Regularity of Harmonic Maps through Partial Differential Equations 33

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, ..., x_n) : |x_i| < 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

(ξ_i)²≤ g_α,β(x)ξ_αξ_β ≤ Λ

N

X

i=1

(ξ_i)²

for all x = (x1, ..., xn) on R and all (ξ1, ..., ξn).

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 C^2,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

L_jk(x, D)u_k(x) = f_j(x), j = 1, ..., N, x ∈ G, (2.4)

be a linear system of partial differential equations subject to boundary conditions of the form

B_rk(x, D)u_k(x) = g_r(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, ..., s_N associated to the equations in the interior, t₁, ..., t_N associated to the functions u₁, ..., u_N, and h₁, ..., h_m associated to the boundary equations. These weights must be chosen so that the order of L_jk is less than s_j+ t_kand the order of B_rkis less than t_k− h_r. Moreover, we can assume max s_j = 0.

Let L⁰_jk(x, Ξ), B_rk⁰ (x, Ξ) be the principal part of the operator L_jk and B_rk respectively, that is, the terms of order exactly s_j + t_k and t_k − h_r respectively, and L(x, Ξ) be the determinant of the matrix whose components are L⁰_jk(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

(s_j+ t_j).

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 τ_s⁺(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 ) = system (2.5) of boundary operators satisfies the complementing condition (with respect to

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

CrQ_rk(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

ϕ_k(x, Du) = 0, k = 1, ..., N, x ∈ G, (2.6)

with boundary values

χ_r(x, Du) = 0, r = 1, ..., m, x ∈ ∂G,

in which the ϕ_k and χ_r 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)v^k := 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 ∈ C^tk(G) and satisfies the equations

(2.7) ϕj(x, Du) = 0, j = 1, ..., N,

on G, where ϕj(x, p) ∈ C^h−sj(Ω), h ≥ 1, and Ω is an open set in (x, p)-space containing all the points (x, Du); we assume that the equations of variation are properly elliptic. Then u_k ∈ H_p^tk+h(D) for each p > 1 and each D ⊂⊂ G an the derivatives D^δu_k, |δ| ≤ h,

satisfy the corresponding differential equations (almost everywhere if |δ| = h). If, also, the ϕ_j ∈ C_µ^h−sj(Ω) then the uk ∈ C_µ^tk+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+ h_r≥ 1, for each r.

Suppose uk∈ C^tk+h0(G)and the uk satisfy 2.7 on G and

χ_r(x, Du) = 0, on ∂G.

Suppose for some h ≥ h0that G is bounded and of class C^t0+h, t0= max t_k, ϕj ∈ C^h−sj(Ω), and χr ∈ C^h+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 ∈ H_p^tk+h(G) for each p > 1 and the D^δu_k satisfy the differential equations on G. If also, the ϕj ∈ C_µ^h−sj(Ω) and χr ∈ C_µ^h+hr(Ω), then u_k ∈ C_µ^tk+h(G). Corresponding results hold in the C^∞ and analytic cases. If h0 ≥ 1, G is of class Cµ^t0+h0, uk ∈ C_µ^tk+h0−1(G), ϕj ∈ C_µ^h0−sj(Ω), and the χr ∈ C_µ^h0+hr(Ω), then uk ∈ C_µ^tk+h0(G).

Metrics on Simplicial Complexes

In this chapter we describe the Riemannian metrics on a 2-dimensional simplicial complex we will be working with as defined in Section 2.2 along with their curvature properties.

The two main classes of metrics we are interested in are Euclidean metrics on the compact simplicial complexes and ideal hyperbolic metrics on the complex punctured at its vertices.

We will also consider metrics conformal to these two classes, which may or may not have conical singularities.

3.1 Euclidean metrics

To define Euclidean metrics on the compact simplicial complex X, we identify each triangle T ∈ C with a Euclidean triangle in such a way that we will induce a well-defined metric on each edge of X. Each edge of a Euclidean triangle is isometric to a segment in R, which is characterized up to isometry by its length. So whenever f : A → B ∈ F is a gluing map, f is an isometry. That is, whenever two triangles are glued together along an edge, the corresponding edges must have the same length.

So at the very least we must specify a length function ` : E[X] → R+ from the set

38

of edges of X to the positive real numbers. And from elementary Euclidean geometry, a Euclidean triangle is uniquely determined (up to isometry) by a triple (a, b, c) of edge lengths satisfying the triangle inequalities: a < b + c, b < c + a, and c < a + b. Once we have a length function satisfying these inequalities on each triangle, we can identify each triangle T ∈ C with a Euclidean triangle with the appropriate edge lengths, and use this identification to induce a metric on T . Moreover, these metrics on the triangles of C glue together and descend to a metric σ on X so that each face of X is isometric to a Euclidean triangle. In the special case where the length of all the edges is the same, our complex is endowed with a flat metric as defined in Section 2.2.1.

It will often be convenient to work in coordinates. Given a Euclidean metric on X, and a triangle T ∈ C, we can construct an isometry φ : ¯T → ∆ to some triangle ∆ ⊂ R². We use such a homeomorphism to induce coordinates in ¯T . If we have a chosen edge A of T , we can arrange that ∆ has one side along the y axis and the remaining vertex in the right half-plane. In this case we introduce the notation T_abc to denote the Euclidean triangle with vertices (0, 0), (0, a), and (b, c) in R².

Definition 3.1.1 (Local model of an edge). Let σ be a Euclidean metric on X. Let e be an edge of X and let Tj enumerate the faces of X incident to e. A system of isometries {φ_j : T_j → ∆_j} is called a local model for e if

1. φj(T_j) lies in the right half-plane for each j,

2. φj(e) is a segment on the y-axis joining (0, 0) and some (0, h), and 3. φj|_e= φ_k|_e for all j, k.

We also comment here about the space of such metrics. Each metric is characterized by E pieces of data, where E = #E[X] is the number of edges in the complex, so the space of such metrics can be identified with a subset of R^E+. Each triangle inequality restricts us to an open subset of R^E+. Since the length function ` ≡ 1, identifying each face with an equilateral triangle, describes a metric satisfying all of the triangle inequalities, our space

of Euclidean metrics is parametrized by a non-empty open subset of R^E+, and thus has dimension E.

Proposition 3.1.2. Let σ be one of the Euclidean metrics on X described above. Then (X\S, σ) is a locally CAT(0) space.

Proof. Each face of X with a Euclidean metric is CAT(0) since it is flat. For a point p on an edge of X, a ball about p that does not reach as far as any of the other edges or vertices is a union of flat half-discs glued along their common convex (geodesic) boundary pieces.

By Corollary 2.4.8 and Lemma 2.4.9, such a set is CAT(0). Hence every point of X\S has a CAT(0) neighborhood.

Remark 3.1.3. The Euclidean metrics we’ve defined may fail to be non-positively curved at the vertices of X. For example, a neighborhood of a vertex may be simply a cone, and if it is a cone of angle < 2π then it fails to have bounded curvature at the cone point. However, following Theorem 2.4.11, if associated to any loop around a vertex v the corresponding faces have angles at v that sum to at least 2π, then the metric is CAT(0) in a neighborhood of v.