simplicial complexes
by
Victòria Gras Andreu
B. Sc., Universitat de Barcelona; Barcelona, Spain, 2013 M. Sc., Universitat de Barcelona; Barcelona, Spain, 2014
M. Sc., Brown University; Providence, RI, 2016
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University
the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.
Date
Georgios Daskalopoulos, Ph. D., Advisor
Recommended to the Graduate Council
Date
Kathryn Mann, Ph. D., Reader
Date
Richard Schwartz, Ph. D., Reader
Approved by the Graduate Council
Date
Andrew G. Campbell Dean of the Graduate School
Victòria Gras Andreu attended Universitat de Barcelona, where she received a Bachelor’s in Mathematics in 2013 having earned the Premi Extraordinari award. In 2014, she com-pleted the Master’s of Science in Advanced Mathematics from the Universitat de Barcelona. Following this, she attended graduate school at Brown University as a La Caixa fellow for the first 2 years. She received the Master’s of Science degree in Mathematics in 2016.
I would like to express my deepest gratitude to my advisor, George Daskalopoulos, for his continuous support of my Ph.D study and research, for his patient guidance, useful advice, critique of the work contained in this thesis, and his assistance in keeping my progress on schedule. I want to thank him for the problems he has suggested, and also for his encouragement to follow my own interests. I could not have imagined a better advisor and mentor.
In addition to my advisor, I would like to thank Kathryn Mann and Richard Scwartz for taking the time to review my thesis and be on my defense committee.
My sincere thanks also goes to Athanase Papadopoulos, for many insightful conversa-tions about the direcconversa-tions of this work, Chikako Mese, for her useful and detailed comments, Graeme Wilkin, for taking the time to listen to the early stages of this project and hosting me at NUS, and Semin Kim, for teaching me so much math at many seminar talks during his time at Brown.
I would like to offer my special thanks to Brian Freidin, my collaborator, academic brother, and friend. This work would not have been possible without his immense knowl-edge, infinite patience, and willingness to help. I am very grateful for all of our discussions and his corrections of my original work.
My special thanks are extended to the staff of the Mathematics Department at Brown University: Audrey Aguiar, Larry Larivee, Lori Nascimento, Carol Oliviera, Doreen Pap-pas, and Sidalia Piriquito. All of them made my time at Brown so easy and enjoyable.
I wish to acknowledge the help provided by Obra Social "La Caixa", who funded the first two and a half years of my Ph.D. study. A special thanks to Emilia Jordi for all her advice on how to navigate that new stage of my life.
During my time at Brown I had the opportunity to interact with many great people, and I would like to thank them here as well. To name a few: Shamil Asgarli, for sharing my passion for doughnuts, Alex Barron, for being my concert buddy, Peihong Jiang, for our shooting adventure, Seoyoung Kim, for our end-of-the-semester Pastiche trip tradition, and Laura Walton, for sharing some of her wisdom with me. I would also like to thank my office-mates, Yang (Sunny) Xiao, Alex McDonough, Cyrus Peterpaul, and Ang Li, for being great working companions and for many insightful conversations that went beyond mathematics. Finally, my deep and sincere gratitude to Ashley Weber and her family, for letting me be part of their wonderful family and giving me a home while I was away from my own.
To my roommates during my time at Brown, Dream and Maya.
I am thankful to all the fantastic people I met during my undergraduate and master’s degrees in Barcelona. I would like to express my appreciation for all the professors that inspired me throughout all those years. Also, to my friends and classmates that always believed in me: Marta Bofill, Gemma Colldeforns, Adrià Màdico, Ivan Martínez, Jordina Orcajo, Francesc Pons, and Edu Soto.
To Bruce, for soundtracking my life.
To Iris Barreda, Laia Estorach, Marina Pallás, and Coral Ricote for a lifetime of in-valuable friendship and support.
To Elchanan Solomon, a pillar of strength for me and an endless source of happiness: הבר הדות.
Last but not least, I am forever grateful to my family for their continuous and unpar-alleled love. Above all, my most profound gratitude to my parents and siblings for always being there for me. Sense vatros res d’això hagués sigut possible.
1 Introduction 1
2 Background 5
2.1 Harmonic maps . . . 5
2.1.1 Harmonic Maps for Riemannian Manifolds . . . 6
2.1.2 Harmonic Maps for Metric Spaces . . . 9
2.2 Riemannian Simplicial Complexes . . . 15
2.2.1 Flat Simplicial Complexes . . . 18
2.2.2 Hyperbolic Simplicial Complexes . . . 21
2.3 Metrics with Conic Singularities . . . 26
2.3.1 Flat Conical Metrics . . . 26
2.3.2 General definition of Conical metrics . . . 28
2.4 Additional results . . . 29
2.4.1 Topological results . . . 31
2.4.2 More on complexes . . . 32
2.4.3 Regularity of Harmonic Maps through Partial Differential Equations 33 3 Metrics on Simplicial Complexes 38 3.1 Euclidean metrics . . . 38
3.2 Hyperbolic metrics . . . 40
3.3 Conformal metrics . . . 43
4 Existence Theory for Conformal Targets 48
4.1 Function Spaces . . . 48
4.2 Existence of Finite Energy Maps . . . 49
4.3 Energy Minimizing Maps . . . 53
4.3.1 Harmonic Replacement . . . 55
4.3.2 Local Regularity . . . 58
4.3.3 Global Existence . . . 63
5 Properties of the Energy Minimizing Maps 66 5.1 Lipschitz and interior regularity . . . 66
5.2 Topological properties . . . 68
5.3 Boundary Regularity of the W1,2(ρ) minimizer . . . . 72
6 Existence Theory for Targets with Cone Type Singularities 79
Bibliography 84
2.1 2-dimensional simplicial complexes with (right) and without (left) singular
edges . . . 17
2.2 Distinguished points. . . 22
2.3 2-dimensional simplicial complexes with complete (left) and incomplete (right) metrics . . . 24
2.4 Link of a vertex . . . 24
2.5 Horocycles in a complete metric . . . 25
4.1 C∞ convex set containing a hexagon . . . . 52
4.2 Compact exhaustion of X . . . . 54
4.3 Local Lipschitz continuity on Euclidean and hyperbolic triangles . . . 62
5.1 Courant-Lebesgue Lemma . . . 70
Introduction
This dissertation is a study of the existence and regularity of harmonic maps between 2-dimensional simplicial complexes. This work begins by defining metrics on complexes and examining the existence of harmonic maps between these complexes that maintain their simplicial structure, that is, mapping faces to faces and edges to edges. We then explore the regularity of these maps. This thesis is based on two papers, both co-authored with Brian Freidin, one of them submitted to publication and one in preparation.
In [CP], Charitos and Papadopoulos study finite 2-dimensional simplicial complexes. They describe how to endow each face (with vertices removed) with the structure of an ideal hyperbolic triangle, define special parameters describing the ways these faces can be glued together to form the complex, and characterize those metrics that are complete. They compute the dimension of the Teichmüller space of complete ideal hyperbolic metrics on such a complex, and describe a compactification of this space in terms of special measured foliations.
The theory of harmonic maps has been applied fruitfully in Teichmüller theory in many ways. To a harmonic map f , one associates its Hopf differential φ(z)dz2, the (2, 0) part of the pull-back of the target metric by f , and the harmonicity of f is equivalent to the
holomorphicity of the Hopf differential. Sampson [S] and Wolf [W] show that for a fixed metric g0on a surface S of genus g, the map that associates a hyperbolic metric g on S with the Hopf differential of the harmonic map f : (S, g0) → (S, g) is a homeomorphism from the Teichmüller space of S to the space of holomorphic quadratic differentials on (S, g0).
In another direction, Gerstenhaber and Rauch propose in [GR] a variational character-ization of Teichmüller mappings; i.e. those that minimize the complex dilatation in their isotopy class, via harmonic maps. In [K], in fact, Kuwert shows that the Teichmüller map is harmonic with respect to a particular singular flat metric in the conformal class of the target. The goal of this thesis is to introduce harmonic map theory into the Teichmüller theory of the ideal hyperbolic complexes of [CP], by constructing harmonic maps on these complexes.
Existence results for harmonic maps go back at least as far as the work of Eells and Sampson [ES] from ’64, where they prove existence, under curvature and completeness hypotheses, by the heat flow method. The study of harmonic maps between singular spaces began with the seminal work of Gromov and Schoen [GS] in ’92, who studied harmonic maps into Riemannian simplicial complexes to prove p-adic superrigidity. Their theory was extended by Korevaar and Schoen [KS] to CAT(0) targets, by Eells and Fuglede [EF] to polyhedral domains, and by Jost [J2] to more general metric-measure domains.
For general singular targets, the most regularity one can hope for is Lipschitz continuity (c.f. [GS],[KS]), and when the domain is simplicial, the map may only be Hölder continuous. In special cases, e.g. harmonic maps from 2-dimensional simplicial complexes into smooth manifolds (c.f. [DM1],[MY]), harmonic maps are smooth, or even analytic, on the faces of the domain.
of [JS], but the result that our maps are diffeomorphisms relies on a local statement for neighborhoods in a simplicial complex, which remains at this time a conjecture.
In Chapter 3, we define the metrics on these simplicial complexes and study their curvature properties. The two main classes of metrics we will be interested in are Euclidean metrics on the compact simplicial complex and ideal hyperbolic metrics on the simplicial complex punctured at its vertices. We will also consider metrics conformal to these two classes, which may or may not have conical singularities.
In Chapter 4, we investigate the existence of energy minimizing maps from a Euclidean or ideal hyperbolic simplicial complex to a simplicial complex with a Euclidean-or ideal hyperbolic-conformally equivalent metric, in the sense of Chapter 3. We construct two different classes of maps: one minimizing energy with respect to all finite energy maps that respect the simplicial structure, and one minimizing energy with respect to those maps that respect the simplicial structure, and whose restriction to each face is a diffeomorphism. We use a harmonic replacement scheme to construct these maps.
Theorem 1.0.1 (c.f. Theorem 4.3.9). There exist energy minimizing mappings u ∈
W1,2(ρ) and uD ∈ D(ρ) in the homotopy class of a finite energy map. That is,
E(u) = inf
v∈W1,2(ρ)E(v),
and
E(uD) = inf
v∈D(ρ)E(v).
In Chapter 5, we study properties of the energy minimizing maps constructed in Chap-ter 4. We prove some topological properties, like properness and degree one (Theorem 5.2.1, Theorem 5.2.3, Theorem 5.2.4). We also show each map is locally Lipschitz continuous (Theorem 5.1.1) and enjoys some extra interior regularity.
Theorem 1.0.2 (c.f. Theorem 5.1.3). Let u ∈ W1,2(ρ) and uD ∈ D(ρ) be the energy
k ≤ ∞. Then for any face T of X, the restrictions u|T and uD|T to the interior of T are
Ck−1 maps. If ρ2τ is analytic, then u|T and uD|T are analytic in the interior of each T .
Moreover the map uD is a diffeomorphism on the interior of each face of X.
We also explore the boundary regularity of the map u ∈ W1,2(ρ).
Theorem 1.0.3 (c.f. Theorem 5.3.6 and Theorem 5.3.7). Let u : (X\S, σ) → (X\S, ρ2τ ) be a harmonic map. If ρ is Ck, then the restriction of u to the closure of each face is Ck−1.
Moreover, if ρ is analytic, then the restriction of u to the closure of each face is analytic.
Finally, in Chapter 6 we study the existence of harmonic maps between simplicial complexes where the target metric may have cone type singularities. That is, we allow the conformal factor on the target to have isolated zeros.
Theorem 1.0.4 (c.f. Corollary 6.0.2). For a cone metric ρ2τ, there exists an energy minimizing map u ∈ W1,2(ρ). That is,
E(u) = inf
Background
In this chapter we describe early work related to the problems addressed in later chapters. It is stated without extensive definitions, but is intended to give the reader an idea of the flavor of the work that predates this thesis. The organization of this chapter is as follows. Section 2.1 describes the classical theory of harmonic maps from Riemannian manifolds to Riemannian manifolds or singular spaces. It also touches upon existence of harmonic diffeomorphism problems. In Section 2.2, we introduce the spaces we focus on in Chapter 3 and give an overview of existence and regularity results of harmonic maps between such spaces. Section 2.3 defines conical singular metrics, which we use in Chapter 6. Finally, in Section 2.4 we collect various technical results that will be instrumental throughout this thesis.
2.1
Harmonic maps
In this section we provide a brief survey of the classical theory of harmonic maps. The basis on harmonic maps from and to Riemannian manifolds is followed by its generalization to singular spaces. It is not intended to be extensive and we will center our attention to
results that will be relevant for us in the latter chapters.
2.1.1 Harmonic Maps for Riemannian Manifolds
Classical Theory
Here we review the classical theory of harmonic maps from and to Riemannian manifolds. We follow [J3].
First we establish basic notations. Let (M, g) and (N, h) be compact Riemannian manifolds of dimension m and n respectively. Let h·, ·iE denote the metric of a bundle E. With respect to local coordinates {x1, ..., xm} on M and {y1, ..., yn} on N ,
gαβ := h ∂ ∂xα, ∂ ∂xβiT M, hij := h ∂ ∂yi, ∂ ∂yjiT N. or also g = gαβdxα⊗ dxβ, h = h ijdyi⊗ dyj. Let √ g :=p det(g).
The Christoffel symbolsMΓγαβ,NΓkij can be written in terms of the metric gαβ, hij and
its derivatives as MΓγ αβ = 1 2g γδ∂gβδ ∂xα + ∂gδα ∂xβ − ∂gαβ ∂xδ , NΓkij = 1 2h kl∂hjl ∂yi + ∂hli ∂yj − ∂hij ∂yl ,
where gαβ, hij denote the components of the inverse of g and h respectively.
Let f : M → N be a C1 map. We can define the energy density of f with xα and fi the local coordinate systems around x and f (x) respectively as
|∇f |2(x) = n X i,j=1 m X α,β=1 gαβ(x)hij(f (x))∂f i ∂xα ∂fj ∂xβ.
This definition is independent of the coordinate system. Indeed, we can also define the energy density intrinsically as follows. The differential of f ,
df = ∂f
i
∂xαdx α⊗ ∂
is a section of the vector bundle T∗M ⊗ f−1T N .
f−1T N is a bundle over M with metric whose components are hij(f (x)), while T∗M
has metric with components gαβ. Then, with ∂x∂fα = ∂f i
∂xα∂f∂i we define the energy density
of f as |∇f |2 = X i,j,α,β gαβ(x)hij(f (x))∂f i ∂xα ∂fj ∂xβ = m X α,β gαβh ∂f ∂xα, ∂f ∂xβif−1T N = hdf, df iT∗M ⊗f−1T N.
Definition 2.1.1. We define the energy of a C1 map f : M → N as
E(f ) := ˆ M |∇f |2dvM, where dvM = √
gdx is the volume form of M.
Definition 2.1.2. A finite energy map f is a harmonic map if it is a critical point of the energy functional E : C1(M, N ) → R.
Theorem 2.1.3. The Euler-Lagrange equations for E are
(2.1) √1 g ∂ ∂xα √ ggαβ(x)∂f i ∂xβ ! + gαβ(x)NΓijk(f (x))∂f j ∂xα ∂fk ∂xβ = 0.
Examples 2.1.4. 1. If N = Rn and hij = δij, then
E(f ) = n X i=1 ˆ ∇f i 2 dvM,
and hence f is harmonic if and only if the components of f are harmonic functions. 2. If M = [0, 1], that is f is a curve in N , then the energy of f is
E(f ) = ˆ 1 0 df dt 2 dt.
3. The identity map id : M → M is harmonic.
The existence and regularity of the unique harmonic map from a complete Riemannian manifold to a complete Riemannian manifold of negative curvature in each homotopy class is a classical result from [ES].
Harmonic Maps and Diffeomorphisms
Properties of harmonic maps have been broadly studied mainly due to the nice classical results for existence and regularity of harmonic maps. One of the properties that has drawn more attention is the existence of harmonic diffeomorphisms. Some of the most famous positive existence results are summarized here.
Theorem 2.1.5. ([SY], Theorem 3.1) Suppose M, N are compact Riemann surfaces with-out boundary with the same positive genus and suppose N has non-positive Riemannian curvature. Let f : M → N be a degree one harmonic map. Then f is a diffeomorphism with positive Jacobian on M.
One can solve a Dirichlet problem in the homotopy class of diffeomorphisms to show existence of a harmonic diffeomorphism in that class. In the case of surfaces with boundary, we have the following result.
One can also show that under suitable regularity of the metric one can solve the Dirichlet problem on some appropriate domain of a Riemannian surface and the resulting map is a diffeomorphism.
Proposition 2.1.7. ([JS], Proposition 1)Let M, N be two compact two dimensional Rie-mannian manifolds with or without boundary and with metrics of class C3. Suppose Ω ⊂ M is a domain homeomorphic to the disc with boundary ∂Ω of class C2+α.
Sup-pose φ : ∂Ω → N is a homeomorphism of ∂Ω onto a C2+α curve which bounds a convex
set contained in the geodesic disc of radius π/2√κ about some point, κ ≥ 0 being an upper bound for the curvature of N. Then there exists a unique harmonic mapping u : Ω → N which assumes the boundary values prescribed by φ, and the map u is a diffeomorphism between Ω and the open disc bounded by φ(∂Ω).
In fact, if the boundary map is a C2 diffeomorphism onto the boundary, the harmonic map will be a diffeomorphism all the way to the boundary.
Theorem 2.1.8. ([J1], Theorem 5.1.1) Suppose Ω is a bounded domain with C2 boundary ∂Ωon some surface M, and that N is another surface. We assume that φ : Ω → N maps Ω homeomorphically onto its image, that φ(∂Ω) is contained in some disc Br(p) with radius
r < π/2κ (where κ2 ≥ 0 is an upper curvature bound on Br(p)) and that the curves φ(∂Ω)
are of class C2 and convex with respect to ∂Ω.
Then there exists a harmonic mapping f : Ω → Br(p) with the boundary values
pre-scribed by φ which is a homeomorphism between Ω and its image, and a diffeomorphism in the interior.
Moreover, if φ|∂Ω is even a C2-diffeomorphism then u is a diffeomorphism up to the
boundary.
2.1.2 Harmonic Maps for Metric Spaces
Sobolev Spaces
A connected open subset (Ω, g) of a Riemannian manifold (M, g) is a Riemannian domain if its metric completion Ω is a compact subset of M .
Definition 2.1.9. Let (Ω, g) be a Riemannian domain and (X, d) a complete metric space For 1 ≤ p < ∞, Lp(Ω, X) is defined as the set of Borel-measurable functions u : Ω → X
with separable range which ˆ
Ω
dp(u(x), Q)dµg< ∞ for some Q ∈ X.
The space Lp(Ω, X) is a Banach space with the metric
Dp(u, v) = ˆ
Ω
dp(u(x), v(x))dµg.
Let (M, g) be a Riemannian manifold of dimension n and (X, d) a metric space. One can extend the definition of energy of an L2(M, X) map f as follows. Let B(x) be the
sphere of center x and radius and σx, the induced measure on B(x). Let M = {x ∈
M : d(x, ∂M ) > }. Define e : M → R by e(x) = ´ y∈B(x) d2(f (x),f (y)) 2 dσx, n−1, for x ∈ M 0 for x ∈ M \M,
Denote by Cc∞(M ) the set of compactly supported smooth functions on M . We define the family of functionals E: Cc∞(M ) → R by setting
E(f )(ϕ) =
ˆ
M
Definition 2.1.10. A map f in L2(M, X)has finite energy if E(f ) := sup ϕ∈Cc(X),0≤ϕ≤1 lim sup →0 E(f )(ϕ) < ∞.
and we define the energy of f as E(f).
Definition 2.1.11. A finite energy map is called harmonic if it is locally energy minimiz-ing.
It is well known (c.f. [KS]) that if f has finite energy, the measures e(x)dx converge
weakly to a measure which is absolutely continuous with respect to the Lebesgue measure. Therefore there exists a function e(x), which we call the energy density, so that e(x)dµ →
e(x)dµ. In analogy to the case of real valued functions, we write |∇f |2(x) in place of e(x). In particular,
E(f ) =
ˆ
M
|∇f |2dµg.
The subspace of L2(M, X) functions with finite energy is denoted by W1,2(M, X).
Theorem 2.1.12 ([KS], Theorem 1.6.1). Let 1 < p < ∞, {fk} ⊂ W1,p(Ω, X). Let fk→ f
in the Lp metric. Write ek for the energy density measure of f
k. Assume there exists
K < ∞ with each E(fk) ≤ K. Then f ∈ W1,p(Ω, X) and its energy-density measure
satisfies
de ≤ lim inf dek as measures.
In [KS], Korevaar and Schoen also developed a trace theory for W1,2(Ω, X), where Ω ⊂
M is a Riemannian domain with boundary. By trace we mean a map tr : W1,2(Ω, X) →
is the graph of a Lipschitz function above some (n − 1)-dimensional hyperplane. With this trace theory we can define the Dirichlet problem for more general spaces.
Theorem 2.1.13. ([KS], Theorem 1.12.2) Let (Ω, g) be a Lipschitz Riemannian domain and let 1 < p < ∞. Any u ∈ W1,p(Ω, X) has a well-defined trace map u (or tr(u)), with
tr(u) ∈ Lp(∂Ω, X). If the sequence {ui} ⊂ W1,p has uniformly bounded energies Eui, and
if {ui} converges in the Lp distance to a map u, then the trace functions of the ui converge
in Lp(∂Ω, X) to the trace of u. Two u, v ∈ W1,p(Ω, X) have the same trace if and only if
d(u, v) ∈ W1,p(Ω, R) has trace zero.
NPC spaces
As in the classical theory, the existence and regularity of harmonic maps has been proven under certain curvature conditions on the target. In this section we define the conditions relevant for our work.
Definition 2.1.14. Let (X, d) be a complete metric space. We say (X, d) is CAT(κ), or that (X, d) has curvature bounded from above by κ, if the following conditions are satisfied: (i) (X, d) is a length space. That is, for any two points P, Q ∈ X, the distance d(P, Q), which for simplicity we will also write as dP Q, is realized as the length of a rectifiable
curve γP Q connecting P to Q. We call such a curve a geodesic.
(ii) Let a = p
|κ|. Let P, Q, R ∈ X (assume dP Q + dQR+ dRP < √πκ for κ > 0) with
Qt defined to be the point on the geodesic γQR satisfying dQQt = tdQR and dQtR =
for κ = 0, and cos(adP Qt) ≥ sin((1 − t)adQR) sin(adQR) cos(adP Q) + sin(tadQR) sin(adQR) cos(adP R) for κ > 0.
If (X, d) does not necessarily satisfy the CAT(κ) conditions, but every point y ∈ X has a neighborhood that does, we say (X, d) is locally CAT(κ).
If a space has curvature bounded above by 0 we also refer to it as non-positively curved (NPC). The definition of CAT(κ) given above, at least the local version, models the notion of sectional curvatures bounded above by κ in a Riemannian manifold.
As a consequence of (ii), geodesics in an NPC space are unique, see for instance [KS]. It is also a well-known fact that if X is locally compact and NPC, then X is simply connected. Conversely, if X is a complete simply connected Riemannian manifold with non-positive sectional curvature, then X is an NPC space with the length metric induced from the Riemannian metric.
Examples 2.1.15. 1. Simply connected Riemannian manifolds of sectional curvature ≤ κ are CAT(κ) spaces.
2. Any CAT(κ) space is CAT(κ0) for κ0 ≥ κ.
3. If X is a CAT(0) and (M, g) is a finite volume Riemannian manifold, L2(M, X) is also CAT(0).
4. R-trees are CAT(κ) for all κ ∈ R. 5. Euclidean buildings are CAT(0) spaces.
Proposition 2.1.16. ([KS], Corollary 2.1.3, part 2.1(iv)) Let (X, d) be an NPC space. Abbreviate the distance d(T, U) by dT U. Let {P, Q, R, S} ⊂ X be an ordered sequence. Let
0 ≤ t ≤ 1 be given and define Pt, Qt to be the point which is the fraction t of the way from
P to S and Q to R respectively (that is on the geodesic joining P and S, and Q and R). Then, for any 0 ≤ α ≤ 1, the following estimate holds
d2(Pt, Qt) ≤ (1 − t)d2P Q+ td2RS− t(1 − t)(α(dSP − dQR)2+ (1 − α)(dRS− dP Q)2).
Existence and Regularity of Harmonic Maps to NPC Spaces
The existence of a solution of the Dirichlet problem from Lipschitz Riemannian domains to NPC spaces, along with some regularity of the solution was proven in [KS].
Theorem 2.1.17. ([KS], Theorem 2.2, Theorem 2.4.6) Let (Ω, g) be a Lipschitz Rieman-nian domain and let (X, d) be a NPC metric space. Let φ ∈ W1,2(Ω, X). Define
Wφ1,2= {u ∈ W1,2(Ω, X)|tr(u) = tr(φ)}.
Then there exists a unique u ∈ W1,2
φ which is stationary for the p = 2 Sobolev energy. In
fact, the energy Eu =´ |∇u|2dµ of u satisfies
Eu= E0≡ inf
v∈Wφ1,2
Ev.
Moreover u is a locally Lipschitz continuous function in the interior of Ω, where the local Lipschitz constant is bounded above by
C E0
min(1, dg(x, ∂Ω)n) !1/2
,
The proof of this result follows the same guidelines as the proof of existence and unique-ness of the Dirichlet’s variational principle in the classical case where X = R. The par-allelogram inequality in Proposition 2.1.16 and the trace theory in Theorem 2.1.13 are instrumental to generalize the proof.
2.2
Riemannian Simplicial Complexes
In this section we define the spaces we will focus on in later chapters. We center our attention to the most relevant properties for our work but we invite the interested reader to check [CP] and [DM1] for more details.
Definition 2.2.1. A 2-dimensional simplicial complex is a topological space X together with two finite or infinite sets C and F that satisfy the following properties:
1. Each T ∈ C is a topological triangle. That is, a topological space homeomorphic to a closed 2-dimensional disc with three distinguished points on the boundary called vertices. The edges of T are the closed segments of ∂T bounded by two vertices and not containing the third.
2. F is a maximal collection of homeomorphisms f : A → B where A ⊂ T and B ⊂ T0
are distinct edges of (possibly identical) triangles T, T0 ∈ C (along with the identity
map on each edge of each triangle). For two edges A, B, there should be at most one map f : A → B in F. The collection F is maximal with respect to two conditions. First, if f : A → B is in F, then so is f−1 : B → A. Second, if f : A → B and
g : B → C are in F, then so is g ◦ f : A → C. The elements of F are called gluing maps.
3. As a topological space, X is the quotient of the disjoint union`
CT of the triangles in Cby the equivalence relation identifying x ∈ A with f(x) ∈ B for each f : A → B ∈ F.
Let π :`
CT → X be the quotient map. The fact that each f ∈ F is a homeomorphism
on each edge of each triangle in C. The image π(T ) of a triangle T ∈ C is called a face of X, the image π(e) of an edge e of T is called an edge of X, and the image π(v) of a vertex v of T is called a vertex of X.
We will also always impose an orientation on each edge of X. Pulling back by π, this puts an orientation on each edge e of each triangle T ∈ C in such a way that the gluing maps f ∈ F are orientation-preserving homeomorphisms.
4. X is path connected.
5. X is locally finite. That is, each edge and each vertex is incident to finitely many faces.
We list now some other notions regarding simplicial complexes.
Remark 2.2.2. A simplicial complex is said to be(locally) 1-chainable if the complement
of the vertices, X\S, is (locally) connected. By the construction of our complexes, since faces are glued to other faces only along edges (and not at vertices alone), our complexes will satisfy this condition as long as they are connected.
Definition 2.2.3. A 2-dimensional simplicial complex is finite if the number of faces is finite.
We will always assume that our complexes are finite.
Definition 2.2.4. Let X be a 2-dimensional simplicial complex. An edge e of X is called singular if e is incident to at least three faces. That is, the preimage of e by the quotient map π consists of three or more distinct edges.
Figure 2.1: 2-dimensional simplicial complexes with (right) and without (left) singular edges
Definition 2.2.5. Let X be a 2-dimensional simplicial complex with C = {Ti}i∈I its set of
triangles and S its set of vertices. We will say that σ is a Riemannian metric on X (resp. X\S) if
• the restriction of σ to each closed face σ|T¯ (resp. σ|T \S¯ ), is a Riemannian metric in
the usual sense, and
• the restriction of σ to each edge is well defined, i.e. if T1, . . . , Tn meet at an edge e
and σi = σ|T¯i denotes the various restrictions, then σi|e= σ1|e for i = 2, . . . , n.
Remark 2.2.6. Given a Riemannian metric σ on X (resp X\S), we induce a distance
function on X (resp. X\S) as follows. Given two points x and y, any piecewise C1 curve joining them has a length as measured by the Riemannian metric in each face of X. The distance between x and y is the infimum of the lengths of such curves.
Let dX denote the distance function on X. For p0 ∈ X the ball of radius r centered around a point p0 is denoted by Br(p0) = {q ∈ X : dX(p0, q) < r} and the union of simplices that contain p0 as st(p0). Also let ρ(p0) = sup{r : Br(p0) ⊂ st(p0)}.
The classical definition of harmonic maps and the Korevaar-Schoen generalization can be extended to maps whose domain is a Riemannian simplicial complex.
Definition 2.2.7. Let X be a 2-dimensional Riemannian simplicial complex, Y a metric space, and f a map from X to Y . The total energy of f : X → Y is the sum of the energy over all the faces of X, namely
A locally energy minimizing map is called harmonic.
2.2.1 Flat Simplicial Complexes
This section follows [DM1].
Definition 2.2.8. Let X be a 2-dimensional simplicial complex in the sense of Defini-tion 2.2.1 with Riemannian metric σ. Let σ|T be the restriction of metric on each face
T of X. We say X is flat if each triangle (T, σ|T) is isometric to (∆, ds20), where ∆ is
an equilateral triangle with vertices (1, 0), (−1, 0), (0,√3) and ds20= dx2+ dy2, and if two
adjacent faces T1, T2 share an edge e, the metrics σT1 and σT2 induce the same distance function on e.
Remark 2.2.9. In Chapter 3 we will generalize Euclidean metrics on 2-dimensional
sim-plicial complexes by allowing our faces to be isometric to triangles that are not necessarily equilateral.
For a flat 2-dimensional simplicial complex we can model Br(p0), r < ρ(p0) by the two cases:
(i) If p0is a face point, we isometrically identify Br(p0) with a disc Drof radius r centered at the origin of R2.
(ii) If p0 is an edge point, we isometrically identify Br(p0) with Xr defined as: take n
copies of the upper half disc D+r = {(x, y) ∈ R2|x2+ y2 < r2, y ≥ 0}, where n is the number of faces contained in st(p0). We can distinguish the copies by labeling them
Dr,i+, i = 1, ..., n and use (xi, yi) to denote the point corresponding to (x, y) ∈ D+r on
the ith copy Dr,i+. Let Xr = ∪ni=1D+r,i/ ∼, where ∼ is defined by (xi, 0) ∼ (xj, 0) for
all i, j and x ∈ [−r, r]. We refer to Xr as an edge piece.
Harmonic Maps from Flat Complexes
important regularity results here.
Theorem 2.2.10. ([DM1], Theorem 3.9) Let f : X1→ Y be a harmonic map where X1 is
an edge piece with N half discs labeled D+
1, · · · , D+N and Y is an NPC space. Let fj = f |Dj+
and fix a conformal structure on each D+
j so that the orientation induced on the x-axis is
the same for each j = 1, · · · N. With this conformal structure, define
(2.2) ϕj = ∂fj ∂x 2 − ∂fj ∂y 2 − 2i * ∂fj ∂x, ∂fj ∂y + . Then Im N X j=1 ϕj(x, 0) = 0 for all −1 < x < 1.
The proof of this result uses a standard variational computation for the energy and the fact that the Hopf differential, given by (2.2), is holomorphic.
Theorem 2.2.11. ([DM1], Theorem 3.10) If f : X1 → Y is a harmonic map into a NPC
space, then ∂f ∂x 2 ≤ 2 πr2E(f ) ∂f ∂y 2 ≤ 2N + 2 πr2 E(f )
at (¯xi, ¯yi) ∈ X1 where 2r is the distance of (¯xi, ¯yi) to ∂X1 and N is the number of faces of
X1.
This follows from Theorem 2.2.10 and a mean value inequality.
Using the former two theorems, one can show the main regularity result.
Theorem 2.2.12. ([DM1], Theorem 3.11) Let X and Y as above and S be the set of vertices of X. If f : X → Y is a harmonic map, then for every p ∈ X\S, there is a constant c independent of p so that
|∇f |2(p) ≤ c
δ2 ˆ
Bδ(p)
where δ = minv∈SdX(p, v)and f is locally Lipschitz continuous with the Lipschitz constant
at p ∈ X dependent only on Ef and δ.
For each point p in the interior of a face, one can construct an edge piece containing p at some bounded distance from its boundary of these ball. Using Theorem 2.2.11 on each of these edge pieces we get the desired inequality.
If the target space enjoys more regularity than just NPC, that is, Y is a complete Riemannian manifold of non-positive Riemannian curvature, the harmonic map can achieve better regularity.
Theorem 2.2.13. ([DM3], Theorem 3)Let f : X1 → Y be a harmonic map where X1 is an
edge piece of a 2-dimensional flat simplicial complex with N half discs labeled D+ 1, ..., D
+
N
and Y is a complete Riemannian manifold of non-positive Riemannian curvature of di-mension m. Let fj = f |
D+j. Let e denote the edge in X1 and e be the -neighborhood of e
containing all the points in X1 such that 0 ≤ y < . For any Lipschitz function η : X1 → R
with compact support and α = 1, ..., m,
lim →0 N X j=1 ˆ Tj∩∂e η(x, )∂f α j ∂y (x, )dx = 0.
The proof of this result is an application of the variational formula for harmonic maps.
Theorem 2.2.14. ([DM3], Theorem 4, Corollary 5)Using notations as in Theorem 2.2.13, for any point p ∈ X1, there exist a neighborhood Ω ⊂⊂ X1 of p so that fjα ∈ C1,β(Ω ∩ Tj),
fjα∈ W2,2(Ω ∩ T
j) and ∂fα
j
∂x ∈ W2,2(Ω ∩ Tj) for α = 1, ..., m, j = 1, ..., N.
Moreover, since f is C1,β near the edges, it follows that
N X j=1
∂fjα
∂y (x, 0) = 0 for all α = 1, ..., m and all (x, 0) ∈ X1∩ e.
disc in a clever way so that Theorem 2.2.13 guarantees that on the x-axis the map is continuous. The higher regularity then follows from standard elliptic theory.
Remark 2.2.15. The former results are actually more general than stated: they hold for
higher dimensional complexes and weighted measures. .
Corollary 2.2.16. Using notations as in Theorem 2.2.13, for every face T , the restriction of fα to the closure of T is C∞ away from the vertices.
In this case the result is only true for 2-dimensional simplicial complexes and follows from a bootstrapping argument.
2.2.2 Hyperbolic Simplicial Complexes
The ideal hyperbolic metrics we describe here are based on those defined in [CP].
Consider the space X\S, and endow each face with the metric of the ideal hyperbolic triangle. To do so we take the upper half-plane model of H2, that is H2= {z ∈ C|Im(z) > 0} = {(x, y) ∈ R2|y > 0} with the Poincaré metric
ds2 = dx 2+ dy2
y2 .
An ideal hyperbolic triangle is the convex hull of three points at the boundary of the hyperbolic plane H2. Since all ideal hyperbolic triangles are isometric, we can take ˜T to
be the convex hull of 0, 1 and ∞. We will denote the hyperbolic metric induced on T by inclusion also by ds2.
For each triangle T ∈ C, choose a diffeomorphism φ : T \S → ˜T . Use this φ to give
coordinates on T , as well as the metric φ∗ds2. This turns T \S into an isometric copy of the ideal hyperbolic triangle. Henceforth any isometry φ : T \S → ˜T can be used to give
Now each edge e of each triangle T is isometric to the real line R. If there is a gluing map f : e → e0, we require that this map be an (orientation preserving) isometry. The collection of isometries {φ : T → ˜T | T ∈ C} together with the isometric gluing maps
induces a Riemannian metric in each face of X\S, which in turn induces a distance metric on X\S in the usual way, via measuring lengths of paths using the Riemannian metrics in faces (see Proposition 1.2 of [CP] for more details).
Definition 2.2.17. An ideal hyperbolic structure or ideal hyperbolic metric σ on X\S is the data of a collection of isometries {φ : T → ˜T | T ∈ C}along with isometric gluing maps f ∈ F, or equivalently the distance metric on X\S that they induce.
Shift parameters
We study in more detail the gluing maps here. For the ideal hyperbolic triangle ˜T , there
is a distinguished point, called the center of the triangle. It can be obtained as the unique fixed point of the full isometry group Isom( ˜T ) ∼= S3, or as the intersection of the three altitudes of the triangle (see Figure 2.2). For the triangle ˜T , the center is at 1+
√ 3i 2 . There are also distinguished points on each edge of ˜T . These points are the feet of the
altitudes that define the center. For the triangle ˜T these points are i, 1 + i, and 1+i2 .
0 1
P2
P1
P3
P
Figure 2.2: Distinguished points.
Put an orientation on the edges of X, which in turn gives an orientation on the edges of ˜
isometries e → R that map distinguished points to 0
Let ˜T , ˜T0 be two ideal hyperbolic triangles and e, e0 be two edges with distinguished points P, P0respectively and let f : e → e0be the gluing map. f will be uniquely determined by an orientation preserving isometry of R, that is a translation of the form ˜f (t) = t + α,
for some α ∈ R. Thus, each gluing map is uniquely determined by the real number α. We call such number the shift or shift parameter. Note that the shift parameter is the algebraic measure of the point f (P ) with respect to the coordinate system of e0. For
e, e0 = {(x, y) ∈ H2 : x = 0}, we map e and e0 isometrically to R via the logarithm of the imaginary part of the points in e and e0. Since the distinguished points lie on the origin of the corresponding coordinate systems, it is clear that α is the image of the origin of the coordinate system corresponding to e in the coordinate system of e0. This construction gives f (iy) = ieαy.
Proposition 2.2.18. Let X be a 2-dimensional simplicial complex and let σ be a hyperbolic structure on X\S. Let T1, T2, T3 be three faces sharing an edge e in X, and let ei =
π−1(e) ∩ Ti be the edge of Ti corresponding to e for i = 1, 2, 3. Let fk,j : ej → ek ∈ F be
the gluing maps for these three edges and let αk,j be the shift parameter of fk,j. Then
α1,2 = −α2,1, α1,2+ α2,3+ α3,1 = 0,
The proof of both of these statements follow from expanding the definition of shift parameters and properties of the logarithm.
Completeness
If X has no singular edges, then X\S is a surface with punctures. It is well known that a hyperbolic metric on a finite type Riemann surface with punctures is complete if and only if the metric at each puncture is that of a cusp. Thus, it seems natural that a metric σ on
X will be complete if and only if all the vertices are cusps. In fact this is the content of
Figure 2.3: 2-dimensional simplicial complexes with complete (left) and incomplete (right) metrics
A hyperbolic cusp can be constructed as the quotient of {z ∈ H2 | Im(z) > 1} by the group of translations generated by z 7→ z + 1. The horizontal curves t 7→ z0+ t are horocycles centered at ∞, and after quotienting by translation they become closed curves around the cusp. Thus a hyperbolic puncture is a cusp if and only if it has a neighborhood foliated by closed horocycles. For more details see [T], for instance Proposition 3.4.18.
In order to describe this condition for completeness on the complex X, we first need to define the link of a vertex. The link of a vertex v of X, denoted lk(v), is a graph whose vertices correspond to edges of X incident to v, and whose edges correspond to faces of
X incident to v. Two vertices of lk(v) are joined by an edge if the corresponding edges of X cobound the corresponding face. It can also be seen that a neighborhood of v in X is
homeomorphic to the cone on lk(v), the set (lk(v) × [0, 1])/(lk(v) × {0}). See for instance Figure 2.4.
Figure 2.4: Link of a vertex
To each vertex z in lk(v) we may associate a sign z = ±1, where z = 1 if the edge
e of X corresponding to z is oriented towards v, and z = −1 if e is oriented away from
v. It is possible that an edge of X will have v at each end. In this case there will be two
Let L be a simple closed curve in the graph lk(v) and let us put an orientation on it. The curve L visits a sequence of vertices and edges in lk(v). Say this sequence is (z1, f1, z2, f2, . . . , z`, f`, z`+1 = z1), where each zj is a vertex in lk(v) corresponding to an
edge ej in X, fj is an edge in lk(v) corresponding to a face Tj in X, and the endpoints of
fj are zj and zj+1. We develop the faces Tj into H2 as follows.
Let φ1 : T1 → ˜T be an isometry so that φ1(e1) is the geodesic joining 0 and ∞ and
φ1(e2) is the geodesic joining 1 and ∞. Inductively choose φj : Tj → rjT , where r˜ jT denotes˜
the ideal hyperbolic triangle with vertices 0, rj, ∞ and so that φj|ej = φj−1|ej − rj−1 and
φj(ej+1) is to the right of φj(ej), for j = 2, . . . , `. Here the scaling factor rj = e−jαj+1,j,
where αj+1,j is the shift parameter associated to the gluing map that glues the triangle Tj to the triangle Tj+1 along the edge ej+1. That is, the triangles scale according to the shift parameter. The completeness of the metric restricted to the faces T1, . . . , T` is equivalent
to the horocycles about ∞ in the developed picture being closed, which in turn means that
φ1(e1) must be glued to φ`(e`+1) by a pure real (horizontal) translation. See Figure 2.5.
˜
T r2T˜ r3T˜ r4T˜ Figure 2.5: Horocycles in a complete metric
This property is captured by the equation
` X j=1
For more details and a slightly different perspective, see [CP].
Definition 2.2.19. Let X be a finite 2-dimensional simplicial complex. The Teichmüller space of X, denoted T (X), is the space of all complete ideal hyperbolic metrics on X\S.
With this characterization of the complete ideal hyperbolic metric one can compute the number of independent parameters which determine the dimension of the Teichmüller space.
Theorem 2.2.20 ([CP], Theorem 3.3). Let X be a finite 2-dimensional simplicial complex, having n vertices v1, ..., vn. Let the rank of π1(lk(vi)) be ri, where lk(vi) is the link of vi
for i = 1, ..., n. Let N be the number of independent gluing maps of X. Then,
dim T (X) = N −
n X i=1
ri.
2.3
Metrics with Conic Singularities
This section introduces metrics on surfaces with conical singularities and discusses the ex-istence of harmonic maps from a Riemannian surface to a surface with conical singularities.
2.3.1 Flat Conical Metrics
Here we define flat metrics with conical singularities on surfaces and study the existence of energy minimizing maps to surfaces with these metrics. The content of this section can be found in Section 1 of [K].
Definition 2.3.1. A flat metric with conical singularities on a surface S is a flat metric on S except at a finite set C of singularities such that near each q ∈ C the surface looks like the cone over a curve of length 2πα, where α = α(q) > 1.
Take local coordinates near a singular point as follows. Let γ : S1 → S2 be a constant speed parametrization of a simple closed curve of length 2πα with α ≥ 1, and let C ⊂ R3 be the cone generated by γ with the origin as vertex. An intrinsic model for C is obtained from the isometric embedding J : (C, ds2) → C ⊂ R3, where
(2.3) J (w) = |w|αγ
w
|w|
.
We can pullback the metric to a metric ds2 on C with
ds2 = α2|w|2(α−1)|dw|2.
Any branch of wα provides a local isometry of ds2 with the standard Euclidean plane. A geodesic segment joining w1, w2 ∈ C is the preimage of a line segment under wα, if the wi
lie in a common sector with angle less than π/α, or the union of the ray segments joining wi
to 0 otherwise. Applying formally the Gauss-Bonnet theorem yields KdA = 2π(1 − α)δ0, where δ0 is the Dirac measure at 0 ∈ C.
An alternative definition of flat conical metrics is therefore:
Definition 2.3.2 ([K], Definition 1). ds2 is a flat metric with conical singularities on S
if for any q ∈ S there is an α = α(q) ≥ 1 and a local parameter w such that ds2 =
α2|w|2(α−1)|dw|2 near q. The finite set of concave vertices is C = {q ∈ S : α(q) > 1}.
Finite energy maps to surfaces with flat conical metrics
We define what it means to have finite energy with this new metrics and discuss the existence of an energy minimizing map.
Let M be a compact Riemann surface with local coordinates z = x + iy and let N be a surface with a flat conical metric.
Definition 2.3.3. For maps f ∈ C0(M, N )with the property that for any open set Ω ⊂ M
the energy of f as
E(f ) =
ˆ
M
2α2|w|2(α−1)(|wz|2+ |wz¯|2)dxdy.
Remark 2.3.4. Note the definition of energy agrees with the one in Section 2.1 in complex
coordinates and allowing the metric to have singularities.
In what follows, by a minimizing or energy minimizing map we mean locally energy minimizing.
We collect here some results that inspired Chapter 6.
Lemma 2.3.5 ([K], Lemma 1). Let ds2 be a flat conical metric as in Definition 2.3.2. Then there is a family of nondegenerate metrics ds2
, > 0, with nonpositive curvature
such that ds2
≥ ds2τ for ≥ τ > 0 and ds2 converges in C0 on N and locally in C∞ on
N \C to ds2 as → 0.
Proposition 2.3.6 ([K], Proposition 1). There exists a minimizing map f in each homo-topy class which is Lipschitz continuous with respect to the metric ds2.
2.3.2 General definition of Conical metrics
We define metrics with conic-type singularities as in Section 1 and 4 of [Tr]. The main difference with the previous section is that in this case we do not require the metric to be flat outside of the singular points.
Definition 2.3.7. A conformal (singular) Riemannian metric on a closed surface S is defined by the local expression
ds2= ρi(zi) |dzi|2,
where zi is a local coordinate on S and ρi is a positive measurable function.
a neighborhood Ui of pi, then there exists a continuous function u : Ui → R, which is of class C2 on Ui\{pi}, and such that in Ui
ds2 = e2u|zi|2βi|dzi|2,
where zi(pi) = 0 and βi> −1 for all i.
The point pi is said to be a conical singularity of angle θi = 2π(βi+ 1) and ds2 is said
to have a singularity of order βi at pi.
Remark 2.3.8. If C is equipped with the metric |z|2β|dz|2, it is isometric to an Euclidean cone of total angle θ = 2π(β + 1). Thus, the definition provided here agrees with the one provided in the previous section.
If ds20is a conformal metric with conical singularities inC and ds21is a smooth conformal metric, by definition there exists a function ρ : S → R such that
ds2= ρds21.
This function is smooth and positive outside of C . If z is a coordinate in a neighborhood of pi (such that z(pi) = 0), then in this neighborhood,
ρ(z) = O(|z|2βi),
for z near 0.
2.4
Additional results
In this section, we collect a variety of topological and analytical results.
Lemma 2.4.1. ([J1], Lemma 3.1.1) Let D be the unit disc in the plane, x0 ∈ D, N a
Riemannian manifold with distance function d, and let δ < 1. Let f be a finite energy map with E(f) ≤ K and Br(x0) denote the ball centered at x0 and radius r. Suppose
∂Br(x0) ∩ D is connected for all r ∈ (δ, √
δ). Then there exists some r ∈ (δ,√δ)for which f |∂Br(x0)∩D is absolutely continuous and
d(f (x1), f (x2)) ≤ 8πK log 1/δ 1/2 , for all x1, x2 ∈ ∂Br(x0) ∩ D.
The next theorem will be instrumental to prove existence of finite energy diffeomor-phisms.
Theorem 2.4.2. (Approximation by diffeomorphisms). ([IKO], Theorem 1.2) Let
Ω, Ω0 be bounded domains in R2 and f : Ω → Ω0 be a homeomorphism of Sobolev class
W1,2(Ω, Ω0). Then for every > 0, there exist a diffeomorphism F : Ω → Ω0 such that
1. d(F, f) ≤ , where d is the distance function in Ω0,
2. E(F ) ≤ E(f).
The following result is a compactness theorem for energy minimizing maps.
Theorem 2.4.3 ([Me], Theorem 13). Let (M, g) be a compact Riemannian manifold with-out a boundary and let {di} be distance functions on X with curvature bounded from above
by κ. Assume X is compact with respect to the metric topology induced by d. Let h : M → X be a continuous map and let fi: M → (X, di) be energy minimizing maps in the homotopy
class of h with fi = h on ∂M if ∂M 6= ∅. Let δi be the pull back distance function of di
under fi, i.e.
δi(·, ·) = di(fi(·), fi(·)).
If the energy of fi is bounded above by K for each i and if di converges uniformly to a
map f0 with respect to d0 so that δi0(·, ·)converges uniformly to d0(f (·), f (·))and the energy of fi0 converges to that of f0.
2.4.1 Topological results
We collect some topological results from Section 13.2 of [DFN].
Let M, N be closed, oriented manifolds of the same dimension n. Let f : M → N be a smooth map, and let y0 ∈ N be a regular value of f , i.e. the complete inverse image of y0 consists of finitely many points x1, ..., xm and if xβi are local coordinates in a neighborhood
of xi, and y0αare local coordinates in a neighborhood of y0, then the Jacobian det(∂y0α/∂xβi) is non-zero at xi for each i = 1, ..., m.
Definition 2.4.4. The degree of a smooth map f : M → N of connected, oriented, closed manifolds, with respect to a regular value y0∈ N is
deg f = X f (xi)=y0 sgn det ∂y α 0 ∂xβi !
It is well known that this definition is independent of the choice of the interior regular value y0.
One can generalize the concept of degree to manifolds with boundary. Let M, N be two manifolds with boundary of the same dimension n. Let f : M → N map the boundary
∂M to the boundary ∂N and the inverse image of an interior point of N to the interior
of M . Note that any map in the relative homotopy class of f will also have this property. One can define the degree of f as in Definition 2.4.4 with respect to a regular value in the interior of N . This definition is also independent of the choice of regular value and is invariant under relative homotopies of f for compact, connected, oriented manifolds M and N .
Theorem 2.4.5 ([DFN], Theorem 13.2.1). The degree of the boundary map f|∂M : ∂M →
∂N is the same as the degree of f; i.e. deg f|∂M = deg f.
Moreover, one can further generalize the definition of degree to manifolds that are not compact in the same way as in Definition 2.4.4 for interior regular points. This definition is also independent of the choice of regular value:
Theorem 2.4.6 ([DFN], Section 13.2 (ii)). The degree of a proper map f : M → N between closed, orientable Riemannian manifolds is independent of the regular value. Moreover, the degree is invariant under proper homotopies relative to the boundary.
2.4.2 More on complexes
As in the classical theory of Sobolev Spaces, we have a Poincaré inequality for complexes.
Theorem 2.4.7. (Poincaré inequality for complexes). ([DM2], Theorem 2.6) Let K be a bounded connected compact subset of (X, σ) and f : X → R a finite energy map with compact support in K. Then, there exists a constant C only depending on K and X such
that ˆ K f2dµσ ≤ C ˆ K |∇f |2dµσ.
Since complexes can be endowed with distance functions, one can study the curvature properties of the complex, that is, when a complex is CAT(κ).
Suppose X1, X2 are locally compact complete length spaces, A1 ⊂ X1, A2 ⊂ X2 are closed convex subsets, and i : A1→ A2 is an isometry. Then we can glue X1 and X2 along
A1 and A2 to obtain a space X1∪iX2 = Y . The structures on X1 and X2 give rise to a length structure on Y and Y is then a locally compact complete length space.
Corollary 2.4.8. ([GH], Chapter 10, Corollary 5)If furthermore X1, X2 are 2-dimensional
simplicial complexes and all triangles in X1 and X2 of perimeter less than 2π/ √
κ are CAT(κ) spaces, then the same is true in Y .
Lemma 2.4.9. ([GH], Chapter 10, Lemma 9) If c0, c1 : [a, b] → B(q) are geodesic arcs,
then |c0(t) − c1(t)| is convex in t. In particular, for q0 and q1 in B(q), there is exactly one
geodesic arc in X from q0 to q1 and this arc depends continuously on q0 and q1. Using these two results [CP] shows
Proposition 2.4.10. ([CP], Proposition 1.4) Let X be an ideal 2-dimensional simplicial complex, with S its set of vertices. Then X\S is a local CAT(-1)-space.
Before we state the next theorem, we need to introduce some notation. Let Dp be the set of directions at p, that is, the set of non-constant geodesic segments c : [0, ] → X such that c(0) = p modulo the relation c ∼ ¯c if ¯c(t) = c(t) for all small t.
Theorem 2.4.11. ([GH], Chapter 10, Theorem 15) If X is a 2-dimensional simplicial complex with a Riemannian metric such that the top dimensional simplices have curvature χ, then X has curvature KX ≤ χ if for all p ∈ X the CAT(-1)-inequality holds for all
triangles in Dp of perimeter < 2π.
2.4.3 Regularity of Harmonic Maps through Partial Differential Equa-tions
In this section we collect some regularity results for maps satisfying certain partial differ-ential equations on different domains.
Definition 2.4.12. ([M1], Definition 3.1) A Riemannian manifold M of dimension n of class C1 is said to be homogeneously regular if there exist numbers λ, Λ, independent of
x0, with 0 < λ ≤ Λ, such that any point x0 in M lies in an open set in M which can be
mapped on the unit hypercube R = {(x1, ..., xn) : |xi| < 1}by a map of class C1 such that
x0 corresponds to the origin and the gi,j(x) satisfy the condition that
λ N X i=1 (ξi)2≤ gα,β(x)ξαξβ ≤ Λ N X i=1 (ξi)2
Examples 2.4.13. A great number of spaces enjoy this property. For instance,
1. Any compact Riemannian manifold. 2. The Euclidean n-space
3. The hyperbolic n-space.
Harmonic maps on such manifolds enjoy some extra regularity.
Theorem 2.4.14. ([M1], Theorem 3.2) Let M be a homogeneously regular Riemannian manifold of class C3. Then the harmonic functions are of class C2,γ on the interior of the
regions of definition for any γ, 0 < γ < 1. Elliptic Partial Differential Equations
Here we provide a short survey on the theory of elliptic partial differential equations we will be interested in. All the concepts and results can be found in [M2].
Let G be a bounded domain with sufficiently smooth non empty boundary and let
Ljk(x, D)uk(x) = fj(x), j = 1, ..., N, x ∈ G,
(2.4)
be a linear system of partial differential equations subject to boundary conditions of the form
Brk(x, D)uk(x) = gr(x), r = 1, ..., m, x ∈ ∂G,
(2.5)
for some m that will be specified below. Assume integer weights are given to the system:
s1, ..., sN associated to the equations in the interior, t1, ..., tN associated to the functions
u1, ..., uN, and h1, ..., hm associated to the boundary equations. These weights must be
Let L0jk(x, Ξ), Brk0 (x, Ξ) be the principal part of the operator Ljk and Brk respectively, that is, the terms of order exactly sj + tk and tk − hr respectively, and L(x, Ξ) be the determinant of the matrix whose components are L0jk(x, Ξ).
Definition 2.4.15. The system (2.4) is elliptic if and only if L(x, Ξ) 6= 0 for any real non-zero Ξ.
Note that L is a homogeneous polynomial of degree
P =
N X j=1
(sj+ tj).
Definition 2.4.16. The system (2.4) is properly elliptic if and only if P is even, say P = 2m and, for each pair Ξ, Ξ0 of linearly independent vectors the equation
L(x, Ξ + τ Ξ0) = 0
has m roots with positive imaginary part and m roots with negative imaginary part.
Let x0 ∈ ∂G, n be the unit normal at x0 and ζ any real vector tangent to ∂G at x0. Let τs+(x0, ζ), s = 1, ..., m, be the roots of L(x0, ζ + τ n) = 0 with positive imaginary part (which exist if the system is properly elliptic). Set
L+0(x0, ζ; τ ) =
m Y s=1
(τ − τs+(x0, ζ)),
And let Ljk(x0, ζ + τ n) be the components of the matrix adjoint to the matrix L0jk(x0, ζ +
τ n). Define Qrk(x0, ζ; τ ) = N X j=1 Brj0 (x0, ζ + τ n)Ljk(x0, ζ + τ n).
Definition 2.4.17. For any x0 ∈ ∂G and any real vector ζ tangent to ∂G at x0, let us
regard L+
0(x0, ζ; τ ) and the elements of the matrix Qrk(x0, ζ; τ ) as polynomials in τ. The
the system (2.4)) if and only if the rows of Q are linearly independent modulo L+ 0(x0, ζ; τ ). That is m X r=1 CrQrk(x0, ζ; τ ) ≡ 0 (mod L+0)
only if the Cr are all 0.
We extend the above definitions to a nonlinear system of differential equations of the form
ϕk(x, Du) = 0, k = 1, ..., N, x ∈ G,
(2.6)
with boundary values
χr(x, Du) = 0, r = 1, ..., m, x ∈ ∂G,
in which the ϕk and χr are analytic functions of their arguments.
Definition 2.4.18. The system (2.6) is elliptic along the solution u if the linear equations of variation Ljk(x, D)vk := d dλ λ=0ϕj (x, Du + λDv) = 0,
form a linear elliptic system. We define properly elliptic and the complementing condition in the same way.
Theorem 2.4.19. ([M2], Theorem 6.8.1) Suppose that uk ∈ Ctk(G) and satisfies the
equations
(2.7) ϕj(x, Du) = 0, j = 1, ..., N,
on G, where ϕj(x, p) ∈ Ch−sj(Ω), h ≥ 1, and Ω is an open set in (x, p)-space containing all
satisfy the corresponding differential equations (almost everywhere if |δ| = h). If, also, the ϕj ∈ C
h−sj
µ (Ω) then the uk ∈ Cµtk+h(D) for each D ⊂⊂ G. If the ϕj are of class C∞
(analytic) on Ω, then the uk are of class C∞ (analytic) on G.
Theorem 2.4.20. ([M2], Theorem 6.8.2)Suppose that h0 is the smallest integer satisfying
the conditions
h0 ≥ 0, h0+ hr≥ 1, for each r.
Suppose uk∈ Ctk+h0(G)and the uk satisfy 2.7 on G and
χr(x, Du) = 0, on ∂G.
Suppose for some h ≥ h0that G is bounded and of class Ct0+h, t0= max tk, ϕj ∈ Ch−sj(Ω),
and χr ∈ Ch+hr(Ω0), where Ω and Ω0 are appropriate neighborhoods in the (x, p)-space
con-taining all the points (x, Du(x)) for x ∈ G. We assume that the linearized equations are properly elliptic and the linearized boundary conditions satisfy the corresponding comple-menting conditions on ∂G. Then uk ∈ Hptk+h(G) for each p > 1 and the Dδuk satisfy
the differential equations on G. If also, the ϕj ∈ C h−sj
µ (Ω) and χr ∈ Cµh+hr(Ω), then
uk ∈ Cµtk+h(G). Corresponding results hold in the C∞ and analytic cases. If h0 ≥ 1, G
is of class Ct0+h0
µ , uk ∈ Cµtk+h0−1(G), ϕj ∈ Ch0
−sj
µ (Ω), and the χr ∈ Cµh0+hr(Ω), then
