Repulsive knot energies and pseudodifferential calculus [Elektronische Ressource] : regorous analysis and regularity theory for O'Hara's knot energy family E_1hn(_1hnα_1hn), α _e63 (2,3) [E (alpha), alpha epsilon (2,3)] / vorgelegt von Philipp Reiter

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• Mathematics

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• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2009
• Nombre de lectures: 47

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RepulsiveKnotEnergies
andPseudodifferentialCalculus
RigorousAnalysisandRegularityTheory
(α)forO’Hara’sKnotEnergyFamily E ,α∈ [2,3)
VonderFakultätfürMathematik,InformatikundNaturwissenschaften
derRWTHAachenUniversityzurErlangungdesakademischenGrades
einesDoktorsderNaturwissenschaftengenehmigteDissertation
vorgelegtvon
Dipl. Math. PhilippReiter
ausHannover
Berichter UniversitätsprofessorDr. HeikovonderMosel
UnivDr. JosefBemelmans
TagdermündlichenPrüfung 13. März2009
DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfügbar.Abstract
(α)In this thesis, we consider J. O’H’s knot functionals E , α ∈ [2,3), proving
∞FréchetdifferentiabilityandC regularityofcriticalpoints.
Usingsomeideasof Z. X.H andfillingmajorgapsinhisinvestigationoftheMöbius
(2)Energy E ,wefurnisharigorousproofofanevenmoregeneralstatement.
(α) 2We start with proving continuity of E on injective and regular H curves, moreover
(α)we establish Fréchet differentiability of E . Among other things, the proof draws
2on the fact that reparametrization of a sequence of curves to arc length preserves H
convergence. Additionally,wederiveseveralformulaeofthefirstvariation.
α−2(α) (α)˜In the second part, we consider the rescaled functional E = length E establish
∞ α 2,3ing a bootstrap argument, which gives C regularity for critical points in H ∩ H
beinginjectiveandparametrizedbyarc length.
The major technique is to introduce fractional Sobolev spaces on a periodic interval
andtostudybilinearFouriermultipliers.
Zusammenfassung
∞In der vorliegenden Schrift werden Fréchet Di fferenzierbarkeit und C Regularität
(α)kritischerPunktefürJ.O’HsKnotenfunktionale E ,α∈ [2,3),bewiesen.
Durch das Schließen gravierender Lücken in einer Untersuchung der Möbius Energie
(2)E vonZ. X.H gelingteinrigoroserBeweiseinerdeutlichallgemeinerenAussage.
(α) 2Wir beginnen mit Stetigkeit von E auf injektiven und regulären H Kurven und er-
(α)halten danach Fréchet Di fferenzierbarkeit von E . Unter anderem beruht der Beweis
auf der Aussage, dass die Reparametrisierung einer Folge von Kurven nach ihrer Bo
2genlänge H Konvergenzerhält. ZudemwerdenmehrereFormelndererstenVariation
hergeleitet.
α−2(α) (α)˜Der zweite Teil widmet sich dem reskalierten Funktional E = Länge E , aus
∞dem man mittels eines Bootstrap Arguments C Regularität für kritische Punkte in
α 2,3H ∩H erhält,dieinjektivundnachBogenlängeparametrisiertsind.
WesentlicheTechnikhierbeiistdieEinführungvonSobolev RäumenreellerOrdnung
aufperiodischenIntervallenunddasStudiumbilinearerFourier Multiplikatoren.Contents
Introduction 5
§0.1 TheTheoryofKnotEnergies . . . . . . . . . . . . . . . . . . . . . . 5
α,p§0.2 On E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
§0.3 Discussionof H’sapproachfortheMöbiusEnergy . . . . . . . . . . 12
§0.4 Exposéofthepresentwork . . . . . . . . . . . . . . . . . . . . . . . 14
(α)1 E isFréchetdifferentiable 19
§1.1 FourierTheoryandFractionalSobolevSpaces . . . . . . . . . . . . . 19
(α) 0 2 d§1.2 E ∈ C (H (R/2πZ,R )) . . . . . . . . . . . . . . . . . . . . . . . 23
ir
2§1.3 Arc lengthreparametrizationpreserves H convergence . . . . . . . . 32
§1.4 FirstVariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
(α) 0 2 d 2 d§1.5δE ∈ C (H (R/2πZ,R )×H (R/2πZ,R )) . . . . . . . . . . . . . 50
ir
§1.6 DerivativeFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
(α) ∞˜2 CriticalpointsofE areC 61
§2.1 MultipliersonFractionalSobolevSpaces . . . . . . . . . . . . . . . 61
§2.2 DerivationoftheEuler LagrangeEquation . . . . . . . . . . . . . . . 70
§2.3 TheMonster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
§2.4 Boot Strapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3Bibliography 81
Index 85
4Introduction
§0.1 TheTheoryofKnotEnergies
Being real life objects, knots always played an important role in technics, culture and
art. Moreover, they sustainably influenced mathematics, especially the development
of topology in the 19th century. Applications of knot theory currently appear in bio
chemistry(proteinmolecules,DNA)andtheoreticalphysics(quantumfieldtheory).
3Traditionally,aknotisrepresentedbyaclosedcurveγ :R/‘Z→R ,asanembedded
mapping of a (periodic) intervalR/‘Z, ‘ > 0, into the three dimensional Euclidean
3spaceR , which is also called a loop. In this text, the term curve will always denote a
closedcurve.
Primarily one is interested in the “knotting type” of a given knot rather than in its
specificshape. Infact,dealingwithropestherearisesarather“canonical”equivalence
relation: Two knots are of the same type if and only if one can be deformed into the
otherwithoutusingscissorsandglue.
In mathematical terms this means that there exists an isotopy between them, i. e. an
3 3 3homeomorphism H : [0,1]×S → [0,1]×S satisfying H(s,x)∈{s}×S , H(0,x)=
3(0,x), and H(1,γ (t)) = (1,γ (t)) for all s∈ [0,1], x∈ S , t∈ R/‘Z. A knot class0 1
containsallcurvesbeing(ambient)isotopictoeachother.
Theproofthattwogivencurvesindeedbelongtothesameknotclasscanbefurnished
by projecting them onto a plane and transforming one into the other using so called
Reidemeistermoves,see[CF77].
In order to show that two given curves do not belong to the same knot class, the con
cept of knot invariants has turned out to be rather successful. A knot invariant is a
5mapping which assigns to each knot an element from some set, e. g. from the reals,
featuring the same value for all knots belonging to the same knot class. However, one
cannotexpectthattwoknotsbeingmappedtothesamevaluebelongtothesameknot
class. Examples for those knot invariants are, e. g., the linking number introduced by
3C. F. G , the fundamental group of the complementR \ Imageγ or various knot
polynomials. Formoredetailedinformationonknotswereferto[BZ03,CF77].
Apart from the investigation of knot invariants there has recently been developed the
approach to search for representatives possessing a particularly “nice” shape within a
given knot class. Besides requirements on its smoothness, such a knot is expected to
lookaslittleentangledaspossible,i.e. differentstrandsofthisrepresentativearewide
apart, having preferably large distances. In order to achieve the latter, one needs to
modelself avoidance phenomena.
During the last twenty years, the investigation of knot energies formed a new sub
field of knot theory, the so called geometric knot theory (or physical knot theory).
Apart from the realization of self avoidance one expected a good modeling of repul
siveforcesof,e.g.,athinfibrechargedwithelectronslyinginaviscousliquid[Fuk88],
the behaviour of protein foldings [KS98], or the motion of knotted DNA structures in
electrophoresisgels[CKS98].
According to J. O’H’s definition [O’H03, Def. 1.1], a knot energy is a functional
that associates to each element of a given class of (closed) curves a real number (or
+∞), is bounded below and self repulsive . The latter means that it blows up on se
quences of embedded curves converging to a curve with a self intersection , i. e. a
non injective curve. By imposing self repulsion one hopes not to run into the danger
of leaving the ambient knot class, e. g. while following a solution of a gradient flow.
Unfortunately,smallknotsmaypulltightinlimitaryprocesseswhichisnot prevented
by O’H’s definition, and, as we will see, even his most famous knot energy, the
MöbiusEnergy,failstopreventthis.
Besides the aim to distinguish between distinct knot classes, the investigation of knot
energies may also have its impact on the sciences, whenever the behavior of fibres is
modeled. The evolution of these objects is obtained by studying the negative gradient
flowassociatedtotherespectiveknotenergy. Ontheotherhand,attractionphenomena
may also be modeled by a corresponding positive gradient flow; see [AFRvdM08]
for an example from mathematical biology discussing interaction between pairs of
filamentsviacross linkers.
The basic technique to produce a knot energy is to penalize pairs of points having a
smallEuclideandistancebutalargeintrinsicdistance. So,tocurveshavingwidespread
arcsanddistantpointsalowerenergyvalueisassignedthantoentangledknotswhose
isotopytypeismoredifficulttodetermine.
The first knot energy should be attributed to S. F [Fuk88] who focussed on
polygonal, i. e. piecewise linear, curves. His idea was to interpret a knot as a non
elastic fiber equipped with electrons, lying in a viscous liquid which absorbs kinetic
6energy. This leads to some motion of the knot reducing its electrostatic energy. With
theaidofagradientflow,Fconstructedanalgorithmwhichassignstoagiven
polygon a new polygon of the same knot type reducing its energy. By iterating this
process he hopes to obtain one of the energy minimizers (within the respective knot
class). In general one expects a limited number of minimizing knots in a given knot
class.
O’H’sKnotEnergies
The first energy on smooth curves, which in contrary to polygons possess a unique
tangent at all points, goes back to O’H. In [O’H91] he defined for curves γ :
3
R/2πZ→R thef