Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

Wheeler, Glen

doi:10.1090/s0002-9939-2014-12351-3

Cited by 12 publications

(5 citation statements)

References 19 publications

(43 reference statements)

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“…Our second application is a new gap result, which can be seen as a companion to the gap results of G. Wheeler and J. McCoy (see theorem 1 in [22], or [27]). While in the latter, it is assumed that the immersion is proper, we instead assume our immersion has bounded energy.…”

Section: Summary Of Main Resultsmentioning

confidence: 93%

Energy quantization for Willmore surfaces and applications

2014

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: We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into R m with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the Möbius group action, below some energy threshold. I IntroductionLet Φ be an immersion from a closed abstract two-dimensional manifold Σ into R m≥3 . We denote by g := Φ * g R m the pull back by Φ of the flat canonical metric g R m of R m , also called the first fundamental form of Φ, and we let dvol g be its associated volume form. The Gauss map of the immersion Φ is the map taking values in the Grassmannian of oriented m − 2-planes in R m given bywhere ⋆ is the usual Hodge star operator in the Euclidean metric.Denoting by π n Φ the orthonormal projection of vectors in R m onto the m − 2-plane given by n Φ , the second fundamental form may be expressed asThe mean curvature vector of the immersion at p is H Φ := 1 2 tr g ( I) = 1 2 I(ε 1 , ε 1 ) + I(ε 2 , ε 2 ) , where (ε 1 , ε 2 ) is an orthonormal basis of T p Σ for the metric g Φ .In the present paper, we study the Lagrangian given by the L 2 -norm of the second fundamental form:An elementary computation gives 1The energy E may accordingly be seen as the Dirichlet Energy of the Gauss map n Φ with respect to the induced metric g Φ . The Gauss Bonnet theorem implies thatwhere K Φ is the Gauss curvature of the immersion, and χ(Σ) is the Euler characteristic of the surface Σ. The energyis called Willmore energy.Critical points of the Willmore energy, comprising for example minimal surfaces 2 , are called Willmore surfaces. Although already known in the XIXth century in the context of the elasticity theory of plates, it was first considered in conformal geometry by Blaschke in [Bla3] who sought to merge the theory of minimal surfaces and the conformal invariance property. This Lagrangian has indeed both desired features : its critical points contain minimal surfaces, and it is conformal invariant, owing to the following pointwise identity which holds for an arbitrary immersion Φ of Σ into R m and at every point of Σ :Using again Gauss Bonnet theorem, the latter implies the conformal invariance of W :This conformal invariance implies that the image of a Willmore immersion by a conformal transformation of R m is still a Willmore immersion. Starting for example from a minimal surface, one may then generate many new Willmore surfaces, simply by applying conformal transformations (naturally, these surfaces need no longer be minimal). In his time, Blaschke used the term conformal minimal for the critical points of W , seeking to insist on this idea of merging together the theory of minimal surface with conformal invariance.An important task in the analysis of Willmore surfaces is to understand the closure of the space of Willmore immersions under a certain level of energy. Because of the non-compactness of the conformal group of transformation of R m , one cannot expect that the spa...

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Section: Summary Of Main Resultsmentioning

confidence: 93%

Energy quantization for Willmore surfaces and applications

2014

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“…Even in the case when W ≡ 0, the term | H| 2 H is already problematic, for it lies in no space that enables us to understand the equation in a distributional sense to the equation. Nevertheless, one may study the problem and obtain estimates, as is done for example in [Whe1] and the references therein. Another approach was originally devised by Tristan Rivière in [Riv1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…While in the latter only identities were derived, the present work brings to fruition the reformulations presented in [Ber2] by obtaining local analytical results for problems of the type (I.5). The present paper should also be seen as a companion to [Whe1], where only a specific class of right-hand sides W were considered. The class of possible right-hand sides will be here significantly expanded.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Analysis of the Inhomogeneous Willmore Equation

Bernard¹,

Wheeler²,

Wheeler³

2018

Preprint

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“…Gap Lemmata are known in a variety of contexts: for the Willmore operator [9], the surface diffusion operator [26], and a family of fourth-order geometric operators [27]. We expect Theorem 2 to enjoy further applications.…”

Section: Introductionmentioning

confidence: 99%

The geometric triharmonic heat flow of immersed surfaces near spheres

2017

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Abstract. We consider closed immersed surfaces in R 3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L 2 . We further use an ε-regularity type result to prove a gap lemma for stationary solutions. Using a monotonicity argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a nonumbilic embedded stationary surface. This allows us to conclude that any solution with initial L 2 -norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to 3 3V 0 /4π, where V 0 denotes the signed enclosed volume of the initial data.

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Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

Cited by 12 publications

References 19 publications

Energy quantization for Willmore surfaces and applications

Energy quantization for Willmore surfaces and applications

Analysis of the Inhomogeneous Willmore Equation

The geometric triharmonic heat flow of immersed surfaces near spheres

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