minimization/maximization, while in the second paper [13] via Morse theory.…”

Section: Introductionmentioning

confidence: 99%

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

Ikoma

Malchiodi

Mondino

2017

Proc. London Math. Soc.

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We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and analyze the sharp asymptotic behavior of degenerating tori under the action of the Möbius group. In this first work we prove two existence results by minimizing or maximizing a suitable reduced functional, in particular we obtain embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) in compact 3-manifolds with constant scalar curvature and in the double Schwarzschild space. In a forthcoming paper new existence theorems will be achieved via Morse theory.

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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, II: Morse Theory

Cited by 5 publications

References 41 publications

Quantization of the Willmore Energy in Riemannian Manifolds

Quantization of the Willmore Energy in Riemannian Manifolds

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

Clifford Tori and the singularly perturbed Cahn–Hilliard equation

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