We will also consider metrics conformal to those described above.
Definition 3.3.1. Let σ be either a Euclidean metric on X or an ideal hyperbolic metric on X\S, and let ρ : X → R>0 be a continuous function that is C3 on the closure of each face of X. We define the metric ρ2σ on each face by taking the Riemannian metric ρ2σ, and by the continuity of ρ these metrics are compatible on the edges of X. We also impose the following constraints on ρ2σ:
1. The volume of X is fixed. That is, if dµσ denotes the volume form of the metric σ, then
X
T ∈C
ˆ
T
ρ2dµσ = X
T ∈C
ˆ
T
dµσ.
2. For every face T and every edge e incident to T , ∇(ρ|T)|e is perpendicular to e.
3. If Kσ is the curvature of the metric σ, then on each face
∆ρ ≥ Kσρ + |∇ρ|2 ρ .
4. Each face is convex with respect to the metric ρ2σ. 5. The complex is complete with respect to the metric ρ2σ.
If we started with a Euclidean metric σ on X, the last condition of completeness is automatically satisfied. If we started with an ideal hyperbolic metric σ on X\S, then it is possible that ρ may blow up at the vertices of X, so long as it does not blow up fast enough to incur infinite area. We now investigate the second and third properties.
Proposition 3.3.2. Given a metric σ, let ρ be a conformal factor as in Definition 3.3.1.
For each face T and each edge e incident to T , e is a geodesic with respect to the metric ρ2σ|T in T .
Proof. For a fixed triangle T ∈ C, choose an isometric embedding φ of T into either R2 if σ is Euclidean or H2 if σ is hyperbolic. By postcomposing with an isometry we may assume that φ takes one of the edges e of T onto the y-axis (using the upper half-space model for H2). We use φ to push forward the metric ρ2σ from T onto its image.
Let γ(t) = (0, f (t)) be a parametrization of φ(e). In order for γ to be a geodesic with respect to the metric ρ2σ, we must have the geodesic equations, the first of which is
0 =¨γ1+ρΓ1ij(γ) ˙γi˙γj
=ρΓ122(0, f (t))(f0(t))2
Here we have used ρΓ to denote the Christoffel symbols of the metric ρ2σ. If Γ represents the Christoffel symbols of σ, then the conformal symbols satisfy
ρΓkij = Γkij +1 ρ
δik∂jρ + δkj∂iρ − σijσkmδij∂kρ.
For the Euclidean metric all Christoffel symbols vanish, and for the Poincaré metric in the upper half-plane model, we have Γ122 = 0. In both cases the metric σ is diagonal. Hence the first of the geodesic equations for γ reduces to
0 = σ22σ11∂xρ.
From condition 2 of Definition 3.3.1, the normal derivative of ρ along e vanishes. In other words, ∂xρ = 0. Thus the first of the geodesic equations is automatically satisfied.
The second of the geodesic equations reads
f00(t) +ρΓ222(0, f (t))(f0(t))2 = 0.
Once the metric σ and the conformal factor ρ are fixed, this is an ordinary differential equation for the function f , which can be solved by classical theory. Therefore some parametrization of the edge e is a geodesic with respect to the metric ρ2σ in the face T .
Proposition 3.3.3. Given a metric σ, let ρ be a conformal factor as in Definition 3.3.1.
The curvature of the metric ρ2σ in each face T is bounded above by 0.
Proof. Fix a face T of X. Let K denote the Gaussian curvature of the metric σ in this metric and Kρthe Gaussian curvature of the metric ρ2σ. Let ∆ denote the Laplacian with respect to the metric σ, which is given by the trace of the Hessian. Then the Gaussian
curvatures satisfy
Kρ = 1 ρ2 h
K − ∆ log ρi
= 1
ρ2
"
K − ∆ρ
ρ +|∇ρ|2 ρ2
#
By the condition 3 of Definition 3.3.1, this curvature quantity is non-positive.
Corollary 3.3.4. The conformal metrics ρ2σ, when ρ is as in Definition 3.3.1, define locally CAT(0) metrics on X\S.
Proof. We have just shown that the metrics on each face are non-positively curved. For a point p on an edge, a ball about p that does not reach as far as any other edge or vertex is a union of NPC half-discs glued along their common convex (geodesic) boundary. By Corollary 2.4.8 and Lemma 2.4.9, such a set is NPC. Hence every point in X\S has a CAT(0) neighborhood.
Remark 3.3.5. Just as in the case of Euclidean metrics, conformal metrics may fail to be CAT(0) at the vertices of the complex.
3.4 Singular metrics
We will want to allow specific types of degenerations of our metrics. The particular degen-erations we will consider are cone-type singularities of non-positive curvature, which we define in analogy to Section 2.3.
Definition 3.4.1. Let X be a finite 2-dimensional simplicial complex and let C = ∪∞n=1{pn} be a discrete set of points lying in the interior of the faces of X. A conformal metric ρ2σ, where ρ may fail to be positive at C , is called a cone metric if it satisfies all five properties from Definition 3.3.1, and in addition at each point q ∈ C there is an α = α(q) > 1 and a positive C3 function φ so that in polar coordinates centered at q the conformal factor ρ
has the form
ρ(x, y) = rα−1φ.
Propositions 3.3.2 and 3.3.3 still apply. For the curvature near a point q ∈ C , the function φ from Definition 3.4.1 satisfying ρ = rα−1φ must satisfy the same inequality as ρ, namely
∆φ ≥ Kφ + |∇φ|2 φ .
Moreover one can see, as in the discussion in Section 2.3.1, that the curvature picks up a Dirac measure at q of mass 2π(1 − α) < 0.
Finally we show that metrics with cone-type singularities are well approximated by the smooth conformal metrics of Definition 3.3.1.
Lemma 3.4.2. Let ρ2σ be a metric with cone-type singularities as in Definition 3.4.1.
There is a family {ρ}>0 of conformal factors satisfying Definition 3.3.1 so that ρ ≥ ρτ for ≥ τ > 0 and so that ρ2σ converges to ρ2σ on the closure of each face as tends to 0.
This convergence is C0 on X and locally in C3 on T \C for each face T . Proof. This result is analogous to Lemma 2.3.5.
By taking an appropriate mollification of ρ near the points ofC we find our family ρ
with the desired property. Concretely, fix a positive test function η supported near the origin in R2. In coordinates near q ∈C , ρ(x, y) = rα−1φ(x, y), so near q define
ρ =3η(r/) + rα−1φ.
The functions ρ are smooth over C , and if η is chosen appropriately and is sufficiently small then the metrics have non-positive curvature. Furthermore it is clear that they satisfy the convergence properties claimed in the statement of the Lemma.
Existence Theory for Conformal Targets
In this chapter we develop the theory of harmonic maps between 2-dimensional simplicial complexes endowed with the Riemannian metrics defined in Chapter 3. We begin by establishing the class of maps we will consider. We prove the existence of finite energy diffeomorphisms in this class which we use to show the existence of an energy minimizing map in the class of finite energy maps and in the class of finite energy diffeomorphisms on each face.