(p.456) Appendix
(p.456) Appendix
We assemble here some technical aspects of the theory of elliptic operators and give a property concerning the critical set of a smooth mapping. We first show the existence of a harmonic function with a given 2jet; this enables us to establish the characterization of harmonic morphisms given in Section 4.2. Then we discuss briefly the notions of polar set and capacity and prove that the critical set of a harmonic function is polar; this implies the same result for harmonic morphisms, as discussed in Section 4.3.
In the next section, we study a class of elliptic equations related to the Yamabe problem. We show that any solution which has no critical points of finite order is smooth; this is needed in Section 11.4 to show that a weakly conformal map between equidimensional Riemannian manifolds is a local diffeomorphism.
In the final section, we show that certain maps, which include horizontally homothetic ones, cannot have any critical points of finite order; this implies that a horizontally homothetic harmonic morphism has no critical points, as mentioned in Corollary 4.5.5.
A.1 ANALYTIC ASPECTS OF HARMONIC FUNCTIONS
The first problem concerns the existence of a harmonic function satisfying given data at a point. We show that, given any smooth function on an open neighbourhood of x _{0} which is harmonic at the point x _{0}, we can find a harmonic function on a possibly smaller open neighbourhood of x _{0} with the same 2jet at x _{0}. It is convenient to formulate this in terms of normal coordinates, as follows.
Lemma A.1.1 (Existence of local harmonic functions) Let (M ^{m}, g) be a Riemannian manifold. Let x _{0} ∈ M and let (x ^{i}) be a system of normal coordinates centred on x _{0}. Then, for any system of constants {C, C _{i}, C _{ij}}_{i, j=1, …,m} with C _{ij} = C _{ji} and ${\sum}_{i=1}^{m}{C}_{ii}}=0,$ there is a harmonic function f defined on an open neighbourhood of x _{0} such that
Proof By using the normal coordinates, we can transfer the problem to R ^{m} endowed with its Euclidean metric, so let f˜ ∈ C ^{∞} (R ^{m}) be any smooth function which satisfies the condition
For ε > 0, z ∈ E and y ∈ B̄, set
Then ψ maps a neighbourhood U ⊆ R × E of (0, 0) into F and satisfies ψ(0, 0) = 0. Furthermore, the partial derivative with respect to the second argument is given by
We now apply the implicit function theorem for Banach spaces; see, e.g., Gilbarg and Trudinger (2001, Theorem 17.6) or Dieudonné (1969, (10.2.1)). To do this, we need to find an inverse to d_{2} ψ(0, 0), i.e., a map A: F → E such that ${\Delta}^{{\mathbb{R}}^{m}}\circ A={\text{Id}}_{F}.$ This is given by the Green function as follows.
Let G _{B} denote the Green function for the unit ball (Example 2.2.4) and set
By the implicit function theorem, there exists a C ^{r−2, α} map V → E, ε ↦ z _{ε} from an open neighbourhood V ⊆ R of 0 such that
Corollary A.1.2 (Existence of local harmonic coordinates) Let M ^{m} be a Riemannian manifold and let x _{0} ∈ M. Then there exist local harmonic coordinates defined on a neighbourhood of x _{0}.
Proof For each j = 1, …, m, set C _{i} = δ _{ij}. By Lemma A.1.1, there exists a harmonic function ${y}^{j}:{U}_{j}\to \mathbb{R}$ on some neighbourhood U _{j} of x _{0}, which satisfies $\partial {y}^{j}/\partial {x}^{i}\left({x}_{0}\right)={\delta}_{ij}.$ Then the functions y ^{j} (j = 1, …, m) form a system of harmonic coordinates defined on a neighbourhood U ⊆ ∩_{j} ∪ _{j} of x _{0}. □
We now discuss the concept of polar set. For our purposes, we only require the notion of polar for closed subsets; this can be described nicely in terms of the classical concept of capacity. See ‘Notes and comments’ for a discussion of ‘polar’ for general subsets.
Definition A.1.3 (Serrin 1964, Section 7) Let K be a compact subset of R ^{m}. The (2)capacity of K is defined by
Remark A.1.4 The notion of qcapacity can be defined for any number q ≥ 1 (cf. Serrin 1964, Section 7).
Definition A.1.5 Let K ⊂ R ^{m} be a compact subset. We say that K is polar if cap(K) = 0. More generally, any closed set A ⊆ R ^{m} is said to be polar if cap (K) = 0 for every compact subset K ⊆ A.
This concept of a polar subset is local and so applies to closed subsets of an arbitrary smooth manifold by taking coordinate charts; invariance under change of coordinates can easily be checked.
The following result identifies many polar sets and can be proved directly from the definition in terms of capacity.
Lemma A.1.6 Let M ^{m} be a Riemannian manifold. Then

(i) a submanifold is polar if and only if its dimension is at most m − 2;

(ii) a countable union of polar sets is polar. □
Proof (i) The first fact follows from the following relationships between zero capacity and Hausdorff dimension (Carleson 1967); see Meier (1986) for an account. For any real number δ (0 ≤ δ ≤ m), let H ^{δ} (K) denote the δdimensional Hausdorff measure of K. (i) If H ^{m−2}(K) is finite, then cap(K) = 0. (ii) Conversely, if cap (K) = 0 then H ^{δ}(K) = 0 for every δ > m − 2.
(ii) That a countable union of polar sets is polar is proved for the Euclidean case in Brelot (1969, Chapter III, §2) and Helms (1975, Theorem 7.6). Results of Hervé (1962) provide the extension to arbitrary Riemannian manifolds. □
(p.459) On a Riemann surface, it is well known that the critical set of a holomorphic function (and so of a harmonic function) consists of discrete points. In higher dimensions, the critical set of a harmonic function is also 'small’, in the sense that is has zero capacity. This property is valid for solutions of more general elliptic equations, and since the result does not appear to be widely known, we give the general case here (the proof was kindly supplied by B. Fuglede).
Theorem A.1.7 (Critical set is polar) Let U be a domain of R ^{m} and suppose that f: U → R is a nonconstant solution to a linear secondorder elliptic equation Lf = 0, where the operator L has the form
Proof For each multiindex I = (i _{1}, …, i _{k}) of order I = k > 1, set
Suppose not. Then there exists x ∈ E _{1} such that rank (∂ _{i}∂_{j}υ(x)) ≤ 1. Then the matrix (∂ _{i}∂_{j}υ(x)) has the form of an outer product (ξ_{i}ξ_{j}) for some vector ξ = (ξ_{1}, …, ξ_{m}). By (A.1.3) and the positive definiteness of (a _{ij}(x)), this means that ξ = 0 and so (∂_{i}∂_{j}υ(x)) is the zero matrix. But this contradicts ∂ _{K}υ(x) = ∂_{K}∂_{J} f(x) = ∂_{I} f(x) ≠ 0.
Thus, rank (∂ _{i}∂_{j}υ(x)) ≥ 2 for all x ∈ E _{1} and there exists a 2 × 2minor of rank 2. For each i, j with 1 ≤ i < j ≤ m, set
By the implicit function theorem, each ${F}_{I}^{ij}$ is an embedded submanifold of U of dimension m − 2, and the proof is completed by applying Lemma A.1.6. □
Since a harmonic function satisfies the linear secondorder elliptic equation given in local coordinates by (2.2.4), we deduce the following immediately.
Corollary A.1.8 Let M be a Riemannian manifold. Then the critical set of a harmonic function f: M → R is polar.
Finally, note the following useful consequence of Sard's theorem.
Lemma A.1.9 A smooth function is constant on each connected component of its critical set.
Proof If the critical set is empty, there is nothing to prove. Otherwise, since the image K of a connected component of the critical set must be connected, it is a point or an interval. However, by Sard's theorem (Remark 2.4.1), K has Lebesgue measure zero, therefore it must be a point.
A.2 A REGULARITY RESULT FOR AN EQUATION OF YAMABE TYPE
Let M be a Riemannian manifold. We study solutions u: M → R to the linear elliptic differential equation
A special case is equation (11.4.2). In that case, up to a constant multiple, f = Scal^{M}−u ^{4/(n−2)} Scal^{N}, which, although dependent on u, is smooth even at points of ∑, since u ^{4/(n−2)} = λ^{2} is the square dilation of a smooth mapping.
For a continuous function u: M → R, say that u has a zero of infinite order at x _{0} ∈ M if, for each positive integer k,
Theorem A.2.1 Let u: M → R be a continuous function which is smooth and satisfies (A.2.1) on the set M \ ∑. Suppose that u vanishes to infinite order on ∑ (i.e., it has no zeros of finite order). Then u is smooth and satisfies (A.2.1) on all of M.
The theorem is a consequence of the following lemma, which is valid for more general equations than (A.2.1). However, for ease of exposition, we shall prove it only in this special case. We thank R. Regbaoui for providing this proof.
(p.461) Lemma A.2.2 Let u: M → R satisfy the hypotheses of the theorem. Then u satisfies equation (A.2.1) weakly, i.e., for all $\psi \in {C}_{0}^{\infty}\left(M\right),$
Proof Let $\psi \in {C}_{0}^{\infty}\left(M\right)$ and let K = supp ψ ∩ ∑; then K is compact, since it is the intersection of a compact subset with a closed subset. For each ∊ > 0, set K _{∊} = {x ∈ M: d(x, K) < ∊} where ‘d(·, ·)’ denotes distance on M and let ρ _{∊} ∈ C ^{∞}(M) be a function with
To show that (A.2.3) is satisfied, we ‘integrate by parts’, i.e., use Green's identities (see Section 2.2), to give
Proof of Theorem A.2.1 Equation (A.2.1) is a linear elliptic differential equation with smooth coefficients. From Lemma A.2.2, the function u is a (continuous) weak solution. The smoothness of u now follows from standard regularity theory for such equations (Schwarz 1966, Chapter VI, Théorème XXIX).
A.3 A TECHNICAL RESULT ON THE SYMBOL
We prove Proposition 4.4.8; in fact, we shall give a more general result (Theorem A.3.4) from which that proposition follows. We follow Fuglede (1982).
Definition A.3.1 Let φ: M → N be a smooth map between Riemannian manifolds. A vector field V defined on M is said to be uniformly nonhorizontal (with respect to φ) if, on the set of points where both V and dφ are nonzero, the angle (taken in the range [0, π/2]) between V and (ker dφ)^{⊥} is bounded below by a constant θ _{0} ∈ (0, π/2].
We shall show that, if φ: M → N is a smooth map with grad dφ^{2} uniformly nonhorizontal, then φ can have no (critical) point of order p with 1 < p < ∞. Recall Definition 4.4.3 of the symbol.
Lemma A.3.2 Let φ: M ^{m} → N ^{n} be a smooth map and let x _{0} ∈ M be a point of finite order. Let σ ≡ σ_{x} _{0}(φ): T _{x} _{0} M → T _{φ}(x _{0})N denote the symbol of φ at x _{0}. Suppose that grad dφ^{2} is uniformly nonhorizontal with respect to φ in a neighbourhood of x _{0}; then grad dσ^{2} is uniformly nonhorizontal with respect to σ on all of T _{x0} M.
Proof In what follows, for any map ψ between Riemannian manifolds, we write U _{ψ} for grad dψ^{2}. Let x _{0} be a point of finite order p, and let U _{φ} be uniformly nonhorizontal in a neighbourhood of x _{0}, with the angle between U _{φ} and (ker dφ)^{⊥} bounded below by θ _{0} ∈ (0, π/2]. If p = 1, then σ is linear, so that dσ^{2} is constant and σ is trivially uniformly nonhorizontal; hence we may suppose that p ≥ 2.
Choose local coordinates (x ^{i}) and (y ^{α}) orthonormal at x _{0} and φ(x _{0}), respectively. Let x be a point where U _{φ} ≠ 0 and dφ ≠ 0. Denote the coordinates of x by ξ = (ξ ^{1},…,ξ ^{m}) and write r = ξ. Let θ = θ(x) denote the angle between U _{φ} and (ker dφ)^{⊥}. Any vector X ∈ (ker dφ _{x})^{⊥} is a linear combination
Thus, from (A.3.1), we see that
Let μ = μ(ξ) ∈ [0, π/2] denote the angle between U _{σ}(ξ) and (ker dσ _{ξ})^{⊥} at a point ξ ∈ R ^{m} where both U _{σ} and dσ are nonzero; then, from (A.3.2),
Lemma A.3.3 Let σ: R ^{m} → R ^{n} be a mapping defined by homogeneous polynomials of degree p ≥ 2. Suppose that σ is of order p at some point ξ _{1} ∈ R ^{m} \{0}, then σ factors as an orthogonal projection R ^{m} → R ^{m−1} followed by a homogeneous polynomial map ρ: R ^{m−1} → R ^{n} of degree p.
Proof Since σ is homogeneous of degree p ≥ 2, by Euler's identity we have 〈ξ, (grad σ)_{ξ}〉 = p σ(ξ), so that σ(ξ _{1}) = 0 and, once more by homogeneity, σ(t ξ _{1}) = 0 for all t ∈ R. We claim that this shows that σ is independent of the coordinate directed along the axis of ξ _{1}, i.e., that σ(t ξ _{1} + y) = σ(y) (y ∈ R ^{m}, t ∈ R). Indeed, Taylor's theorem (in its exact form for polynomials) gives
By a rotation of the coordinates, we can suppose that ξ _{1} is parallel to (0,…, 0, 1), so that σ(ξ) is independent of the last coordinate ξ ^{m}. Now define ρ: R ^{m−1} → R ^{n} by
Theorem A.3.4 Let φ: M → N be a smooth map between Riemannian manifolds. If grad dφ^{2} is uniformly nonhorizontal with respect to φ, then φ can have no points of order p with 1 < p < ∞.
Proof We first show that the theorem is true when dim M = 1. In that case, we can take M to be an open interval of R and dφ _{ι}^{2} = φ′(t)^{2} (t ∈ M). The hypothesis of the theorem means that d(φ′(t)^{2})/dt = 0 whenever φ′(t) ≠ 0. But then d(φ′(t)^{2})/dt = 2(d(φ′(t))/dt)φ′(t) = 0 for all t, so that φ′(t) is constant; hence, if φ′(t _{0}) = 0 for some t _{0} ∈ U, then φ′ is identically zero and every point is of infinite order.
We shall proceed by induction on dim M. Suppose that the theorem is true for dim M = m − 1 for some m ≥ 2. We shall show that it is true for dim M = m. To do this, we do an induction on p, starting with p = 2; as in the proof of Lemma A.3.2, we shall write U _{σ} for grad dσ^{2}.
Let x _{0} ∈ M be a point at which the mapping φ has order p with 1 < p < ∞, and let σ = σ _{x0} (φ) denote the symbol of φ at x _{0}. After choosing local coordinates orthonormal at x _{0} and φ(x _{0}), we can regard the symbol as a map σ: R ^{m} → R ^{n}. By Lemma A.3.2, U _{σ} is uniformly nonhorizontal with respect to σ. Hence, by our inductive hypothesis if p > 2, and trivially if p = 2, σ cannot have any points of order q with 1 < q < p. If, on the other hand, σ is of order p at some point ξ _{1} ∈ R ^{m} \ {0}, then, by Lemma A.3.3, after a suitable rotation of the coordinates,
(p.465) By homogeneity, there exist, constants α,β > 0 such that
A.4 NOTES AND COMMENTS
Section A.1

1. We have adapted the proof of Alinhac and Gérard (1991) given for more general elliptic equations to prove existence of local harmonic functions. Note that it is essential to use C ^{r, α} spaces rather than C ^{r} spaces to ensure that the formula (A.1.2) gives an inverse (see Gilbarg and Trudinger 2001, Chapter 4). An alternative proof is given by Bers (1955). Greene and Wu (1975) show that every noncompact Riemannian manifold M of dimension m can be embedded by means of harmonic functions in a Euclidean space of dimension 2m + 1. This theorem gives another method for proving the existence of local harmonic coordinates.
Loubeau (2000) applies the existence theorem of Alinhac and Gérard to construct local pharmonic coordinates on a manifold.

2. The integral used to define capacity in Definition A.1.3 is precisely (twice) the Dirichlet integral. Capacity derives from the classical notion in electrostatic theory of a condenser formed by conductors separated by a dielectric (Feynman, Leighton and Sands 1963, 1964). Let M _{1} and M _{2} be two smooth closed hypersurfaces in Euclidean space R ^{m}, with M _{1} enclosing M _{2}. Let u be a harmonic function defined on the domain D between M _{1} and M _{2}. The condenser capacity is the number defined by
$${C}_{{M}_{1},{M}_{2}}=\{\begin{array}{c}\frac{1}{\left(m2\right){\sigma}_{m}}{\displaystyle {\int}_{D}{\left\text{grad}\text{}u\right}^{2}}\text{d}x\text{}\text{}\left(m\ge 3\right),\\ \frac{1}{2\pi}{\displaystyle {\int}_{D}{\left\text{grad}\text{}u\right}^{2}}\text{d}x\text{}\text{}\text{}\text{}\text{}\text{}\text{}\left(m=2\right),\end{array}$$ 
3. Let D be a compact domain of a Riemannian manifold M. Let f _{1}, f _{2} ∈ C ^{2}(D) be functions with f _{1} harmonic on D and f _{2} superharmonic on D (Δf _{2} ≤ 0) such that (p.466) f _{2}(x) ≥ f _{1}(x) for all x ∈ ∂D. Then f _{2} ≥ f _{1} in D. This fact easily follows from the minimum principle (see Section 2.2), since, if we let f = f _{2} − f _{1}, then Δf ≥ 0 and f ≥ 0 on ∂D. By the minimum principle, f has no minimum in D unless it is constant; it follows that f ≥ 0 in D. This property enables us to define the notion of 'superharmonic’ for functions f: M → (−∞, ∞] which are only lower semicontinuous (i.e., the set {x ∈ M: f(x) > c} is open for every c ∈ R). Then, as mentioned in ‘Notes and comments’ to Section 2.2, a subset A (not necessarily closed) of a Riemannian manifold is polar if there is a lower semicontinuous superharmonic function f which is infinite on A (and possibly elsewhere) but not identically infinite; see, e.g., Brelot (1969) for a description for Euclidean space and Hervé (1962) for an account of the necessary extensions required to deal with arbitrary Riemannian manifolds. The equivalence, for closed subsets, of this definition with Definition A. 1.5 is proved for Euclidean space in Helms (1975, Theorem 7.33). This equivalence carries over to arbitrary Riemannian manifolds by applying results of Hervé (1962). In fact, Hervé shows that the notions of ‘superharmonic function’ and ‘polar’ can be defined with respect to an arbitrary elliptic operator L, and that polar sets relative to L coincide with polar sets relative to Δ (Hervé 1962, Theorem 36.1). Note that f is said to be subharmonic if –f is superharmonic.

4. Removable singularity theorems for general elliptic equations were established by Serrin (1964). A direct proof that a set with vanishing capacity represents a removable singularity for a harmonic function is given by Carleson (1967). The case of more general elliptic equations is also dealt with by Meier (1983). The problem has been studied more generally for harmonic maps by Meier (1986), and Eells and Polking (1984); the case of minimal submanifolds is also considered by Meier (1986).
Section A.2
As mentioned in Section 11.4, Theorem A.2.1 has no application to the study of horizontally weakly conformal mappings when dim M > dim N; in fact, it is unknown whether such mappings can have critical points of infinite order. Such maps may certainly have critical points of finite order.
Section A.3
For horizontally homothetic maps, in contrast to the last note, there can be no critical points of finite order, but it is still unknown whether there can be critical points of infinite order.