Chord arc submanifolds of arbitrary codimension [Elektronische Ressource] / vorgelegt von Simon Blatt

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• Mathematics

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• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2008
• Nombre de lectures: 19
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

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Chord-Arc Submanifolds
of Arbitrary Codimension
Von der Fakultat¨ fu¨r Mathematik, Informatik und
Naturwissenschaften
der Rheinisch-Westfal¨ ischen Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker
Simon Blatt
aus Mannheim
Berichter: Universit¨atprofessor. Dr. Heiko von der Mosel
Universitatsprofessor Dr. Pawe l Strzelecki¨
Tag der mu¨ndlichen Pru¨fung: 08. Februar 2008
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online
verfu¨gbar.iiiii
Kurzfassung Abstract
In dieser Arbeit werden von Stephen In this work we extend the studies of
Semmes begonnenen Untersuchungen Stephen Semmes concerning hypersur-
von Hyperflac¨ hen, die eine Bogen- faces which satisfy a chord-arc condi-
Sehnen-Bedingung mit einer kleinen tion with small constant to geometric
Konstante erfullen,¨ auf geometrische objects of higher codimension and the-
Objekte hoherer Kodimension ausge- reby we open this subject to questions¨
weitet und dabei insbesondere die To- arising in the field of geometric knot
pologie dieser Objekte untersucht. theory.
Wir betrachten eingebettete, zu- We consider embedded, connec-
sammenhangende und vollstandige Un- ted, and complete submanifolds of¨ ¨
termannigfaltigkeiten des euklidischen the Euclidean space without boundary
Raumes ohne Rand, die durch den which contain the point infinity.
Punkt unendlich gehen. First, we show that such submani-
In der Arbeit wird zun¨achst ge- folds satisfy a chord-arc condition with
zeigt, dass solche Untermannigfaltig- small constant and a certain Ahlfors re-
keiten genau dann eine Bogen-Sehnen- gularity condition, if and only if the
Bedingung und eine gewisse Ahlforsre- BMO-norm of the normal spaces is
gularit¨at mit kleiner Konstant erfullen,¨ small and the submanifolds satisfies a
wenn die BMO-Norm der Normalen- Reifenberg flatness condition with a
raume¨ klein ist und eine Reifenberg- small constant.
Flachheitsbedingung mit einer kleinen The main tool hereby is that such
Konstante gilt. submanifolds contain big portions of
Das Haupthilfsmittel dabei und fur the graph of continuous differentiable¨
die weiteren Untersuchungen ist, dass functions.
die Untermannigfaltigkeiten große Tei- Using an extension of an approxi-
le von Graphen stetig differenzierbarer mation technique due to Semmes and a
Funktionen enthalten. new extension theorem for isotopies, we
Mittels einer Versch¨arfung einer von then show that these submanifolds are
Semmes entwickelten Approximations- diffeomorphic to spheres and are unk-
technik und eines neuen Fortsetzungs- notted.
satzes fur¨ Isotopien zeigen wir, dass
solche Untermannigfaltigkeiten diffeo-
morph zu einer Sphare¨ und unverkno-
tet sind.iv
Acknowledgment
First of all, I want to express my deepest gratitude to the advisor of this thesis,
Prof. Dr. Heiko von der Mosel. He did not hesitate to give me the opportunity
to write my Ph.D. thesis at the RWTH Aachen and recommended me to be
a scholar of the graduate school ”Hierarchie und Symmetrie in mathematischen
Modellen”, although he hardly knew me at this time. He enthusiastically received
all my knew insights and encouraged me to take the next step. To cut a long
story short, he was and is an excellent mentor.
Special thanks go to all my colleagues at the Institut fur Mathematik of the¨
RWTH Aachen, for making me feel at home in Aachen from the first day on and
for not only being colleagues but friends.
I am deeply indebted to Philipp Reiter for sharing with me his knowledge
regarding knot theory and knot energies in many fruitful discussions and for
inspiring and proofreading some early parts of this work.
Moreover, I want to thank
• Frank Roeser and my father, Prof. Dr. Hans-Peter Blatt, for proofreading
the final version of this manuscript,
• Prof. Dr. Pawe l Strzelecki, for pointing out to me the work of Haeflinger
in [Hae62a, Hae62b] and excepting to be the second referee of this Ph.D.
thesis,
• the Deutsche Forschungsgemeinschaft for supporting this thesis through a
grant of the Graduiertenkolleg 775, ”Hierarchie und Symmetrie in mathe-
matischen Modellen”,
• Prof. Dr. Gerhard Hiß, the speaker of the graduate school mentioned
above, who always had an open ear for both administrative and mathe-
matical questions.
Last but not least, I want to thank my parents, Elvira and Hans-Peter Blatt,
and my sisters Sarah and Sophie and my brother Markus for their multifarious
support and, of course, Martina Herrmannsdorfer for keeping me on the right¨
track, encouraging, and loving me.Contents
1 Introduction 1
2 Analytic Foundations 7
2.1 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Local Doubling Spaces . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Hardy-Littlewood Maximal Theorem . . . . . . . . . . . . 8
2.2.2 Inequality of John and Nirenberg . . . . . . . . . . . . . . 12
2.2.3 Example of a Local Doubling Space . . . . . . . . . . . . 19
2.3 Lipschitz Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Topological Foundations 23
13.1 Extending C Isotopies . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Local Extension Lemma . . . . . . . . . . . . . . . . . . . 28
3.1.3 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . 31
3.2 Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Number of Preimages . . . . . . . . . . . . . . . . . . . . 36
03.2.2 Approximation of C Mappings . . . . . . . . . . . . . . . 39
3.2.3 Proof of Theorem 3.12 . . . . . . . . . . . . . . . . . . . . 43
3.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Chord-Arc Submanifolds 51
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Chord-Arc Submanifolds . . . . . . . . . . . . . . . . . . . 51
4.1.2rc Constants . . . . . . . . . . . . . . . . . . . . 59
14.2 Big Portions of C Graphs . . . . . . . . . . . . . . . . . . . . . . 66
4.3 γ Small Implies η Small . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 η I γ . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Approximation of Chord-Arc Submanifolds . . . . . . . . . . . . 99
5 Unknottedness 121
Nomenclature 137
Index 139
vvi CONTENTS
Bibliography 141Chapter 1
Introduction
In 1991 Stephen Semmes published three articles [Sem91a, Sem91b, Sem91c] in
which he extended the well-known chord-arc condition for curves to hypersur-
faces of the Euclidean space. These articles had a deep impact in various fields
of mathematics like the study of harmonic measures and the regularity of free
boundaries (cf. [KT97, KT99, KT02, KT03, CKL05, KT06]) or in the search
for a sufficient criterion for the existence of bi-Lipschitz parameterizations of
two-dimensional manifolds (cf. [Tor95, Fu98, BL03]).
2Let us recall that a rectifiable Jordan curveσ⊂R is called achord-arc curve
with chord-arc constant η˜(σ) if
d (x,y)σ
η˜(σ) := sup − 1<∞.
x,y∈σ |x−y|
Here,|·| denotes the Euclidean norm and d (x,y) the distance of the points xσ
and y along σ. A domain bounded by a chord-arc curve is called a chord-arc
domain. In [Sem91a], Semmes considered complete, connected, and embedded
2 nC hypersurfaces Γ⊂R without boundary. Furthermore, he assumed that Γ∪
2 n n∼{∞} is aC hypersurface ofR ∪{∞} S . Among other things, this guarantees=
that Γ goes through infinity and that Γ is an orientable manifold that divides
nthe ambient space R into two connected components Ω and Ω . Semmes+ −
extended the definition of the chord-arc constant of curves to hypersurfaces by
setting
( )˛ ˛ ˛ ˛
n−1˛ ˛ ˛ ˛d (x,y) H (Γ∩B (x))Γ R˛ ˛ ˛ ˛η˜(Γ) := max sup − 1 , sup − 1 ,
˛ ˛ ˛ ˛n−1|x−y| ω Rx=y∈Γ x∈Γ,R>0 n−1
kwhered is the geodesic distance on Γ,H thek-dimensional Hausdorff measure,Γ
andω denotes the volume of ak-dimensional ball with radius one. Furthermore,k
1
62 CHAPTER 1. INTRODUCTION
he defined
(
Z
1 n−1
γ˜(Γ) := max sup |ν−ν |dH ,B (x)Rn−1H (Γ∩B (x))x∈Γ,R>0 R Γ∩B (x)R
˛ ˛!)˙ ¸
˛ ˛x−y,ν˛ B (x) ˛R
sup sup ˛ ˛ ,
˛ ˛Rx∈Γ,R>0 y∈Γ∩B (x)R
where ν denotes the unit normal and
Z
1 n−1
ν := ν(z)dH (z).B (x)R n−1H (Γ∩B (x))R Γ∩B (x)R
So γ controls the BMO norm of the unit normal and contains some Reifenberg
condition. Finally, Semmes introduced two other constants α(Γ) and β(Γ) that
reflect the boundary behavior of Clifford holomorphic functions on Ω and Ω+ −
(cf. [Sem91a, p. 200] for more details). His main theorem in this context is that
all four constantsα(Γ),β(Γ),γ˜(Γ), andη˜(Γ) are small if any of them is sufficiently
small. Thus, he proved analogs to some of the well-known relations between
the chord-arc constant for curves, the geometry of and the operator theory on
such curves, and function theory on the corresponding chord-arc domains (cf.
[Pom78, TV80, CM83, Dav82, JK82, Sem86, Sem88]). We will use the term
chord-arc hypersurface with small constant for hypersurfaces for which any of
the above constants and hence all of these