Scale-invariant geometric curvature functionals, and characterization of Lipsch...

(1)

Scale-Invariant Geometric Curvature Functionals, and

Characterization of Lipschitz- and C ¹ -Submanifolds

Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades

eines Doktors der Naturwissenschaften genehmigte Dissertation

vorgelegt von Bastian Käfer, M. Sc.

aus Aachen

Berichter: Univ.-Prof. Dr. Heiko von der Mosel (RWTH Aachen University) AOR Priv.-Doz. Dr. Alfred Wagner (RWTH Aachen University) Prof. Dr. Paweł Strzelecki (Uniwersytet Warszawski)

Tag der mündlichen Prüfung: 31. Mai 2021

Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek online verfügbar.

(2)

(3)

Abstract

In this thesis, we investigate the connection of local flatness and the existence of graph representations of certain regularity for subsets ofRⁿ with arbitrary dimensionm6n. In this process, we formulate sufficient conditions providing local graph representations of classC^0,1andC¹. We identify sets satisfying those local representations at each point as Lipschitz- and C¹-submanifolds, respectively. Based on the concept ofδ-Reifenberg- flat sets, we introduce a characterization for the class ofm-dimensionalC¹-submanifolds ofRⁿ.

We apply the gained information in the study of two families of geometric curvature functionals for different classes of m-dimensional admissible sets. Reifenberg-flatness remains to be a crucial tool to achieve additional topological and analytical properties assuming finite energy. The first class of functionals is given by the tangent-point energies T P^(k^,^l) with focus on the scale-invariant casek =l+2m. We prove that admissible sets with locally finite energy are embedded submanifolds ofRⁿwith local graph representations satisfying Lipschitz continuity. In a second step, using a technique of S. Blatt, we characterize the energy space ofT P^(k^,^l)for alll > mandk∈[l+2m, 2l+m)as submanifolds of classC^0,1∩W^k−m^l ^,^l. In contrast to the first step, the proof of this characterization requires a priori given graph representations by Lipschitz functions in order to guarantee the existence of tangent planes for H^m-almost all points in the computation of the tangent-point energy.

Following the work of R. B. Kusner and J. M. Sullivan, we then define a family E^τ of Möbius-invariant energies form-dimensional subsets ofRⁿ. As forT P^(k^,^l), locally finite E^τ-energy for admissible sets provides local graph representations satisfying Lipschitz continuity. We also prove that for τ > 0, each locally compact C^0,1∩W¹⁺^1+τ¹ ^,⁽¹^+τ)m- submanifold ofRⁿ has locally finiteE^τ-energy.

(4)

(5)

Zusammenfassung

In der vorliegenden Dissertation wird der Zusammenhang lokaler Flachheit mit der Existenz von Graphendarstellungen gewisser Regularität für Teilmengen des Rⁿ mit beliebiger Dimension m 6 n untersucht. Es werden hinreichende Bedingungen for- muliert, die lokale Graphendarstellungen der KlasseC^0,1oderC¹garantieren. Mengen, die eine solche lokale Darstellung an jedem Punkt ermöglichen, werden als Lipschitz- bzw. C¹-Untermannigfaltigkeit identifiziert. Darüber hinaus lässt sich die Klasse der m-dimensionalenC¹-Untermannigfaltigkeiten desRⁿ basierend auf dem Prinzip vonδ- Reifenberg-flachen Mengen charakterisieren.

Die erzielten Beobachtungen werden für die Untersuchung zweier Familien von geome- trischen Krümmungsfunktionalen für verschiedene Klassenm-dimensionaler zulässiger Mengen verwendet. Dabei bleibt die Reifenberg-Flachheit ein entscheidendes Hilfsmit- tel, um aus der Annahme endlicher Energie weitere topologische und analytische Eigen- schaften zu erhalten. Die erste Familie bildet die Tangenten-Punkt EnergieT P^(k^,^l), wobei der Fokus der Betrachtungen auf dem skalierungsinvarianten Fallk= l+2mliegt. Es lässt sich zeigen, dass zulässige Mengen mit lokal endlicher Energie eingebettete Unter- mannigfaltigkeiten des Rⁿ sind, die lokale Lipschitz-stetige Graphendarstellungen be- sitzen. Weiterführend wird eine Beweistechnik von S. Blatt angewendet, um den Raum endlicher Energie vonT P^(k^,^l)für allel > mundk∈[l+2m, 2l+m)zu charakterisieren.

Dieser ist gegeben durch die Klasse der C^0,1 ∩W^k−m^l ^,^l-Untermannigfaltigkeiten. Im Gegensatz zu den zuvor gemachten Beobachtungen verlangt der Beweis dieser Charak- terisierung die Existenz von Graphendarstellungen mittels Lipschitz-stetiger Funktio- nen, um fürH^m-fast alle Punkte die Existenz einer Tangentialfläche für die Berechnung der Tangenten-Punkt Energie sicherzustellen.

Basierend auf einer Arbeit von R. B. Kusner und J. M. Sullivan wird eine Familie E^τ von Möbius-invarianten Energien fürm-dimensionale Teilmengen desRⁿ definiert. Wie bereits fürT P^(k^,^l)beobachtet impliziert lokal endlicheE^τ-Energie für zulässige Mengen ebenfalls die Existenz lokaler Lipschitz-stetiger Graphendarstellungen. Außerdem lässt sich fürτ >0 zeigen, dass jede lokalkompakteC^0,1∩W¹⁺^1+τ¹ ^,⁽¹^+τ)m-Untermannigfaltigkeit desRⁿ lokal endlicheE^τ-Energie besitzt.

(6)

(7)

Acknowledgements

I want to express the deepest gratitude to my advisor Univ.-Prof. Dr. Heiko von der Mosel who introduced me to many fascinating topics already during my undergraduate studies. While working on this thesis, he provided me with lots of helpful advice, as well as encouraged my preference for the sometimes unnecessary usage of explicit constants and values in my proofs.

Moreover, I want to thank Priv.-Doz. Dr. Alfred Wagner and Prof. Dr. Paweł Strzelecki for accepting to be additional referees for this thesis.

I also want to thank all of my colleagues at the Institut für Mathematik and the former and honorable members of office 118.1 for an atmosphere of support and friendship. A special thanks goes to Jan for many years full of mathematical and non-mathematical discussions.

Finally, I am grateful to my family for their support and encouragement throughout my whole life. Especially, I want to thank my wife Ann-Kathrin, this thesis and much more would not be possible without her constant support.

The author’s work was partially funded by DFG Grant no. Mo 966/7-1 Geometric curvature functionals: energy landscape and discrete methods (project no. 282535003).

(8)

(9)

Introduction

In our daily life, we permanently are surrounded by objects and surfaces of different kinds and shapes. It is only natural that investigating properties of surfaces and classifying them in order to describe each element of a given class in a comparable way is a vast field in science, especially in mathematics. Mathematically, this classification may be realized by the concept ofsubmanifoldsof Euclidean space where we distinguish classes with respect to regularity. We understand a submanifold as a subset that can locally be described by graph patches of functions with the corresponding regularity. This description of submanifolds is, in fact, equivalent to other common definitions in the dif- ferentiable case (see e.g. [27], [29], [47]). However, for Lipschitz-submanifolds there are different ways to characterize these objects (cf. [52]).

It turns out that the localflatnessof an object is a remarkable tool to decide whether it is a submanifold or not and, obviously, flatness will affect the smoothness as well. The first question that immediately arises is: “When do we call an object flat?” Intuitively, something is flat if it can locally be approximated well by a plane. But what does this meanexactly?

A tabletop seems to be the perfect example of a flat surface. However, since the table isnotinfinitely large, at one of its edges it is far from beingequalto any plane. If this does not bother us too much, i.e. if we only focus on distances of points in the set to an approximating plane, then we end up with the concept of P. Jones’ so-calledβ-numbers introduced in [34]. For theGrassmannianG(n,m)ofm-dimensional subspaces ofRⁿ, a setΣ⊂Rⁿ, a compact subsetK⊂Σ, and a scaler >0, theβ-number is defined by

βK(r) := 1 r sup

p∈K inf

P∈G(n,m) sup

q∈Σ∩Br(p)

dist(q,p+P). (1)

In fact, ifΣis a tabletop, there holdsβK(r)≡0 regardless ofKandr. In general, an object may be considered flat if the correspondingβ-numbers are bounded from above by a small constant. However, flatness is a highly local concept and a surface that does not look flat on a large scale may have smallβ-numbers on a smaller scale. For instance, planet Earth is approximately a sphere nevertheless humankind believed for centuries that the world is flat. In fact, each sufficiently small part of planet Earth’s surface is well-approximated by a part of a plane. Therefore, the Earth is alocallyflat object. Though each hill in the landscape or even each puddle shows that it isnotequal to a plane no matter how small

(12)

we choose the piece of its surface. The same effect may occur for other objects as well. A human standing in front of a cathedral may describe its shape as fascinating, complex, or by referring to the period of time it was constructed, but most likely not to be flat.

However, the fly sitting on the dome may swear that it sees a flat plane.

In the mathematical approach of flatness, there is an additional difficulty: thedimension. Hence, a curve is flat with respect to dimension one if it locally looks like a straight line and a setΣ⊂Rⁿis considered to bem-dimensionally flat if it is locally well-approximated bym-planes for fixed 16m6n.

By replacing the Euclidean distance with the bilateralHausdorff-distancein the definition ofβ-numbers, one obtains the more restrictiveθ-numbers:

θ_K(r) := 1 r sup

p∈K inf

P∈G(n,m)distH (Σ∩B_r(p),(p+P)∩B_r(p)) (2) The set Σis considered flat in this stronger sense if the θ-numbers are bounded from above by some constantδ ∈ (0, 1)for all compact subsetsK ⊂ Σand all radiir less or equal to a scalerK. In particular, a setΣsatisfying this condition is calledδ-Reifenberg- flatwhich refers to the remarkable work of E. R. Reifenberg on the Plateau problem for m-dimensional surfaces [59]. Notice that the tabletop is not flat with respect to theθ- number but the surface of planet Earth is flat for both definitions.

Many topological properties are provided solely byδ-Reifenberg-flatness and, in fact, ifδis sufficiently small thenΣis a topological manifold. This result is widely known as Reifen- berg’s topological disk lemma (see [59, Chapter 4], [51, Theorem 10.5.1], [62], [31]). Ad- ditionally, for sufficiently smallδ, we can bound the Hausdorff-dimension which, in fact, may even be fractional (cf. [68], [20], [30]). Many mathematicians started to combine analytical conditions with Reifenberg-flatness in order to enhance regularity. In particular, T. Toro established in [67] local bi-Lipschitz parametrizations ofΣ, and assuming an additional stronger control on theβ-numbers defined in (1), G. David, C. Kenig, and T. Toro constructed local graph representations forΣof classC^1,^κ, which means thatΣis aC^1,^κ-submanifold ofRⁿ(see [19, Proposition 9.1]). Furthermore, Reifenberg-flatness is a fundamental tool in other fields as well, such as PDEs, geometric measure theory, and harmonic analysis. For instance, it can provide control of the boundary in the study of partial differential equations (e.g. [16], [17], [49]). Moreover, the existence of an exten- sion operator for domains with sufficiently Reifenberg-flat boundary is shown in [48]. It is also considered in the context of (harmonic) measures (e.g. [19], [38], [37]) and, related to this, for zero sets of harmonic polynomials (see [4], [5]).

Different but related types of local flatness are assumed in the definition ofm-dimensional admissible sets as a starting point of investigations of geometric energy functionals, for instance thetangent-point energy(see [66], [8]) and integral Menger curvature (see [64], [39], [40], [10], [41], [18]). In fact, the requirement of finite energy enhances the

(13)

flatness and provides geometrical and analytical properties, frequently resulting in local graph representations of a certain regularity. At least for the tangent-point energies

T P^(k^,^l)(Σ,P) :=

Z

Σ

Z

Σ

dist(q,p+P(p))^l

|q−p|^k dH^m(q)dH^m(p) (3)

withm-planesP(p)attached top, a connection of the functional and Reifenberg-flatness is apparent. For both, it is crucial that the distance of points in a neighbourhood ofpto an affinem-plane is small in comparison to the distance topitself. Notice that the formula given in (3) generalizes the mixed tangent-point energies introduced for curves in [11] to arbitrary dimensions. The classical tangent-point energy of [65], [66] can be restored by settingk=2land inserting the multiplicative factor 2^l.

The study of both aforementioned energy functionals started with the investigation of closed, one dimensional, and injective curves γ: [0, 1] → R³ (e.g. [63], [65], [43], [11]), which can be identified asknots. The most prominent energy in geometric knot theory is the so-calledMöbius energyintroduced by J. O’Hara in [53] some thirty years ago. The name is based on its invariance under Möbius transformations, i.e., under inversions in spheres and hyperplanes, as first proven by M. Freedman, Zh.-Xu He, and Zh. Wang in [25]. The interest attracted by this special kind of symmetry was followed by numerous results for different aspects of the Möbius energy. For example, variational and gradient formulas were studied in [28], [60], [32], [33], [61]. Moreover, the round circle was identified as global minimizer (see [25], [1]) and the regularity of minimizers and critical points has been studied in [25], [28], [60], [12], [13]. Additionally, investigations of the L²-gradient flow were presented in [28], [7], [9]. However, almost all studies of the Möbius energy are focused on the one dimensional case. There are only a few discussions on how to generalize the functional for higher dimensional surfaces (cf. [45], [46], [3], [57]) and even less concerning the effects of finite energy (see [56]).

In general, the existence and study of energy functionals for higher dimensional sets provides a method to evaluate and compare the quality of different objects. This is interesting from a theoretical point of view but also for applications to, e.g., architecture, computer graphics, and engineering in the search of optimal shapes.

In this thesis, we constructsufficientconditions for the existence of localC^0,1- andC¹- graph representations, and wecharacterizeC¹-submanifolds of Rⁿ using the concept of δ-Reifenberg-flat sets. Moreover, applying geometric estimates, we can show that admissible sets Σ ⊂ Rⁿ with locally finite tangent-point energy areδ-Reifenberg-flat for any δ ∈(0, 1). Here, the possibly unbounded setΣhas locally finite energy if and only if the energy value ofΣ∩BN(0)is finite for allN ∈ N. Combining the techniques concerning Reifenberg-flatness with the study of geometric curvature functionals, we can construct local graph representations of Lipschitz regularity for wider classes of admissible, non- smooth subsets ofRⁿ. In particular, in order to achieve these graph patches, we use the structure of the integrand in (3) to find approximating planes as well as to bound the angles between them. Here, the critical, scale-invariant casek=l+2mwhich is left out

(14)

in most present studies is explicitly included and in the focus of interest. Moreover, we can extend S. Blatt’s characterization of the energy-space of the classical tangent-point energy (see [8]) for the generalized functional as in (3) for alll > m,k∈[l+2m, 2l+m) which, in particular, includes the scale-invariant casek=l+2m.

Motivated by the approach of R. B. Kusner and J. M. Sullivan in [45], [46], we introduce Möbius-invariantenergy functionals

E^τ(Σ,P) :=

Z

Σ

Z

Σ

sup

v∈P(q)∩Sⁿ⁻¹

π^⊥_P(p)(v) −_|_q−p² _|₂hv,q−piπ^⊥_P(p)(q−p)

|p−q|²^m

(1+τ)m

dH^m(p)dH^m(q). (4)

The invariance of the energies E^τ is based on the equality of the integrand in (4) to sin(θ_m(p,q))⁽¹^+τ)m/|p−q|²^m, where θ_m(p,q)denotes the largest principal angle in the definition of the so-calledconformal angleθ(p,q)used in [45], [46]. In fact,θ(p,q)and θ_m(p,q)describe the intersection angle of twom-dimensional spheres completely deter- mined by the pointsp,qand them-planesP(p),P(q). We prove that locally finite energy provides local graph representations by Lipschitz functions for a wide class of admissible, non-smooth sets. Finally, as a first step to analyze the energy space of these newly defined functionals, we observe that a suitableτ-dependentfractional Sobolev regularityis sufficient to guarantee a finite energy value. In fact, this is exactly the class of regularity we would expect to characterize the energy space due to S. Blatt’s results for the Möbius energy for one dimensional curves in [6], i.e., for the special caseτ=1 andm=1.

This thesis is structured as follows: InChapter1, we present the concept ofm-planes as elements of theGrassmannian. Moreover, we introduce two different notions of the anglebetween twom-planes inRⁿ and compare both definitions with each other. Here, orthogonal projectionsand their properties are of crucial importance. For later use, we investigate various conditions allowing to establish an upper bound for the angle between twom-planes.

In Chapter 2, we introduce β- and θ-numbers and discuss basic properties. After definingδ-Reifenberg-flatness, we recall Reifenberg’s seminal topological disk lemma as well as additional conditions leading to a Lipschitz continuous parametrization or local graph representations of classC^1,^κ. We give a new definition ofsequential Reifenberg- flatnesswhich demands to check the boundedness of (2) only for a sequence tending to zero in a controlled way. In fact, this concept is equivalent to Reifenberg-flatness up to some multiplicative constant. Then, following ideas of G. David, C. Kenig, and T. Toro in [19], we show two statements for the orthogonal projection of Σonto approximating planes: Those projections are locally surjective and if the planes satisfy an additional condition on their angles, then the projections are locallyinjectiveas well.

The achieved localbijectivityof the orthogonal projection is crucial inChapter3and allows to construct local graph representations. The additional angle condition even es- tablishesLipschitz continuityfor these graph patches. On the other hand, a priori given graph representations of classC^0,1guarantee Reifenberg-flatness as well as a similar con-

(15)

dition on the angles between approximating planes. However, in this case we get much better constants in our estimates and, therefore, we do not obtain an equivalent condition. Nevertheless, if we assumeδ-Reifenberg-flatness for allδ∈(0, 1)and convergence of approximating planes as the radius tends to zero, then we achieve a characterization of C¹-submanifolds ofRⁿ. Moreover, we present some other sufficient conditions for the existence of local graph representations of classC¹and even a sufficient and necessary one which all are related to the integral structure of the aforementioned energy functionals.

Chapter4introduces non-smoothadmissible setsin order to study effects of finite energy in the upcoming chapters. We recall the class A(δ)as defined in [66] and present a slightly modified class A(δ,ξ) which allows improved results for the scale-invariant tangent-point energy. Moreover, we define two new admissibility classes A^m(α) and A^m(α,M)and discuss examples and basic properties following directly from these definitions. Here, e.g., immersed, compact m-dimensional C¹-manifolds, or finite unions of those, as well as countable unions of α-Lipschitz graphs and subsets of those which contain for each point at least one graph patch with locally uniform size satisfy the admissibility conditions. The two types of admissibility indicated byA andA^mare neither disjoint nor is one contained in the other.

InChapter5, we investigate admissible sets with locally finitetangent-pointenergy.

For setsΣ ∈ A(δ), we can use results concerning Reifenberg-flatness and obtain local graph patches of class C^1,^κ in the sub-critical casek ∈ (l+2m, 2l+m). In the scale- invariant case, we can only achieve thatΣis a topological manifold but we can improve this to a C^0,1-submanifold if we claim additional conditions on the planes given in the definition of the admissibility class. In particular, every set contained inA(δ,ξ)satisfies those conditions. Local graph representations of classC^0,1are also provided if we require Σ ∈A^m(α)for sufficiently smallα >0. We conclude this chapter by characterizing the energy space ofT P^(k^,^l) fork ∈[l+2m, 2l+m)andl > mbased on S. Blatt’s technique in [8].

Chapter6introduces theMöbius-invariantenergyE^τas in (4). We compare this energy with the ones presented by R. B. Kusner and J. M. Sullivan in [45] and [46], as well as with the tangent-point energy. Then, we use techniques analogous to the previous chapter in order to investigate sets contained inA^m(α,M)with locally finiteE^τ-energy, resulting in local graph representations for those sets by Lipschitz functions. In a final step, we investigate sufficient conditions guaranteeing finite energy.

Although we conjecture that the energy space ofE^τis characterized by sets with local graph representations satisfying Lipschitz continuity and a suitable fractional Sobolev regularity similarly to the Möbius energy in the one dimensional case, the question if locally finite E^τ-energy provides such regularity exceeding Lipschitz is still open. How- ever, the presented results might be a starting point for a systematic investigation of the Möbius-invariant family of energiesE^τ.

(16)

The tangent-point energies and in particular the scale-invariant case also preserve open aspects. For instance, the additionally granted regularity of critical points is ana- lyzed in detail for curves but still open for higher dimensional surfaces.

The statements of Lemma 2.13, Lemma 2.15, including the auxiliary Lemma 2.14, and parts of Chapter3are already published in [35]. The main result of Sections6.2and 6.3as well as methods in the discussion of classA^m(α,M)and corresponding examples are included in a joint paper with H. von der Mosel in [36]. The techniques presented there were developed in the progress of this thesis.

(17)

The Grassmannian, Planes and 1

Angles

We will start this thesis by introducing the definitions of m-planes, which in our view are linear subspaces of an ambientRⁿ, and angles between such planes. Here and throughout the upcoming chapters,Rⁿ denotes then-dimensional Euclidean space. The aforementioned planes and angles are basic concepts, but of crucial importance for almost every result of this thesis. Once we are able to describem-planes, we become interested in the relations between two of them and how those can differ. Therefore, we focus on one specific angle function which will define a metric on the set of all m-dimensional planes. We will not only compare this angle function with the different concept of principal angles, but also observe many conditions that guarantee lower and upper bounds for the angle between two givenm-planes. All of the used assumptions will occur in the following chapters and the presented bounds are needed to accomplish the later results.

Definition 1.1. Forn, m∈Nandm6n, theGrassmannianG(n,m)denotes the set of allm-dimensional linear subspaces onRⁿ.

Elements of the GrassmannianG(n,m)will be calledm-planesor justplanes, if the dimensions mandnare fixed. For this chapter and the following ones, we will always assume n andmto be positive integers satisfying 1 6 m 6 n. We think of mas the dimension of an observed plane or surface, whilenis the dimension of the ambient space.

Since we view planes as subsets of Rⁿ, the Grassmannian is strongly connected with orthogonal projections onto correspondingm-planes.

Definition 1.2. For P ∈ G(n,m), the orthogonal projection ofRⁿ ontoP is denoted by πP. Furthermore,π^⊥_P :=π_P⊥ =idRⁿ−πP shall denote theorthogonal projection onto the linear subspace perpendicular toP.

Notice that in general the dimensions ofPandP^⊥ are not equal. In fact, we have:

dim P^⊥

=n−dim(P) =n−m for all P∈G(n,m).

Using orthogonal projections, we can define a well-known metric on the Grassman- nian.

(18)

Definition 1.3. For two planesP₁,P₂∈G(n,m), theangleis defined by (P₁,P₂) :=kπP₁−πP₂k:= sup

x∈Sⁿ⁻¹

|πP₁(x) −πP₂(x)|.

Here,k · kdenotes the norm for linear operators fromRⁿ toRⁿ.

Remark 1.4. With this definition of the angle between twom-planes, we obtain some useful properties.

1. (·,·)is a metric onG(n,m). Moreover, it is a well-known fact, that if we equip the Grassmannian with the angle-metric, then(G(n,m),(·,·))is a compact set. In particular,G(n,m)is a compact manifold (see e.g. [50, Lemma 5.1]).

2. For two planesP₁,P₂∈G(n,m)holds

06(P₁,P₂)61.

In particular, (P₁,P₂) = 0, if and only if P₁ = P₂ and (P₁,P₂) = 1 is equivalent to P^⊥

1 ∩P₂6={0}respectivelyP^⊥

2 ∩P₁6={0}.

There exist various different presentations for this angle-metric. The following lemma contains a collection of those, which will be uses in upcoming chapters. We will leave out the proof. It can be found, together with this statement, in [2].

Lemma 1.5(8.9.3 in [2]). LetP₁,P₂∈G(n,m), then kπP₁−πP₂k=kπ^⊥_P

1−π^⊥_P

2k=kπ^⊥_P

1◦πP₂k=kπP₁◦π^⊥_P

2k=kπ^⊥_P

2◦πP₁k=kπP₂◦π^⊥_P

1k. Notice that due to the identity(P₁,P₂) =kπ^⊥_P

1 ◦π_P

2kand the definition of the norm for linear operators, we have

(P₁,P₂) = sup

y∈P₂∩Sⁿ⁻¹

π^⊥_P

1(y)

= sup

x∈P₁∩Sⁿ⁻¹

π^⊥_P

2(x)

. (1.1)

The knowledge of the angle between two m-planes provides many properties, which are important in the later chapters. Typically, we want the angles between to planes to be small, such that we can compare them to each other. In fact, even the bound(P₁,P₂)<1, i.e. P^⊥₁ ∩P₂ = {0}= P₁∩P^⊥₂ (see Remark1.4), is already enough to guarantee that the orthogonal projection ofP₂ontoP₁is a linear isomorphism. This is shown as a part of Lemma 2.2 in [42]

Lemma 1.6. AssumeP₁,P₂ ∈ G(n,m)satisfy(P₁,P₂) < 1. Then the restriction of the orthogonal projection ontoP₁toP₂, i.e. π_P

1|P₂:P₂→P₁, is a linear isomorphism.

Notice that (·,·) is not the only possible way to define the angle between two m- dimensional planes. A different, also common way is the concept of principal angles (see e.g. [26, Section 12.4.3]). For m-planes, this is not a single value, but defined as

(19)

a sequence ofmincreasing real numbers in [0,π/2]. The first principal angle between P₁,P₂∈G(n,m)is given by

cos(θ₁) := sup

x∈P₁∩Sⁿ⁻¹

sup

y∈P₂∩Sⁿ⁻¹

|hx,yi|. (1.2)

SinceP₁∩Sⁿ⁻¹andP₂∩Sⁿ⁻¹are compact sets, there exists anx₁respectivelyy₁in the corresponding set, such that this angle can be obtained by

cos(θ₁) =hx₁,y₁i.

Now, we can iterate this idea for all k ∈ {2, . . . ,m}and receive the remaining principal angles by

cos(θk) := sup

x∈P1∩Sn−1 x⊥span(x1 ,...,xk−1)

sup

y∈P2∩Sn−1 y⊥span(y1 ,...,yk−1)

|hx,yi|. (1.3)

Again, there arexk ∈P₁∩Sⁿ⁻¹andyk ∈ P₂∩Sⁿ⁻¹perpendicular to span(x₁, . . . ,xk−1) and span(y₁, . . . ,y_k−₁), respectively, satisfying

cos(θ_k) =hx_k,y_ki.

This definition guarantees θ₁ 6 · · · 6 θ_m andθ_i ∈ [0,π/2] for all i ∈ {1, . . . ,m}. The collection of vectors(x₁, . . . ,xm)and(y₁, . . . ,ym)are called principal vectors forP₁and P₂. Notice that these always form an orthonormal basis of the correspondingm-plane.

In addition, there is a strong correlation between the principal angles and the angle as defined in Definition1.3. The angle(P₁,P₂)is equal to the sine of the principal angle with the highest value, i.e. θ_m. This connection is also mentioned in [26, Section 12.4.3].

Lemma 1.7. ForP₁,P₂∈G(n,m)andθmdefined as in (1.2) and (1.3), we have

sin(θm) =(P₁,P₂).

For the proof, we will also follow the idea of [26, Section 12.4.3].

Proof. Let(x˜₁, . . . , ˜x_m)be an orthonormal basis ofP₁and(y˜₁, . . . , ˜y_m)an orthonormal basis ofP₂. Consequently, there exist unit vectors ˜xm+1, . . . , ˜xn, ˜ym+1, . . . , ˜yn∈Rⁿsuch that (x˜₁, . . . , ˜x_n)and(y˜₁, . . . ˜y_n)are both orthonormal bases ofRⁿ. Now we define matrices by

X:=

x˜₁|· · ·|x˜n

∈R^n×n, Y :=

y˜₁|· · ·|y˜n

∈R^n×n, X₁:=

x˜₁|· · ·|x˜m

∈R^n×m, Y₁:=

y˜₁|· · ·|y˜m

∈R^n×m, X₂:=

x˜m+1|· · ·|x˜n

∈R^n×(n−m) and Y₂:=

y˜m+1|· · ·|y˜n

∈R^n×(n−m).

(20)

LetA^T denote the transpose of a matrix A. Since we can write π^⊥_P

1(z) = X

2X^T

2 ·z and π_P

2(z) =Y

1Y^T

1 ·z, for allz∈Rⁿ, we get kπ^⊥_P

1◦π_P

2k=kX₂X^T₂ ·Y₁Y₁^Tk

=kX^T ·X₂X^T₂ ·Y₁Y₁^T ·Yk

=

0 X^T

2

!

· Y₁|0

=

0 0

X^T

2Y

1 0

!

=kX^T₂Y₁k_R^m.

Here,k · k_R^m denotes the norm for linear operators fromR^m toR^m. For this calculation, we have used the orthogonality ofXandY. Moreover, we have

X^TY= X^T

1Y

1 X^T

1Y

2

X^T

2Y

1 X^T

2Y

2

!

∈O(n),

i.e. X^TY is an orthogonal matrix. Therefore, for all z = (z₁, . . . ,z_m, 0, . . . , 0) = (z˜, 0) ∈ Rⁿ∩Sⁿ⁻¹, we obtain

1= X^TY·z

2=

X^T₁Y₁ ·z˜

2+

X^T₂Y₁ ·z˜

2, which implies

X^T

2Y

1 ·z˜ =

q 1−

X^T

1Y

1 ·z˜ ². Using this and Lemma1.5yields

(P₁,P₂) = X^T₂Y₁

= sup

z∈˜ R^m∩S^m−1

X^T₂Y₁ ·z˜ =

r

1− inf

z∈˜ R^m∩S^m−1

X^T

1Y

1 ·z˜

2.

According to [26, Theorem 8.6.1] holds inf

z∈R˜ ^m∩S^m−1

X^T₁Y₁ ·z˜

=σ_m(X^T₁Y₁), whereσ_m(X^T

1Y

1)denotes the smallest singular value ofX^T

1Y

1. The only thing left to show is thatσm(X^T

1Y

1) = cos(θm). To see this, letU =

u₁|· · ·|u_m

, V =

v₁|· · ·|v_m

∈ O(m)be orthogonal matrices, such that X^T

1Y

1 = UΣV^T, with the diagonal matrix Σ = diag(σ₁(X^T₁Y₁), . . . ,σm(X^T₁Y₁))andσ₁(X^T₁Y₁)>· · ·>σm(X^T₁Y₁)>0, is the singular value decomposition ofX^T

1Y

1. For the existence of this decomposition, which is called SVD in the literature, and also a detailed discussion about its properties, we refer to [26, Section 2.5.3].

Now, we can definex_i :=X₁·u_i ∈P₁andy_i :=Y₁·v_i ∈ P₂for all i ∈{1, . . . ,m}. More- over, due to the orthogonality ofUandV,(x₁, . . . ,xm)is an orthonormal basis ofP₁and

(21)

(y₁, . . . ,ym)is one ofP₂. The decomposition ofU^TX^T

1Y

1Valready guarantees an identity for the scalar product of ˜x_iwith ˜y_j. In particular, we obtain

hxi,y_ji=



 σ_i(X^T

1Y

1) for i=j, 0 for i6=j. Forx∈P₁∩Sⁿ⁻¹ andy∈P₂∩Sⁿ⁻¹, we can writex=Pm

i=1ai·xiandy=Pm

j=1bj·yj. Then, we can calculate the scalar product ofxwithyand obtain

|hx,yi|=

*X_m

i=1

ai·xi, Xm j=1

bj·yj

+

6 Xm i=1

Xm j=1

|ai|·|bj|·|hxi,yji|

= Xm i=1

|ai|·|bi|·σi(X^T₁Y₁)6σ₁(X^T₁Y₁)

=|hx₁,y₁i|.

This estimate shows thatx₁andy₁realise the maximal value of th scalar product, that can occur forx∈P₁∩Sⁿ⁻¹andy∈P₂∩Sⁿ⁻¹. According to the definition ofθ₁(see (1.2)), this implies

σ₁(X^T₁Y₁) =cos(θ₁).

With the same idea, we can show iteratively, that for all k ∈ {2, . . . ,m}and two points x∈P₁∩Sⁿ⁻¹∩span(x₁, . . . ,xk−1)^⊥andy∈P₂∩Sⁿ⁻¹∩span(y₁, . . . ,yk−1)^⊥there holds

|hx,yi|6σk(X^T₁Y₁) =|hxk,yki|.

Due to (1.3), the right hand side is equal to

σk(X^T₁Y₁) =cos(θk). In particular, we haveσm(X^T

1Y₁) =cos(θm), which implies (P₁,P₂) =

q

1−cos(θ_m)²=sin(θ_m).

Remark 1.8. As already mentioned, every collection of principal vectors is an orthonormal basis of the correspondingm-plane. Moreover, in Lemma1.7, we have seen that for givenP₁,P₂∈G(n,m)we can construct principal vectors(x₁, . . . ,xm)and(y₁, . . .ym)such that

hx_i,y_ji=





cos(θ_i) for i=j, 0 for i6=j.

An advantage of the principal angles is that they give full information about the rela- tion of two givenm-planes, which can not be obtained from a single value. Nevertheless,

(22)

for our further observation, we will rely on Definition1.3. One reason to focus on(·,·) instead of the principal angles is the simplicity to describe the position of twom-planes by just one parameter. As mentioned before, we will lack some information, which could be provided by the missing principal angles, but these are not necessary for the results of the following chapters. Additionally, we can describe the angle with linear operators of orthogonal projections. The geometric methods presented in Chapter2 and 3seem to be more connected to this definition of the angle. Moreover, some arguments used to study the energy functionals in Chapter5 and6will be accomplished by using similar projections.

In the rest of this chapter, we present some estimates for the angle between twom- planes. Each estimate provides an upper bound for(·,·)and we will handle all settings occurring in the later chapters. We start with two results from [41]. In the first situation, we have an orthonormal basis of a plane, which does not differ to much from another plane of the same dimension. Then, the angle between bothm-planes is bounded in terms of this distance.

Lemma 1.9(Proposition 2.5 in [41]). LetP₁,P₂ ∈ G(n,m)and let(e₁, . . . ,e_m)be some orthonormal basis ofP₁. Assume that for eachi = 1, . . . ,mwe havedist(ei,P₂) 6 θ for someθ∈(0, 1/√

2).Then there exists a constantC^A

1 =C^A

1(m)such that (P₁,P₂)6C^A₁θ.

The constantC^A

1 is given explicitly in [41]. With the constant C^A₀ =8(m−1) +16

m−X3 i=0

3ⁱ(m−i−2), (1.4)

depending only on the dimensionm, we can set C^A₁ :=2m 2+C^A₀

. (1.5)

For a more general setting, we adopt a definition of an almost orthogonal basis from [41]

as well.

Definition 1.10(Definition 2.3 in [41]). LetP ∈G(n,m)and let (x₁, . . . ,xm)be a basis ofP. For fixed radiusρ >0 and two constantsε,δ∈(0, 1),(x₁, . . . ,xm)is called a(ρ,ε,δ)- basisofP, if

(1−ε)ρ6|xi|6(1+ε)ρ for i∈{1, . . . ,m} and

|hxi,xji|6δρ² for i6=j.

Comparing an almost orthogonal basis of anm-plane with an arbitrary basis of an- otherm-plane, we obtain the following statement.

(23)

Lemma 1.11(Proposition 2.6 in [41]). Let(x₁, . . . ,xm)be a(ρ,ε,δ)-basis ofP₁∈G(n,m) with constants ρ > 0, ε ∈ (0, 1/2), and δ ∈ (0, 1). Let (y₁, . . . ,y_m) be some basis of P₂ ∈ G(n,m), such that|xi−yi| 6 θρ for someθ ∈ (0, 1/√

2−1/4)and i ∈ {1, . . . ,m}. Furthermore, assume that

CÂ₁(m)(ε+CÂ₀δ)< 1 2. Then there exists a constantCÂ₂ =CÂ₂(m,ε,δ)such that

(P₁,P₂)6C^A₂θ.

Once again, the constant can be given explicitly:

CÂ₂ := CÂ₁ 1−CÂ

1(ε+C^A

0δ), withC^A

0 andC^A

1 as in (1.4) and (1.5).

Notice that both constants,C^A

1 andC^A

2, may not be sharp. For our application it is sufficient to bound the angle in terms of the distance of elements of the basis to the other plane, which is realised byθ. The explicit constants are less important than the fact that we have this kind of control at all.

Most of the time, we are interested in angles between planes which approximate a sub- setΣ ⊂Rⁿ. We want to study topological and analytical properties ofΣand hope that the angle between the planes provide some new results. As we will see in later chapters, control of angles between approximating planes is an important tool, in order to studyΣ. The crucial ingredients for a bound of these angles are the quality of the approximation and of course, the used distance function.

Whenever we say anm-plane approximatesΣatp, we claim that there is a radiusr >0 such that the distance of allq∈Σ∩B_r(p)to the plane is small compared tor, i.e. there exist aδ1 such that

sup

q∈Σ∩Br(p)

dist(q,p+P)6δr.

Here, dist(·,·)is defined by dist(q,p+P) := infx∈P|q− (p+x)|. SinceP ∈G(n,m)is a linear space, we can rewrite this as

dist(q,p+P) =

π^⊥_P(q−p)

. (1.6)

This kind of approximation is formalized by the so-calledβ-numbers(see Definition2.1), introduced by P. Jones in [34]. We will discuss these numbers as an indicator of the ap- proximability ofΣin Chapter2, in particular Definition2.1.

If we have two planes approximatingΣat the same point, we need an additional topological property and obtain the following lemma.

(24)

Lemma 1.12. Let Σ ⊂ Rⁿ and p ∈ Σsuch that there exist a radius r > 0, constants λ∈(0, 1),δ₁,δ₂∈[0, 1], and twom-planesP₁,P₂∈G(n,m)satisfying

sup

q∈Σ∩Br(p)

dist(q,p+P₁)6δ₁r and sup

q∈Σ∩Br(p)

dist(q,p+P₂)6δ₂r. If for allx∈∂B_λr(0)∩P₁there exists aq∈Σ∩B_r(p)withπ_P

1(q−p) =x, then it holds (P₁,P₂)6 δ₁+δ₂

λ .

With the notation of Chapter2, we can rewrite the conditions of Lemma1.12as β(p,P₁,r)6δ₁, β(p,P₂,r)6δ₁ and ∂Bλr(0)⊂πP₁((Σ∩Br(p)) −p). Notice that it is not important, whether the last condition holds forP₁orP₂. Proof. Forx∈∂Bλr(0)∩P₁there is aq∈Σ∩Br(p)such that

πP₁(q−p) =x.

Due to the requirements of the statement and identity (1.6), we obtain δ₂r>

π^⊥_P

2(q−p)

= π^⊥_P

2 πP₁(q−p) +π^⊥_P

1(q−p)

>

π^⊥_P

2(πP₁(q−p)) −

π^⊥_P

2 π^⊥_P

1(q−p)

>

π^⊥_P

2(x) −

π^⊥_P

1(q−p)

>

π^⊥_P

2(x) −δ₁r.

Taking the supremum over allx∈∂Bλr(0)∩P₁yields sup

x∈∂Bλr(0)∩P₁

|π^⊥_P

2(x)|6(δ₁+δ₂)r. We use the linearity of the orthogonal projectionπ^⊥_P

2 and the formula for the angle given by Lemma1.5, respectively (1.1) and get

(P₁,P₂) = sup

x∈∂B₁(0)∩P₁

|π^⊥_P

2(x)|= 1

λr sup

x∈∂Bλr(0)∩P₁

|π^⊥_P

2(x)|6δ₁+δ₂ λ .

Instead of assuming the existence of certain points in Σ with nice projections and prescribing small distance of Σ to both planes, we can also require smallness for the distance ofΣto one of them-plane and for the distance of points in the other plane to Σ.

(25)

Lemma 1.13. Letp₁,p₂∈Σ⊂Rⁿand0< r₁6r₂,δ₁,δ₂∈(0, 1/2), andP₁,P₂∈G(n,m) such that|p₁−p₂|< r₁/2,

dist(p₁+x,Σ∩B_r

1(p₁))6δ₁r₁ for all x∈P₁∩B_r

1(0) and dist(q,(p₂+P₂)∩B_r

2(p₂))6δ₂r₂ for all q∈Σ∩B_r

2(p₂). Then, we have

(P₁,P₂)6C^A₁ · 2 1−2δ₁

δ₁+2r₂ r₁δ₂

.

Notice that if we fixr₂and letr₁tend to zero in Lemma1.13, then the upper bound of the angle betweenP₁andP₂will increase to infinity. Hence, the result of this statement guarantees a useful upper bound only for radii, which can be compared to each other.

Proof. Let(e₁, . . . ,em)be an orthonormal basis ofP₁. For an arbitrary, sufficiently small ε >0, i.e. ε∈(0,(1−2δ₁)/2), we define

x₀:=p₁ and xi:=p₁+1−2δ₁−2ε

2 r₁·ei for i∈{1, . . . ,m}.

Due to supx∈P₁∩Br

1(0)dist(p₁+x,Σ∩Br₁(p₁))6δ₁r₁, for alli ∈{1, . . . ,m}there exists a qi∈Σ∩Br₁(p₁)such that

|qi−xi|6(δ₁+ε)r₁.

Additionally, we defineq₀:=p₁. Then for alli∈{1, . . . ,m}, there holds

|q_i−p₁|6|q_i−x_i|+|x_i−p₁|6(δ₁+ε)r_i+1−2δ₁−2ε 2

r₁= r₁ 2. This leads to

|qi−p₂|6|qi−p₁|+|p₁−p₂|< r₁ 2 +r₁

2 =r₁6r₂ for all i∈{1, . . . ,m} and |q₀−p₂|=|p₁−p₂|< r₁

2 < r₂,

which guaranteeqi∈Σ∩Br₂(q₂)for alli∈{0, . . . ,m}.

Since supq∈Σ∩B_r

2(p₂)dist(q,(p₂+P₂)∩B_r

2(p₂)) 6 δ₂r₂, for alli ∈ {0, . . . ,m}, we obtain the existence of pointsyi∈(p₂+P₂)∩Br₂(p₂)such that

|yi−qi|6δ₂r₂.

Here, we have used the structure ofp₂+P₂as an affine plane to guarantee the existence of they_i. In fact, we can definey_ias the projection ofq_ionto the affine planep₂+P₂, i.e.

yi:=p₂+πP₂(qi−p₂). Fori∈{1, . . . ,m}, we have

x˜i:= xi−x₀

|xi−x₀| =ei∈P₁ and ˜yi:= yi−y₀

|xi−x₀| ∈P₂.

(26)

The former estimates imply

|x˜i−y˜i|= 2

(1−2δ₁−2ε)r₁|xi−x₀−yi+y₀|

6 2

(1−2δ₁−2ε)r₁(|xi−qi|+|q₀−x₀|+|qi−yi|+|y₀−q₀|)

6 2

(1−2δ₁−2ε)r₁((δ₁+ε)r₁+0+δ₂r₂+δ₂r₂)

6 2

1−2δ₁−2ε

δ₁+ε+2 r₂ r₁δ₂

. If

2 1−2δ₁

δ₁+2r₂ r₁δ₂

6 1

√ 2

, then we can chooseεsufficiently small such that

|x˜i−y˜i|< 1

√ 2

as well. This allows to apply Lemma1.9in this setting and therefore yields (P₁,P₂)6C^A₁ · 2

1−2δ₁−2ε

δ₁+ε+2r₂ r₁δ₂

. Taking the limitε→0 verifies the statement in the first case.

If

2 1−2δ₁

δ₁+2r₂ r₁δ₂

> 1

√ 2

, then by (1.5), we findC^A

1 >4. Hence,

(P₁,P₂)616C^A₁ · 2 1−2δ₁

δ₁+2r₂ r₁δ₂

, which concludes the proof.

It is possible to replace the distances used above with theHausdorff-distance, i.e. we assume

distH (Σ∩B_r_i(p_i),(p_i+P_i)∩B_r_i(p₁))6δ_ir_i for i∈{1, 2}.

The Hausdorff-distance for two setsA,B⊂Rⁿ is defined by dist_H (A,B) :=max

sup

a∈Adist(a,B), sup

b∈Bdist(b,A)

. (1.7)

Therefore, it is a stronger assumption for Σ to be well approximated by affine planes with respect to the Hausdorff-distance, than the requirements of Lemma1.13. Due to this observation, we adopt the result in the following lemma assuming small Hausdorff- distance, since this will be a common setting in the upcoming chapters.

(27)

Lemma 1.14. Letp₁,p₂∈Σ⊂Rⁿand0< r₁6r₂,δ₁,δ₂∈(0, 1/2), andP₁,P₂∈G(n,m) such that|p₁−p₂|< r₁/2,

distH (Σ∩B_r

1(p₁),(p₁+P₁)∩B_r

1(p₁))6δ₁r₁, and dist_H (Σ∩B_r

2(p₂),(p₂+P₂)∩B_r

2(p₂))6δ₂r₂. Then, we have

(P₁,P₂)6C^A₁ · 2 1−2δ₁

δ₁+2r₂ r₁δ₂

.

With the notation of later chapters (see Definition2.2), we can summarize the requirements of Lemma1.14, concerning the approximation ofΣ, by

θ(p₁,P₁,r₁)6δ₁ and θ(p₂,P₂,r₂)6δ₂.

If we assume thatΣhas even more structure and we know that the set can locally be written as the graph of a Lipschitz-function, then even stronger estimates hold. Obvi- ously, this requirement is more restrictive than the assumptions in the previous lemmata.

In Example2.4, we will see that any suchΣsatisfies the conditions of Lemma1.12and Lemma1.14. The following lemma is an immediate result of 8.9.5 in [2].

Lemma 1.15. Letα∈(0, 1),P∈G(n,m), and assumeu∈C^0,1(P,P^⊥)satisfiesLip(u)6α.

ForΣ=graph(u)⊂Rⁿand allx,y∈Pwithx+u(x) =p, y+u(y) =q∈Σsuch thatTpΣ andTqΣexist, then there holds

(T_p(Σ),P)6kDu(x)k and

(T_p(Σ),T_q(Σ))6kDu(x) −Du(y)k6 s

1+α²

1−α² ·(T_p(Σ),T_q(Σ)).

Due to Rademacher’s theorem (see e.g. [24, Theorem 3.1.6]), we obtain this estimate for almost allx,y∈P. If evenu∈C¹(P,P^⊥), then this holds true for allx,y∈P. Addi- tionally, we can use the continuity ofDuand ensure the existence of an arbitrary small bound for the angle betweenTpΣandTqΣ, as long aspandq, respectivelyxandy, are sufficiently close to each other.

Notice that foru:P→P^⊥, we identifyP∈G(n,m)andP^⊥∈G(n,n−m)as subsets ofRⁿ to define the graph ofuas the set

graph(u) :={x+u(x) | x∈P}. (1.8) This is different to the commonly used introduction by(x,u(x))∈P×P^⊥. Obviously, due toP×P^⊥=∼ Rⁿ, both concepts lead to the same set.

Scale-invariant geometric curvature functionals, and characterization of Lipsch...