Local Palais–Smale sequences for the Willmore functional

Bernard, Yann; Rivière, Tristan

doi:10.4310/cag.2011.v19.n3.a5

Cited by 13 publications

(19 citation statements)

References 32 publications

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“…In this case, it was shown in [BR1] 13 that the system (II.71) yields that Φ is smooth across the unit disk. In the case when θ 0 ≥ 2, we have shown in the previous section that…”

Section: Ii5 When the Residue C 0 Vanishes: Point Removabilitymentioning

confidence: 95%

Singularity removability at branch points for Willmore surfaces

Bernard

Rivière

2013

Pacific J. Math.

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I OverviewThe Willmore energy of an immersed closed surface Φ : Σ → R m≥3 is given bywhere H denotes the weak mean curvature vector, and dvol g is the area form of the metric g induced on Φ(Σ) by the canonical Euclidean metric on R m . Critical points of the Lagrangian W for perturbations of the form Φ+t ξ, where ξ is an arbitrary compactly supported smooth map on Σ into R m , are known as Willmore surfaces. Not only is the Willmore functional invariant under reparametrization, but more importantly, it is invariant under the group of Möbius transformations of R m ∪{∞}. This remarkable property prompts the use of the Willmore energy in various fields of science. A survey of the Willmore functional, of its properties, and of the relevant literature is available in [Ri3]. Owing to the Gauss-Bonnet theorem, we note that the Willmore energy (I.1) may be equivalently expressed aswhere I is the second fundamental form, and χ(Σ) is the Euler characteristic of Σ, which is a topological invariant for a closed surface. From the variational point of view, Willmore surfaces are thus critical points of the energyIt then appears natural to restrict our attention on immersions whose second fundamental forms are locally square-integrable.We assume that the point-singularity lies at the origin, and we localize the problem by considering a map Φ : D 2 → R m≥3 , which is an immersion of D 2 \ {0}, and satisfyingBy a procedure detailed in [KS2], it is possible to construct a parametrization ζ of the unit-disk such that Φ•ζ is conformal. To do so, one first extends Φ to all of C\{0} while keeping a bounded image and the second fundamental form squareintegrable. One then shifts so as to have Φ(0) = 0, and inverts about the origin so as to obtain a complete immersion with square-integrable second fundamental form. Calling upon a result of Huber [Hu] (see also [MS] and [To]), one deduces that the image of the immersion is conformally equivalent to C. Inverting yet once more about the origin finally gives the desired conformal immersion 1 , which we shall abusively continue to denote Φ. It has the aforementioned properties (i)-(iii), and moreover,1 which degenerates at the origin in a particular way, see (I.9).Away from the origin, we define the Gauss map n viawhere (x 1 , x 2 ) are standard Cartesian coordinates on the unit-disk D 2 , and ⋆ is the Euclidean Hodge-star operator. The immersion Φ is conformal, i.e.where λ is the conformal parameter. An elementary computation shows thatHence, by hypothesis, we see that n ∈ W 1,2 (D 2 \ {0}). In dimension two, the 2-capacity of isolated points is null, so we actually have Φ ∈ W 1,2 (D 2 ) and n ∈ W 1,2 (D 2 ) (note however that Φ remains a non-degenerate immersion only away from the singularity). Rescaling if necessary, we shall henceforth always assume thatwhere the adjustable parameter ε 0 is chosen to fit our various needs (in particular, we will need it to be "small enough" in Proposition A.1).For the sake of the following paragraph, we consider a conformal immersion Φ : D 2 → R m , whi...

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“…In this case, it was shown in [BR1] 13 that the system (II.71) yields that Φ is smooth across the unit disk. In the case when θ 0 ≥ 2, we have shown in the previous section that…”

Section: Ii5 When the Residue C 0 Vanishes: Point Removabilitymentioning

confidence: 95%

Singularity removability at branch points for Willmore surfaces

Bernard

Rivière

2013

Pacific J. Math.

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“…Observe next that Altogether, we see that a conformally-constrained Willmore immersion, just like a "plain" Willmore immersion (i.e., with ≡ 0) satis es the system (3.1). In fact, it was shown in [3] that to any smooth solution ⃗ Φ of (3.1), there corresponds a transverse, traceless, symmetric 2-form satisfying (3.3).…”

Section: Conformally-constrained Willmore Immersionsmentioning

confidence: 96%

“…Standard Wente estimates may thus be performed on (1.5). The system becomes subcritical and regularity statements ensue [3,51]. Furthermore, one veri es that (1.5) is stable under a weak limiting process, which has many nontrivial consequences [3,5].…”

Section: Introductionmentioning

confidence: 97%

“…As mentioned earlier, the Willmore energy also plays a distinguished role in conformal geometry, where it has given rise to many interesting problems and where it has stimulated too many elaborate works to be cited here. We content ourselves with mentioning the remarkable tour de force of Fernando Marques and André Neves [41] to solve the celebrated Willmore conjecture stating that, up to Möbius transformations, the Cli ord torus⁴ minimizes the Willmore energy amongst immersed tori in ℝ 3 . Aiming at solving the Willmore conjecture, Leon Simon initiated the "modern" variational study of the Willmore functional [63] when proving the existence of an embedded torus into ℝ ≥3 minimizing the 2 norm of the second fundamental form.…”

Section: Introductionmentioning

confidence: 99%

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Noether’s theorem and the Willmore functional

Bernard

2016

Advances in Calculus of Variations

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“…Remark III.2. A natural question arising from Proposition III.6 is if actually the system (III.73) is equivalent to the frame energy equation (III.67); in analogy with the situation in the Willmore framework (see [2]- [31]- [33]) we expect this not to be the case. More precisely we expect the system (III.73) to be equivalent to the conformal-constrained Willmore equation.…”

Section: Iii3 a System Of Conservation Laws Involving Jacobian Nonlimentioning

confidence: 99%

A frame energy for immersed tori and applications to regular homotopy classes

Mondino

Rivière

2016

J. Differential Geom.

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The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean 3 ≤ m-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As first result, a Willmore-conjecture type lower bound is established : namely for every torus immersed in R m , m ≥ 3, and any moving frame on it, the frame energy is at least 2π 2 and equalty holds if and only if m ≥ 4, the immersion is the standard Clifford torus (up to rotations and dilations), and the frame is the flat one. Smootheness of the critical points of the frame energy is proved after the discovery of hidden conservation laws and, as application, the minimization of the Frame energy in regular homotopy classes of immersed tori in R 3 is performed.Math. Class.

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Local Palais–Smale sequences for the Willmore functional

Cited by 13 publications

References 32 publications

Singularity removability at branch points for Willmore surfaces

Singularity removability at branch points for Willmore surfaces

Noether’s theorem and the Willmore functional

A frame energy for immersed tori and applications to regular homotopy classes

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