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Harmonic Morphisms Between Riemannian Manifolds$

Paul Baird and John C. Wood

Print publication date: 2003

Print ISBN-13: 9780198503620

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198503620.001.0001

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(p.456) Appendix

(p.456) Appendix

Source:
Harmonic Morphisms Between Riemannian Manifolds
Publisher:
Oxford University Press

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 2-jet; 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 2-jet 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 0M 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 i = 1 m C i i = 0 , there is a harmonic function f defined on an open neighbourhood of x 0 such that

(A.1.1)
f ( x 0 ) = C , f x i ( x 0 ) = C i , 2 f x i x j ( x 0 ) = C i j ( i , j = 1 , , m ) .

Proof By using the normal coordinates, we can transfer the problem to R m endowed with its Euclidean metric, so let C (R m) be any smooth function which satisfies the condition

( I f ˜ ( 0 ) ) | I | 2 = ρ 0 ( C , C i , C i j ) i , j = 1 , , m .
Here, as in Section 4.4, for a multi-index I = (i 1, …, i k), we write 1 to mean the partial derivative I = k / x 1 i 1 x k i k . For a fixed integer r ≥ 2 and number (p.457) α with 0 < α < 1, define Banach spaces
E = { z C r , a ( B ¯ ) : I z ( 0 ) = 0 for all I , 0 | I | 2 } , F = { v C r 2 , α ( B ¯ ) : v ( 0 ) = 0 } ;
here denotes the closed unit ball {x ∈ R m: |x| ≤ 1} and C r(·) denotes the subspace of C r(·) whose derivatives of order r are Hölder continuous with exponent α. Define the function Ψ = Ψ((x i), u, (u i), (u ij)) (i, j = 1, …,m) by
Ψ = g i j ( x ) ( u i j Γ i j k ( x ) u k ) .

For ε > 0, zE and y, set

ψ ( ε , z ) ( y ) = Ψ ( ε y , f ˜ ( ε y ) + ε 2 z ( y ) , d f ˜ ( ε y ) + ε d z ( y ) , d 2 f ˜ ( ε y ) + d 2 z ( y ) ) .
Note that ψ(ε, z)(y) computes the Laplacian of x (x) + ε 2 z(x/ε) at the point y = x/ε.

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

d 2 ψ ( 0 , 0 ) = | I | = 2 Ψ u I ( 0 , ρ 0 ) I ;
this is precisely the Laplacian on R m, i.e.,
d 2 ψ ( 0 , 0 ) = Δ m = 2 x 1 2 + + 2 x m 2 .

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 d2 ψ(0, 0), i.e., a map A: FE such that Δ m A = 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

(A.1.2)
G B [ v ] ( x ) = B G B ( x , y ) v ( y ) d y ;
this determines a mapping A 0: υG B[υ] such that ( Δ m A 0 ) ( v ) = v . However, A 0(υ) may not be in the Banach space E. We therefore define the modified map A: FE by
( A v ) ( y ) = ( A 0 v ) ( y ) ( A 0 v ) ( 0 ) d ( A 0 v ) 0 ( y ) 1 2 d 2 ( A 0 v ) 0 ( y ) ,
i.e., we subtract the terms up to second order in the Taylor expansion of (A 0 υ) (y). Then E and furthermore ( Δ m A ) ( v ) = v , since the Laplacians of the constant and linear terms are obviously zero, and for the quadratic term we have
Δ m ( d 2 ( A 0 v ) 0 ( y ) ) = Tr d 2 ( A 0 v ) 0 = Δ m ( A 0 v ) 0 = v ( 0 ) = 0.

By the implicit function theorem, there exists a C r−2, α map VE, ε ↦ z ε from an open neighbourhood V ⊆ R of 0 such that

ψ ( ε , z ε ) = 0 ( ε V ) .
(p.458) Choose a non-zero ε and set
f ( x ) = f ˜ ( x ) + ε 2 z ε ( x / ε ) ,
so that f is C r, α. As remarked above, this is harmonic. That f is of class C follows from the smoothness of harmonic functions (see Section 2.2). □

Corollary A.1.2 (Existence of local harmonic coordinates) Let M m be a Riemannian manifold and let x 0M. 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 on some neighbourhood U j of x 0, which satisfies y j / x i ( x 0 ) = δ i j . Then the functions y j (j = 1, …, m) form a system of harmonic coordinates defined on a neighbourhood U ⊆ ∩jj 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

cap ( K ) = inf { K | d ψ | 2 d x : ψ C 0 ( m ) , ψ 1 on K } .
Here C 0 ( ) denotes the space of C functions of compact support.

Remark A.1.4 The notion of q-capacity 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 KA.

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

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

  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 ≤ δm), let H δ (K) denote the δ-dimensional 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 non-constant solution to a linear second-order elliptic equation Lf = 0, where the operator L has the form

L = i , j = 1 m a i j ( x ) 2 x i x j + i = 1 m b i ( x ) x i
for some smooth functions a ij, b i on U with the matrix (a ij (x)) positive definite for each xU. Then the set E of all critical points of f can be covered by a countable family of (m − 2)-dimensional embedded submanifolds of U and so is polar.

Proof For each multi-index I = (i 1, …, i k) of order |I| = k > 1, set

E I = { x U : I f ( x ) 0 , and J f ( x ) = 0 for | J | < | I | } ,
so that E IE. By the unique continuation property for elliptic operators (Aronszajn 1957), as I varies over all multi-indices of order k ≥ 2, the sets E I cover E. Now fix a multi-index I of order |I| > 1 and write I as I = J K with |K| = 2, so that |J| = |I| − |K| (≥ 0). Let xE I. Then x is a critical point for the function υ = j f, since, on writing i = / x i , etc., we have iυ(x) = i j f (x) = 0 (i = 1, …, m) and the order of i j is 1 + |J| = |I| − 1. Since L∂ j f j L f contains only derivatives of f of order at most |J| < |I| and these vanish at x, we have
L v ( x ) = L J f ( x ) = J L f ( x ) = 0.
Because iυ(x) = 0, it follows from L f = 0 that
(A.1.3)
i , j = 1 m a i j ( x ) i j v ( x ) = 0 ( x E I ) .
We claim that rank ( ijυ(x)) ≥ 2 for all xE 1.

Suppose not. Then there exists xE 1 such that rank ( ijυ(x)) ≤ 1. Then the matrix ( ijυ(x)) has the form of an outer product (ξiξj) for some vector ξ = (ξ1, …, ξm). By (A.1.3) and the positive definiteness of (a ij(x)), this means that ξ = 0 and so (∂ijυ(x)) is the zero matrix. But this contradicts Kυ(x) = ∂KJ f(x) = ∂I f(x) ≠ 0.

Thus, rank ( ijυ(x)) ≥ 2 for all xE 1 and there exists a 2 × 2-minor of rank 2. For each i, j with 1 ≤ i < jm, set

F I i j = { x U : i v ( x ) = j v ( x ) = 0 , | i 2 v ( x ) i j v ( x ) j i v ( x ) j 2 v ( x ) | 0 } .
(p.460) We have established that,
E I 1 i < j m F I i j ( | I | > 1 ) , and E | I | > 1 E I .

By the implicit function theorem, each F I i j 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 second-order 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.2.1)
Δ u = u f ,
where f: M → R is a given smooth function. We shall demand that u: M → R be continuous on M, and smooth on M \ ∑, where ∑ = {xM: u(x) = 0}.

A special case is equation (11.4.2). In that case, up to a constant multiple, f = ScalMu 4/(n−2) ScalN, which, although dependent on u, is smooth even at points of ∑, since u 4/(n−2) = λ2 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 0M if, for each positive integer k,

(A.2.2)
| u ( x ) | = O ( d ( x , x 0 ) k ) ,
where d(x, x 0) denotes the distance from x to x 0. Note that, by Taylor's theorem, a smooth function u has a zero of infinite order at x 0 if and only if it vanishes at x 0, together with all its derivatives. (Note that this accords with Definition 4.4.3.)

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 ψ C 0 ( M ) ,

(A.2.3)
M u Δ ψ v M = M u f ψ v M .

Proof Let ψ C 0 ( M ) 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 = {xM: d(x, K) < } where ‘d(·, ·)’ denotes distance on M and let ρ C (M) be a function with

ρ ε = { 0 if x K ε , 1 if x M \ K 3 ε ,
which satisfies the derivative bounds
| ρ ε | c K / ε and | Δ ρ ε | c K / ε 2 ,
for some constant c K which depends only on K. The existence of such a ‘bump’ function is standard; see the proof of Theorem 1.5.4 in Hörmander (1976) (in particular, equation (1.5.10) of this book gives the bounds on the derivatives). Note that, as → 0, the function ρ tends to χ M\K in a weak sense, where χ M\K denotes the characteristic function of the set M \ K.

To show that (A.2.3) is satisfied, we ‘integrate by parts’, i.e., use Green's identities (see Section 2.2), to give

(A.2.4)
M \ K ε u Δ ψ ρ ε v M = M \ K ε < u , ψ > ρ ε v M M \ K ε u < ψ , ρ ε > v M .
Now
| M \ K ε u < ψ , ρ ε > v M | K 3 ε \ K ε | u | | ψ | ( c K ε ) v M 0 as ε 0 ,
by (A.2.2). On the other hand, integration by parts twice more gives
M \ K ε u , ψ ρ ε υ M = M \ K ε Δ u ψ ρ ε υ M + M \ K ε u , ρ ε ψ υ M = M \ K ε Δ u ψ ρ ε υ M M \ K ε ψ , ρ ε υ M M \ K ε u ψ ρ ε υ M M \ K Δ u ψ υ M as ε 0 ,
again by applying (A.2.2). Let → 0 in (A.2.4), then we conclude that
M \ K u Δ ψ v M = M \ K u f ψ v M .
(p.462) But on K we have
K u Δ ψ v M = K u f ψ v M ,
since both sides vanish. Hence (A.2.3) is satisfied, and the proof is complete.

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: MN be a smooth map between Riemannian manifolds. A vector field V defined on M is said to be uniformly non-horizontal (with respect to φ) if, on the set of points where both V and dφ are non-zero, 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: MN is a smooth map with grad |dφ|2 uniformly non-horizontal, 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 mN n be a smooth map and let x 0M be a point of finite order. Let σ ≡ σx 0(φ): T x 0 MT φ(x 0)N denote the symbol of φ at x 0. Suppose that grad |dφ|2 is uniformly non-horizontal with respect to φ in a neighbourhood of x 0; then grad |dσ|2 is uniformly non-horizontal 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 non-horizontal 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 non-horizontal; 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

X = r p 2 α = 1 n t α grad ϕ α ( t α ) ,
(p.463) where we include the factor r p−2 for later convenience. Then, since |U φ| sin θ measures the distance of U φ from the horizontal space (ker dφ), we have
(A.3.1)
| U ϕ | 2 sin 2 θ 0 | U ϕ | 2 sin 2 θ | U ϕ X | 2 = | U ϕ r p 2 α = 1 n t α grad ϕ α | 2
for all t = (t 1,…,t n) ∈ R n. Now, by Taylor's theorem and Lemma 4.4.1,
ϕ α ( x ) = σ α ( ξ ) + O ( r p + 1 ) ( α = 1 , , n ) , ϕ α x i ( x ) = σ α ξ i ( ξ ) + O ( r p ) ( i = 1 , , m , α = 1 , , n ) .
By smoothness of the respective metrics, we have g ij(x) = δ ij + O(r) and h αβ(φ(x)) = δ αβ + O(r), so that
| d ϕ | 2 ( x ) = | d σ | 2 ( ξ ) + O ( r 2 p 1 ) , U ϕ ( x ) = U ϕ ( ξ ) + O ( r 2 p 2 ) .

Thus, from (A.3.1), we see that

(A.3.2)
| U σ | 2 sin 2 θ 0 | U σ r p 2 α = 1 n t α grad σ α | 2 + O ( r 4 p 5 )
for all t = (t 1,…,t n) ∈ R n.

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 non-zero; then, from (A.3.2),

sin 2 μ = min t n | U σ ( ξ ) r p 2 t α grad ( σ α ) ξ | 2 | U σ ( ξ ) | 2 sin 2 θ 0 + O ( r ) .
Hence, on a suitable neighbourhood of the origin in T x0 M,
sin 2 μ sin 2 ( 1 2 θ 0 ) ,
so that μ 1 2 θ 0 . By the homogeneity of σ, this inequality holds on the whole of R m.

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

σ ( ξ 1 + y ) = σ ( ξ 1 ) + s ( y ) ,
where s is the symbol of σ at ξ 1. But
σ ( t ξ 1 + y ) = t p σ ( ξ 1 + t 1 y ) = t p { σ ( ξ 1 ) + s ( t 1 y ) } = t p s ( t 1 y ) = s ( y ) ;
(p.464) taking the limit as t → 0 shows that σ(y) = s(y). Thus,
σ ( ξ 1 + y ) = σ ( ξ 1 ) + σ ( y ) = σ ( y ) .
But now, by homogeneity, σ(t ξ 1 + t y) = σ(t y); so, on replacing t y by y, we conclude that
σ ( t ξ 1 + y ) = σ ( y ) ( y m , t ) .

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

ρ ( ξ 1 , , ξ m 1 ) = σ ( ξ 1 , , ξ m 1 , ξ m )
where ξ m ∈ R is arbitrary. Then σ has the form claimed.

Theorem A.3.4 Let φ: M → N be a smooth map between Riemannian manifolds. If grad |dφ|2 is uniformly non-horizontal 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 (tM). 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 0U, 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 0M 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 non-horizontal 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,

σ ( ξ 1 , , ξ m ) = ρ ( ξ 1 , , ξ m 1 )
for some homogeneous polynomial map ρ: R m−1 → R n. It is easily checked that, at any point (ξ 1,…,ξ m−1) where U ρ and dρ are both non-zero, the angle θ between U ρ and (ker dρ) is the same as the angle between U σ and (ker dσ) at (ξ 1,…, ξ m−1, 0); hence θ ≥ θ 0 for some θ 0 > 0. But, since ρ has order p with 1 < p < ∞ at the origin of R m−1, this contradicts our inductive hypothesis on the dimension m. Thus, σ has order 1 at every point of R m \ {0} and so
d σ ξ 0 ( ξ m \ { 0 } ) .
We now explore what this means.

(p.465) By homogeneity, there exist, constants α,β > 0 such that

(A.3.3)
α | ξ | 2 p 2 | ( d σ ) ξ | 2 β | ξ | 2 p 2 ( ξ m ) ;
indeed, we may choose α and β to be the maximum and minimum, respectively, of |dσ|2 on the unit sphere S m−1. Now, by homogeneity and Euler's identity, if U σ(ξ) = 0 for some ξ ∈ R m \ {0}, then |(dσ)ξ|2 = 0. It follows that U σ ≠ 0 in R m \ {0}, so that E = {ξ ∈ R m: |(dσ)ξ|2 = 1} is a compact smooth hypersurface. Consider the homogeneous polynomial | σ | 2 = σ 1 2 + + σ n 2 of degree 2p. Then the restriction of |σ|2 to E has a maximum at some point ξE and at that point, grad |σ|2 is orthogonal to E. But U σ is also orthogonal to E, so that grad |σ|2 and U σ are linearly dependent at ξ. Now U σ(ξ) ≠ 0 and grad |σ|2 = 2σ 1 grad σ 1 +…+2σ n grad σ n ∈ (ker dσ). Since U σ is uniformly non-horizontal, grad |σ|2 = 0 at ξ. In particular,
0 = < ξ , grad | σ | 2 > = 2 p | σ ( ξ ) | 2 ,
and therefore σ vanishes on E. Now, by (A.3.3), any ray through the origin hits E, so that, by homogeneity, σ vanishes identically on R m, a contradiction to the definition of symbol. Thus, φ can have no points of order p with 1 < p < ∞, and the induction step is complete, establishing the theorem.

A.4 NOTES AND COMMENTS

Section A.1

  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 non-compact 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 p-harmonic coordinates on a manifold.

  2. 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 = { 1 ( m 2 ) σ m D | grad u | 2 d x ( m 3 ) , 1 2 π D | grad u | 2 d x ( m = 2 ) ,
    where σ m = 2π m/2/(mΓ(m/2)) denotes the volume of the unit ball in R m. If, now, M 1 is a sphere of radius r, the limit as r → ∞ is called the Newtonian capacity of the set K bounded by M 2—this measures the electrostatic capacity of an isolated conductor K.

  3. 3. Let D be a compact domain of a Riemannian manifold M. Let f 1, f 2C 2(D) be functions with f 1 harmonic on D and f 2 superharmonic on Df 2 ≤ 0) such that (p.466) f 2(x) ≥ f 1(x) for all x∂D. Then f 2f 1 in D. This fact easily follows from the minimum principle (see Section 2.2), since, if we let f = f 2f 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 {xM: 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. 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.