Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 433–458

FINITE ENERGY MAPS FROM RIEMANNIAN POLYHEDRA TO METRIC SPACES

Bent Fuglede

University of Copenhagen, Department of Mathematics Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark; fuglede@math.ku.dk

Dedicated to the memory of Professor Heinz Bauer

Abstract. It is shown that every map of finite energy in the sense of Korevaar and Schoen into a complete metric space Y (not necessarily locally compact) is quasicontinuous, the domain space being an admissible Riemannian polyhedron. Assuming that Y is a geodesic space of upper bounded Alexandrov curvature, two inequalities are obtained for the energy of certain maps associated with a given pair of maps. One of these inequalities is due to T. Serbinowski (unpublished) and applied to establish existence and uniqueness of the solution to the variational Dirichlet problem for harmonic maps into Y .

1. Introduction and preliminaries

In this article the hypothesis of local compactness of the target space is omitted in certain results about maps of finite energy established in [EF], [F1], and [F2].

We first show (Theorem 1) that every finite energy map φ: X → Y is quasicontinuous, i.e., continuous relative to the complement of open subsets of X of arbitrarily small capacity, cf. [EF, p. 153]. The domain space is a Riemannian manifold, or more generally an admissible Riemannian polyhedron (X, g), dim X = m. The target is a complete metric space (Y, d_{Y}). Under the extra hypothesis that closed balls in Y be compact, the result was obtained in [EF, Theorem 9.1] by a non-constructive compactness argument.

Basically following Serbinowski [Se] (Thesis, unpublished) we next establish existence and uniqueness of the solution to the variational Dirichlet problem for harmonic maps of X into suitable balls in Y , assuming that Y has upper bounded Alexandrov curvature (Theorem 2). See Section 2 for a precise formulation.

Referring to [EF, Chapter 4] we recall that a (Lipschitz) polyhedron X is defined as a metric space which is Lip homeomorphic to a connected locally finite simplicial complex. Admissibility means that X is dimensionally homogeneous and that (if m ⩾ 2) any two m-simplexes of X with a common face σ (dim σ = 0, 1, …, m − 2) can be joined by a chain of m-simplexes containing σ, any two consecutive ones of which have a common (m − 1)-face containing σ.

2000 Mathematics Subject Classification: Primary 58E20, 49N60; Secondary 58A35.

The polyhedron X becomes a Riemannian polyhedron when endowed with a Riemannian metric g, defined by giving on each open m-simplex s of X a nondegenerate Riemannian metric g|_{s}. We require that g be simplexwise smooth, i.e., each g|_{s} shall be smooth, and shall extend smoothly to the affine span of s, cf. [EF, Remark 4.1]. The associated volume measure on X is denoted by μ_{g} = μ, the intrinsic (Riemannian) distance on X by d_{X}^{g} = d_{X}, and the closed ball with centre x ∈ X and radius r by B_{X}(x, r).

Based on the work of Korevaar and Schoen [KS] a concept of energy of a map φ of (X, g) into a metric space (Y, d_{Y}) is developed in [EF, Chapter 9]. The map φ is supposed first of all to be measurable with separable essential range, and to be of class L_{loc}^{2}(X, Y) in the sense that the distance function d_{Y} (φ( ·), y) is of class L_{loc}^{2}(X, μ) for some and hence (by the triangle inequality) for any point y ∈ Y . The approximate energy density e_{ε}(φ) ∈ L_{loc}^{1}(X, μ) is then defined for ε > 0 at every point x ∈ X by

(1.1)  e_{ε}(φ)(x) = ∫_{B_{X}(x,ε)} \frac{d_{Y}^{2} (φ(x), φ(x′))}{ε^{m+2}} dμ(x′).

The energy of φ: (X, g) → (Y, d_{Y}) is defined as

(1.2)  E(φ) = sup_{f∈C_{c}(X,[0,1])}(lim sup_{ε→0} ∫_{X} fe_{ε}(φ) dμ) (⩽ ∞),

where C_{c} stands for continuous functions of compact support. W_{loc}^{1,2}(X, Y) denotes the space of all maps X → Y for which E(φ|_{U}) < ∞ for every relatively compact connected open set U ⊂ X (equivalently: the above lim sup is finite for every f). If X is compact then (1.2) reduces to

E(φ) = lim sup_{ε→0} ∫_{X} e_{ε}(φ) dμ.

If φ ∈ W_{loc}^{1,2}(X, Y) (and only then), there exists a non-negative function e(φ) ∈ L_{loc}^{1}(X, μ), called the energy density of φ (more precisely: the 2-energy density), such that e_{ε}(φ) → e(φ) as ε → 0, in the sense of weak convergence as measures:

(1.3)  lim_{ε→0} ∫_{X} fe_{ε}(φ) dμ = ∫_{X} fe(φ) dμ

for every f ∈ C_{c}(X). In the affirmative case it follows from (1.2), (1.3) that

E(φ) = ∫_{X} e(φ) dμ.

For the above assertions, see Steps 2, 3, and 4 of the proof of [EF, Theorem 9.1]. These steps are independent of the general requirement in [EF] that also the target of maps X → Y shall be locally compact. However, the proof of the second part of Step 1, leading to quasicontinuity of φ, required compactness of closed balls in Y .^{1} In the present article we show that this extra hypothesis can be omitted; and that appears to be new even when the domain is a manifold.

A function u: X → R is of class W_{loc}^{1,2}(X, R) in the above sense (with Y = R) if and only if u ∈ W_{loc}^{1,2}(X) as defined in [EF, p. 63] (cf. [F1, footnote 2] for the uniqueness of ∇u). And if that is the case, the energy density of u equals

(1.4)  e(u) = c_{m}|∇u|^{2} = c_{m}g^{ij}∂_{i}u ∂_{j}u  a.e. in X ,

with the usual summation convention. Here c_{m} = ω_{m}/(m + 2), ω_{m} being the volume of the unit ball in R^{m}. See [EF, Corollary 9.2], which is based on [KS, Theorem 1.6.2] (where X is a Riemannian domain in a Riemannian manifold), and is also a particular case of [EF, Theorem 9.2].

2. Formulation of results

A version of a μ-measurable map φ arises when φ is redefined on some μ-nullset.

Theorem 1. Every map φ ∈ W_{loc}^{1,2}(X, Y) of an admissible Riemannian polyhedron (X, g) into a complete metric space (Y, d_{Y}) has a quasicontinuous version. When dim X = 1, φ has a Hölder continuous version with exponent \frac{1}{2} .

The proof of this theorem, given in Section 3, includes the following explicit description of a quasicontinuous version φ^{∗}: X → Y of φ: For any point a in X less a certain set P (likewise explicitly described) of capacity 0, φ^{∗}(a) is the essential radial limit of φ(x) as x → a along small rays issuing from a in almost all directions. In the proof of the theorem (for m ⩾ 2) we employ the fine topology of H. Cartan on R^{m} —the weakest topology in which every subharmonic function in R^{m} is continuous. The fine topology is stronger than the metric topology. We shall also use the following lemma on finite energy maps R^{m} → Y , likewise established in Section 3, and drawing on Korevaar–Schoen’s study of directional energies [KS].

Lemma 1. Let φ be a map of finite energy from R^{m}, m ⩾ 2, into a complete metric space (Y, d_{Y}). In terms of the 2-energy density e(φ) ∈ L^{1}(R^{m}) suppose that

∫_{R^{m}} |x|^{1−m} \sqrt{e(φ)(x)} dx < ∞.

1 At this point in [EF], local compactness of Y should be read as closed balls in Y being compact. Furthermore, the reference to Remark 7.5 should be to Remark 7.6.

For almost every open ray R in R^{m} issuing from 0, the restriction φ|_{R}: R → Y has finite 1-energy, and therefore possesses a version of bounded variation up to the point 0.

In the proof of Theorem 1 we shall furthermore need the potential theoretic notion of “thinness” of a set, introduced by Brelot in 1939: A ⊂ R^{m} is thin at a point x_{0} ∈ R^{m} \ A if either x_{0} ∉ Ā or there exists a superharmonic function u ⩾ 0 in a neighbourhood ω of x_{0} in R^{m} such that

(2.1)  u(x_{0}) < lim inf_{ω∩A∋x→x_{0}} u(x),

cf. [Br1, Sections 3, 11] or [Br2, p. 2]. It was this concept that led Cartan to introducing the fine topology, having observed that the complements of the sets A ⊂ R^{m}\{x_{0}} which are thin at x_{0} are precisely the fine neighbourhoods of x_{0} , cf. [Br1, Théorème 5] or [Br2, p. 3].—All this remains in force with R^{m} replaced more generally by an admissible Riemannian polyhedron (X, g), cf. [EF, Chapter 7].

In the rest of this section, the target (Y, d_{Y}) is a complete geodesic space (again not necessarily locally compact) of Alexandrov curvature ⩽ 1. (The case of curvature ⩽ K for a constant K > 0 reduces to the case K = 1 by rescaling the metric on Y .) All maps are assumed to have essential range contained in a closed geodesically convex ball B = B_{Y} (q, R) in Y of radius R < \frac{1}{2}π and satisfying bipoint uniqueness (i.e., geodesics in B shall be uniquely determined by their endpoints and shall vary continuously with them). For the above concepts see [EF, Chapter 2], [F1], [F2], and literature quoted there.

By way of preparation to Theorem 2 below we bring, in the following two propositions, two inequalities connected with a pair of finite energy maps φ_{0}, φ_{1}: X → B. Their distance function u(x) = d_{Y} (φ_{0}(x), φ_{1}(x)), x ∈ X, is of class W_{loc}^{1,2}(X) because φ_{i} ∈ L_{loc}^{2}(X, Y) (hence u ∈ L_{loc}^{2}(X)), and u has finite Dirichlet integral satisfying

(2.2)  ∫_{X} |∇u|^{2} dμ ⩽ 2c_{m}(E(φ_{0}) + E(φ_{1})),

as seen from (1.1), (1.2), and (1.4), by application of the triangle inequality:

|u(x) − u(x′)|^{2} ⩽ (∑_{i=0}^{1} d_{Y} (φ_{i}(x), φ_{i}(x′)))^{2} ⩽ 2 ∑_{i=0}^{1} d_{Y}^{2} (φ_{i}(x), φ_{i}(x′)).

Proposition 1. For any two finite energy maps φ_{i}: X → B, i = 0, 1, and any function κ: X → [0, 1] with finite Dirichlet integral, the map φ_{κ} = (1 − κ)φ_{0} + κφ_{1}: X → B has finite energy satisfying

(2.3)  E((1 − κ)φ_{0} + κφ_{1}) ⩽ C(E(φ_{0}) + E(φ_{1}) + ∫_{X} |∇κ|^{2} dμ),

where B = B_{Y} (q, R), and where C depends on R and dim X = m only.

By abuse of notation, the map (1−κ)φ_{0} +κφ_{1} is defined at a point x ∈ X as the point of the geodesic segment [φ_{0}(x), φ_{1}(x)] at distance κ(x)d_{Y} (φ_{0}(x), φ_{1}(x)) from φ_{0}(x). The proof of Proposition 1 is given in Section 4.

Proposition 2 (Serbinowski’s inequality, [Se]). Given two finite energy maps φ_{0}, φ_{1}: X → B with midpoint map φ_{1/2} = \frac{1}{2}φ_{0} + \frac{1}{2}φ_{1} and distance function u = d_{Y} (φ_{0}, φ_{1}) (∈ W_{loc}^{1,2}(X)), consider the finite energy map φ̂_{1/2}: X → B given by

(2.4)  φ̂_{1/2} = (1 − η)φ_{1/2} + ηq,

where the function η: X → [0, 1[ of class W_{loc}^{1,2}(X) is defined in terms of the function ϱ := d_{Y} (φ_{1/2}( ·), q) ∈ W_{loc}^{1,2}(X) by

(2.5)  sin[(1 − η)ϱ] = sin ϱ cos(\frac{1}{2}u)

at points x ∈ X where ϱ(x) > 0, and by 1 − η = cos(\frac{1}{2}u) elsewhere. Then

(2.6)  c_{m} cos^{8} R |∇\frac{tan(\frac{1}{2}u)}{cos ϱ}|^{2} ⩽ \frac{1}{2}e(φ_{0}) + \frac{1}{2}e(φ_{1}) − e(φ̂_{1/2})

μ-a.e. If u ∈ W_{0}^{1,2}(X \ bX) then d_{Y} (φ_{1/2}, φ̂_{1/2}) = ηϱ ∈ W_{0}^{1,2}(X \ bX).

By integration, (2.6) yields^{2}

(2.7)  c_{m} cos^{8} R ∫_{X} |∇\frac{tan(\frac{1}{2}u)}{cos ϱ}|^{2} dμ ⩽ \frac{1}{2}E(φ_{0}) + \frac{1}{2}E(φ_{1}) − E(φ̂_{1/2}).

Proposition 2 is an analogue of the inequality [EF, (11.2)], cf. [KS, (2.2iv)], expressing strict convexity of the energy of maps into a simply connected complete geodesic space of nonpositive curvature. The proof of Proposition 2 is based on the theory of directional energies in [KS] and is given in Section 5 below, essentially following [Se] (though of necessity invoking Proposition 1 above).

Theorem 2 is about the variational Dirichlet problem for energy minimizing maps X → Y ; and the domain space X is therefore required to be compact with nonvoid boundary bX (the union of all (m − 1)-simplexes of X contained in only one m-simplex). We denote by W^{1,2}(X, B) the class of all maps X → B of finite energy. The trace tr_{bX} φ on bX of a map φ ∈ W^{1,2}(X, B) is defined by collecting the traces tr_{σ} φ of φ on the various (m−1)-simplexes σ of bX. (If s denotes the open m-simplex of X having σ as a face, tr_{σ} φ is defined as in [KS, Section 1.12] applied to Ω = s, Γ = σ.) Then tr_{bX} φ is defined ℋ_{m−1} -a.e. on bX (ℋ_{m−1} denoting (m − 1)-dimensional Hausdorff measure). In terms of a quasicontinuous version of φ (cf. Theorem 1), we have tr_{bX} φ = φ|_{bX} ℋ_{m−1} -a.e. on bX, cf. [F2, Section 2] (this will not be used in the present paper).

2 In [Se], X is a Riemannian domain (in a Riemannian manifold). That leads to (2.6) for the present admissible Riemannian polyhedron X, simply by application to each open m-simplex of X, the (m − 1)-skeleton being a μ-nullset. In [Se], and hence (sic!) in [EF], the inequality (2.7) is unfortunately mis-stated, the “hat” over φ_{1/2} being missing (thereby invalidating the inequality, even in the case of geodesics φ_{0}, φ_{1} on the standard 2-sphere). The proof given in [Se] pertains of course to the correct version as stated above.—The parenthetical statement in [EF, p. 201] about avoiding directional energies if R < π/4 is dubious in the case of discontinuous maps.

Given a map ψ ∈ W^{1,2}(X, B), consider the subclass

(2.8)  \begin{aligned}W_{ψ}^{1,2}(X, B) & = \{φ ∈ W^{1,2}(X, B) : tr_{bX} φ = tr_{bX} ψ ℋ_{m−1}-a.e. \} \\ & = \{φ ∈ W^{1,2}(X, B) : d_{Y} (φ, ψ) ∈ W_{0}^{1,2}(X \ bX)\}. \end{aligned}

For the latter equation see [F2, Lemma 1(b)] (applied to Γ = bX), which of course also shows that, for any two maps φ_{0}, φ_{1} ∈ W_{ψ}^{1,2}(X, B), the distance function d_{Y} (φ_{0}, φ_{1}) is of class W_{0}^{1,2}(X \ bX) because tr_{bX} φ_{0} = tr_{bX} φ_{1} ℋ_{m−1} -a.e.

Theorem 2 (Serbinowski [Se]). For any map ψ ∈ W^{1,2}(X, B) there exists a unique map φ ∈ W_{ψ}^{1,2}(X, B) of least energy.

This map φ is called the solution to the variational Dirichlet problem, or the variational solution to the Dirichlet problem. Its existence and uniqueness (μa.e.) was established in [F2], assuming for existence that Y be locally compact. Serbinowski’s proof [Se] of existence and uniqueness of the variational solution, without local compactness of Y , is based on Proposition 2 above, but is otherwise quite similar to the proof by Korevaar–Schoen of the analogous result for targets of nonpositive curvature [KS, Theorem 2.2]. The proof in the present setting with a polyhedral domain is also similar, again in view of Proposition 2: Write d_{Y} (φ_{0}, φ_{1}) = u and

E_{0} = inf{E(φ) : φ ∈ W_{ψ}^{1,2}(X, B)}.

For uniqueness, let φ_{0}, φ_{1} ∈ W_{ψ}^{1,2}(X, B). Then φ_{1/2} ∈ W_{ψ}^{1,2}(X, B), by (2.8) and Proposition 1, because d_{Y} (φ_{0}, φ_{1/2}) = \frac{1}{2}u ∈ W_{0}^{1,2}(X \ bX). It follows similarly that φ̂_{1/2} ∈ W_{ψ}^{1,2}(X, B), and so E(φ̂_{1/2}) ⩾ E_{0} , because d_{Y} (φ_{1/2}, φ̂_{1/2}) = ηϱ ∈ W_{0}^{1,2}(X \ bX), by the final assertion of Proposition 2. (Alternatively, consider the traces of these functions and maps on bX.) When φ_{0}, φ_{1} are minimizers, i.e., E(φ_{0}) = E(φ_{1}) = E_{0} , (2.7) shows that the function tan(\frac{1}{2}u)/ cos ϱ of class W_{0}^{1,2}(X \bX) is constant, hence equals 0 μ-a.e. because 1 ∉ W_{0}^{1,2}(X \bX) by the Poincaré inequality [F2, Lemma 1(c)]. Consequently, u = d_{Y} (φ_{0}, φ_{1}) = 0 μ-a.e., and so φ_{0} = φ_{1} μ-a.e.

For existence consider a minimizing sequence (φ_{i}) in W_{ψ}^{1,2}(X, B). Write

u_{ij} = d_{Y} (φ_{i}, φ_{j}), φ_{ij} = \frac{1}{2}φ_{i} + \frac{1}{2}φ_{j}, ϱ_{ij} = d_{Y} (φ_{ij}, q),

and define η_{ij} as in (2.5) (now with η, ϱ, u replaced by η_{ij}, ϱ_{ij}, u_{ij}). Defining φ̂_{ij} = (1 − η_{ij})φ_{ij} + η_{ij}q, cf. (2.4), we then have φ_{ij}, φ̂_{ij} ∈ W_{ψ}^{1,2}(X, B), by the same argument as above, and hence

E(φ_{i}) + E(φ_{j}) − 2E(φ̂_{ij}) ⩽ E(φ_{i}) + E(φ_{j}) − 2E_{0},

so that the left-hand member converges to 0 as i, j → ∞. By (2.7) this implies

lim_{i,j→∞} ∫_{X} |∇\frac{tan(\frac{1}{2}u_{ij})}{cos ϱ_{ij}} |^{2} dμ = 0.

It follows by the quoted Poincaré inequality that tan(\frac{1}{2}u_{ij})/ cos ϱ_{ij} → 0 in L^{2}(X). The same therefore applies to u_{ij} itself, and so (φ_{i}) has a limit φ in the complete metric space (L^{2}(X, B), D), where D^{2}(φ′, φ′′) := ∫_{X} d_{Y}^{2} (φ′, φ′′) dμ for φ′, φ′′ ∈ L^{2}(X, B), cf. [KS, Section 1.1]. Because the energy functional is lower semicontinuous, [EF, Lemma 9.1] (this does not depend on local compactness of the target), we conclude that φ ∈ W^{1,2}(X, B) and that E(φ) ⩽ E_{0} . According to [F2, Lemma 1(a)] (extending [KS, Theorem 1.12.2]) it follows that tr_{bX} φ_{i} → tr_{bX} φ in L^{2}(bX, B) (complete, with metric analogous to D above), and so tr_{bX} φ = tr_{bX} ψ (=tr_{bX} φ_{i}). Altogether, φ ∈ W_{ψ}^{1,2}(X, B), and we conclude that E(φ) = E_{0} .

Remark 1. The solution to the above variational problem is known to have a Hölder continuous version in X \bX (continuous up to the boundary bX if tr_{bX} ψ is continuous), provided that either R < \frac{1}{4}π (rather than R < \frac{1}{2}π) or that Y is locally compact, [EF, Theorem 11.4], [F1, Theorem 2], [F2, Theorem 3].

3. Proof of Lemma 1 and Theorem 1

Proof of Lemma 1. For 0 < α < β < ∞ consider the Euclidean Riemannian domain Ω_{αβ} = {x ∈ R^{m} : α < |x| < β} and the unit vector field

ω(x) = x/|x|, x ∈ Ω_{αβ},

with associated directional 2-energy density ^{ω}e(φ) = |φ_{∗}(ω)|^{2} ⩽ C(m)^{2}e(φ) Lebesgue a.e. in Ω_{αβ}, where |φ_{∗}(ω)|(x) denotes the 1-energy density of φ in the direction ω(x), and C(m) depends on m only, [KS, Theorem 1.11]. In particular, |φ_{∗}(ω)| ∈ L^{2}(Ω_{αβ}). Denoting by S^{m−1} the unit sphere in R^{m}, with surface measure σ, we therefore have

∫_{S^{m−1}} dσ(ξ) ∫_{α}^{β} r^{m−1}r^{1−m}|φ_{∗}(ω)|(rξ) dr ⩽ C(m) ∫_{Ω_{αβ}} |x|^{1−m}\sqrt{e(φ)(x)} dx.

For α → 0, β → ∞ this leads, by the hypothesis of the lemma, to

(3.1)  ∫_{0}^{∞} |φ_{∗}(ω)|(rξ) dr < ∞  for σ -a.e. ξ ∈ S^{m−1}.

We show that (again for σ -a.e. ξ ∈ S^{m−1} ) the map φ_{ξ}: R_{+} → Y , defined Lebesgue a.e. by φ_{ξ}(r) = φ(rξ), has finite 1-energy, and its 1-energy density e(φ_{ξ}) satisfies

(3.2)  e(φ_{ξ})(r) ⩽ |φ_{∗}(ω)|(rξ)

for Lebesque a.e. r ∈ R_{+} . It will then follow that φ_{ξ} possesses a version of bounded variation on every interval ]0, β[ (0 < β < ∞), cf. [KS, Lemma 1.9.2], applied in dimension 1.

For 0 < 4ε < β − α the approximate 1-energy density ^{ω}e_{ε}(φ)(x) of φ at x = rξ in the direction ω(x) = x/|x| = ξ is defined on Ω_{α+ε,β−ε} for σ -a.e. ξ ∈ S^{m−1} by

ωe_{ε}(φ)(rξ) = \frac{1}{ε}d_{Y} (φ(rξ), φ((r + ε)ξ)), α + ε < r < β − ε.

As shown by Serbinowski [Se, Lemma 2.5] (his proof is reproduced below in the proof of Lemma 2(a) in Section 5),

d_{Y} (φ(rξ), φ((r + ε)ξ)) ⩽ ∫_{0}^{ε} |φ_{∗}(ω)|((r + t)ξ) dt

for σ -a.e. ξ ∈ S^{m−1} and for Lebesgue a.e. r ∈]α + 2ε, β − 2ε[ . For any function f ∈ Cc^{+}( ]α, β[ ) it follows for small ε > 0 that

(3.3)  \begin{aligned}∫_{α}^{β} ^{ω}e_{ε}(φ)(rξ)f(r) dr & ⩽ \frac{1}{ε} ∫_{0}^{ε} dt ∫_{α}^{β} |φ_{∗}(ω)|((r + t)ξ)f(r) dr \\ & → ∫_{α}^{β} |φ_{∗}(ω)|(rξ)f(r) dr \end{aligned}

as ε → 0, the inner integral of convolution type on the right of the inequality being continuous in t on ]0, ε[ in view of (3.1). Consequently, the lim sup of the left-hand member of (3.3) for ε → 0 is no bigger than ∫_{α}^{β} |φ_{∗}(ω)|(rξ)f(r) dr < ∞, and so φ_{ξ} has indeed (for σ -a.e. ξ ∈ S^{m−1} ) finite energy on ]α, β[ , with energy density e(φ_{ξ}) satisfying (3.2) there. Using (3.1) fully, this shows that E(φ_{ξ}) < ∞, and that (3.2) holds for a.e. r ∈ R_{+} .

Proof of Theorem 1. The assertions are easily reduced to the case where E(φ) < ∞, i.e., \sqrt{e(φ)} ∈ L^{2}(X). The case m = 1 is contained in [KS, Lemma 1.9.2], so suppose m ⩾ 2. Topological notions relative to the Cartan fine topology on X (mentioned early in Section 2) are indicated by adding “fine(ly)”. Referring to [EF, Proposition 7.8] we recall that a map φ: X → Y is quasicontinuous if and only if φ is finely continuous quasi-eveywhere, i.e., everywhere except in some polar set. A polar set is the same as a finely discrete and hence finely closed set; it is also the same as a set of capacity 0. A polar set has μ-measure 0, and a nonvoid finely open set has μ-measure > 0, cf. [F1, Section 8, Lemma 4].

Case 1. Let X = R^{m} with the Euclidean Riemannian metric. The set

(3.4)  P′ = \{a ∈ X : ∫_{X} |a − x|^{1−m}\sqrt{e(φ)(x)} dx = ∞\}

is polar, as shown by Deny [De2] with \sqrt{e(φ)} replaced by any function f ∈ L^{2}(R^{m}), f ⩾ 0. Consider a point a ∈ X \ P′ and a fine neighbourhood U of a in X. As shown in [De1], U contains for σ -a.e. ξ ∈ S^{m−1} a straight line segment [a, a + ϱ(ξ)ξ], ϱ(ξ) > 0. According to Lemma 1 we may assume that the map r ↦ φ(a + rξ) has a version of bounded variation over ]0, ϱ(ξ)], and so there exists

(3.5)  φ^{∗}(a, ξ) := ess lim_{0<r→0} φ(a + rξ) ∈ Y

for σ -a.e. ξ ∈ S^{m−1} , (Y, d_{Y}) being complete.

Denote by φ^{∗}(a, S^{m−1}) the σ -essential range of the map ξ ↦ φ^{∗}(a, ξ) of S^{m−1} into Y ; it is independent of U.

In order next to prove that φ^{∗}(a, S^{m−1}) consists of a single point, choose a dense sequence (z_{n}) in Y , and write for brevity

(3.6)  d_{Y} (φ(x), z_{n}) = v(x, z_{n}).

By [EF, Corollaries 9.1, 9.2], v( · , z_{n}) ∈ W^{1,2}(X, R) = W^{1,2}(X), and hence v( · , z_{n}) has a quasicontinuous version v^{∗}( · , z_{n}) (see e.g. [EF, proof of Theorem 7.2]). Thus there is a μ-nullset N such that, for all n,

(3.7)  v( · , z_{n}) = v^{∗}( · , z_{n})  in X \ N,

and a polar set P′′ ⊂ X such that v^{∗}( · , z_{n}) is finely continuous in X \ P′′, [EF, Proposition 7.8(c)]; then P := P′ ∪ P′′ is likewise polar.

Henceforth, let a ∈ X \P. Given ε > 0 and y ∈ φ^{∗}(a, S^{m−1}), choose i = i(y) so that d_{Y} (y, z_{i}) < ε; then

(3.8)  d_{Y} (φ^{∗}(a, ξ), z_{i}) < ε

for points ξ ∈ S^{m−1} forming a set of σ -measure > 0.

For U above take now any one of the following finely open sets containing a:

(3.9)  U_{n,ε} = {x ∈ X \ P : |v^{∗}(x, z_{n}) − v^{∗}(a, z_{n})| < ε}.

By (3.6), (3.7), and (3.9),

(3.10)  |d_{Y} (φ(x), z_{n}) − v^{∗}(a, z_{n})| < ε

for every x = a+rξ ∈ U_{n,ε} \N; and hence for σ -a.e. ξ ∈ S^{m−1} and for (Lebesgue) a.e. small r > 0, by [De1], as noted above (because a+rξ ∉ N for σ -a.e. ξ ∈ S^{m−1} and a.e. r > 0). For r → 0 this leads by (3.5) for σ -a.e. ξ ∈ S^{m−1} to

(3.11)  |d_{Y} (φ^{∗}(a, ξ), z_{n}) − v^{∗}(a, z_{n})| ⩽ ε.

Applying (3.11) with n = i from (3.8), and combining with (3.8), gives v^{∗}(a, z_{i}) < 2ε. For any other point y′ ∈ φ^{∗}(a, S^{m−1}) choose similarly j = j(y′) so that d_{Y} (y′, z_{j}) < ε and hence v^{∗}(a, z_{j}) < 2ε. For x = a + rξ ∈ U_{i,ε} ∩ U_{j,ε} \ N (≠ ∅) we thus obtain from (3.10)

d_{Y} (z_{i}, z_{j}) ⩽ d_{Y} (φ(x), z_{i}) + d_{Y} (φ(x), z_{j}) ⩽ v^{∗}(a, z_{i}) + v^{∗}(a, z_{j}) + 2ε < 6ε.

Consequently, d_{Y} (y, y′) < d_{Y} (z_{i}, z_{j}) + 2ε < 8ε. This holds for any two y, y′ ∈ φ^{∗}(a, S^{m−1}) and for any ε > 0. Thus y = y′, and φ^{∗}(a, S^{m−1}), defined for a ∈ X \ P, reduces indeed to a point φ^{∗}(a) ∈ Y ; hence (3.5) holds with φ^{∗}(a, ξ) replaced by φ^{∗}(a).

Again for a ∈ X\P and for given ε > 0, choose i so that now d_{Y} (φ^{∗}(a), z_{i}) < ε. Inserting φ^{∗}(a, ξ) = φ^{∗}(a) in (3.11) we obtain as above v^{∗}(a, z_{i}) < 2ε. For x ∈ U_{i,ε} \ N, cf. (3.9), we therefore have d_{Y} (φ(x), z_{i}) < 3ε in view of (3.10). Hence, for such x,

d_{Y} (φ(x), φ^{∗}(a)) ⩽ d_{Y} (φ(x), z_{i}) + d_{Y} (φ^{∗}(a), z_{i}) < 4ε.

Consequently, for every a ∈ X \ P,

(3.12)  φ(x) → φ^{∗}(a)  as x → a finely through X \ N.

It follows by (3.6), (3.7), and (3.12) that, for every a ∈ X \ P and every n,

(3.13)  v^{∗}(a, z_{n}) = fine lim_{X\N∋x→a} d_{Y} (φ(x), z_{n}) = d_{Y} (φ^{∗}(a), z_{n}).

If a ∈ X \ (N ∪ P) this implies φ(a) = φ^{∗}(a) because

d_{Y} (φ(a), z_{n}) = v(a, z_{n}) = v^{∗}(a, z_{n}) = d_{Y} (φ^{∗}(a), z_{n})

for every n, and because a point of Y is uniquely determined by its distances from the points z_{n} of a dense sequence in Y .

For a ∈ X\P and for given ε > 0 choose n = n(ε) so that d_{Y} (φ^{∗}(a), z_{n}) < ε; and let x ∈ U_{n,ε} \ P. By (3.13) applied with a replaced by x, by (3.9), and by (3.13) as it stands, it follows that

\begin{aligned}d_{Y} (φ^{∗}(x), φ^{∗}(a)) & ⩽ d_{Y} (φ^{∗}(x), z_{n}) + ε = v^{∗}(x, z_{n}) + ε \\ & < v^{∗}(a, z_{n}) + 2ε = d_{Y} (φ^{∗}(a), z_{n}) + 2ε < 3ε. \end{aligned}

Thus φ^{∗} is finely continuous at every point a of the finely open set X \ P. Regardless of how φ^{∗} is defined in P we conclude that φ^{∗} is a quasicontinuous version of φ, cf. [EF, Proposition 7.8(c)], because φ^{∗}(a) = φ(a) for a not in the μ-nullset N ∪ P.

Case 2. Let X be the union of closed halfspaces X_{1}, …, X_{k} (in copies X̃_{1}, …, X̃_{k} of R^{m}), disjoint save for a common boundary hyperplane—a copy X_{0} of R^{m−1} . Let τ_{i} denote reflection of X̃_{i} in X_{0} . We endow X with the Euclidean Riemannian structure and associated intrinsic distance d_{X}, cf. [EF, Section 4]. Claims:

(i) A set A ⊂ X_{i} is polar relative to X if and only if A is polar relative to X_{i}, or equivalently: relative to X̃_{i} = X_{i} ∪ τ_{i}(X_{i}) (= R^{m}).

(ii) A set A ⊂ X_{i} is thin at a point x_{0} ∈ X_{i} if and only if A is thin at x_{0} relative to X_{i}, or equivalently: relative to X̃_{i}. Note that, in case k = 2, X̃_{i} is the same as the space X itself. A permutation

π of {1, …, k} induces an isometry π: X → X such that π|_{X_{j}} (j ∈ {1, …, k}) is the obvious identification of X_{j} with X_{π(j)} , leaving X_{0} pointwise fixed. Every neighbourhood in X of a point of X_{0} contains an open neighbourhood ω of that point such that ω is symmetric, i.e., π(ω) = ω for every π.

Ad (i). Suppose first that A ⊂ X_{i} is polar in X. Clearly, A\X_{0} is then polar in X_{i} as well, so we may assume that A ⊂ X_{0} . There exists a superharmonic function u ⩾ 0 on a symmetric neighbourhood ω in X, as above, such that u = ∞ on A ∩ ω. For any permutation π, u ∘ π has the same properties because A ⊂ X_{0} , and so has u^{∗} := ∑_{π} u∘π (⩾ u). It remains to prove that u^{∗}, restricted to ω_{i} := ω ∩ X_{i}, is superharmonic relative to X_{i}. Any λ ∈ Lipc^{+}(ω_{i}) extends by symmetry across X_{0} to a symmetric (i.e., permutation invariant) function

λ^{∗} ∈ Lipc^{+}(ω), and

k ∫_{ω_{i}}⟨∇u^{∗}, ∇λ⟩ dμ = ∫_{ω}⟨∇u^{∗}, ∇λ^{∗}⟩ dμ ⩾ 0

because u^{∗} is superharmonic and symmetric in ω relative to X. This shows that (u^{∗})|_{ω_{i}} is weakly superharmonic, and even superharmonic. Indeed, there is a (unique) superharmonic function ũ on ω_{i} and a μ-nullset N ⊂ ω_{i} such that ũ = u^{∗} in ω_{i}\N. Actually we may take N ⊂ ω_{i}∩X_{0} because u^{∗} is superharmonic in the open set ω_{i} \ X_{0} . According to [EF, Proposition 7.8(d)] it follows for every x_{0} ∈ ω ∩ N = ω_{i} ∩ N that

(3.14)  ũ(x_{0}) = lim inf_{ω_{i}\N∋x→x_{0}} ũ(x) = lim inf_{ω_{i}\N∋x→x_{0}} u^{∗}(x) = u^{∗}(x_{0}).

The first, respectively last, equation (3.14) holds because ũ, respectively u^{∗}, is superharmonic in ω_{i}, respectively ω, and because ω_{i} \N is not thin at x_{0} relative to X_{i}, respectively X, cf. (2.1) (for then the nullset N would be a fine neighbourhood of x_{0} in ω_{i}, respectively the isometric images ω_{j} \N of ω_{i} \N, j = 1, …, k, would likewise be thin at x_{0} relative to X, and so would their union ω \ N, i.e., N would again be a fine neighbourhood of x_{0} in X). We conclude that ũ = u^{∗} holds not only in ω_{i} \ N, but also in ω_{i} ∩ N, by (3.14), and so u^{∗} = ũ is indeed superharmonic in all of ω_{i} relative to X_{i}.

Conversely, if A ⊂ X_{i} is polar relative to X_{i} then A \ X_{0} is polar also in X, so we may assume again that A ⊂ X_{0} . There exists a superharmonic function u_{i} ⩾ 0 in some open neighbourhood ω_{i} in X_{i} of a given point of X_{0} such that u_{i} = ∞ on A ∩ ω_{i}. By symmetry across X_{0} , ω_{i} extends to a symmetric open set ω ⊂ X with ω ∩ X_{i} = ω_{i}, and u_{i} extends to a symmetric weakly superharmonic function u ⩾ 0 on ω with u = ∞ on A ∩ ω = A ∩ ω_{i}. It is shown much like above (replacing now ω_{i}, u^{∗} by ω, u in (3.14) and exploiting the symmetry of ω and u) that u is actually superharmomnic in ω.

Ad (ii). Thinness being a local property, we may assume that x_{0} ∈ X_{0} . Suppose first that A ⊂ X_{i} is thin at x_{0} relative to X, and let u ⩾ 0 denote a superharmonic function on a symmetric open neighbourhood ω of x_{0} in X such that

(3.15)  u(x_{0}) < lim inf_{A∩ω∋x→x_{0}} u(x),

cf. (2.1). Exactly as in the proof of (i), u^{∗} := ∑_{π} u ∘ π ⩾ 0 is likewise superharmonic on ω; and u^{∗}, restricted to ω_{i} := ω ∩ X_{i}, is weakly superharmonic, and indeed superharmonic. From (3.15) follows the same with ω replaced by ω_{i} and u by u^{∗}|_{ω_{i}}. Because A ∩ ω_{i} = A ∩ ω this is seen by adding over all permutations π the inequalities

(u ∘ π)(x_{0}) ⩽ lim inf_{A∩ω∋x→x_{0}}(u ∘ π)(x),

valid by lower semicontinuity of u and hence of u ∘ π, noting that there is sharp inequality for π = id (the identity permutation) by (3.15) as it stands. It follows that A is indeed thin at x_{0} relative to X_{i}.

Conversely, if A ⊂ X_{i} is thin at x_{0} ∈ X_{0} relative to X_{i}, there exists a superharmonic function u_{i} ⩾ 0 on an open neighbourhood ω_{i} of x_{0} in X_{i} such that (3.15) holds with ω, u replaced by ω_{i}, u_{i}. Exactly as in (i), by symmetry across X_{0} , ω_{i} extends to a symmetric open neighbourhood ω of x_{0} in X with ω ∩ X_{i} = ω_{i}, and u_{i} extends to a superharmonic function u ⩾ 0 on ω. Then (3.15) holds as it stands because u(x_{0}) = u_{i}(x_{0}) and u = u_{i} on ω ∩ A = ω_{i} ∩ A. Consequently, A is indeed thin at x_{0} relative to X.

Thus prepared, we now establish Theorem 1 in the present Case 2. The set

(3.16)  P′ = \{a ∈ X : ∫_{X} d_{X}(a, x)^{1−m} \sqrt{e(φ)(x)} dx = ∞\}

(cf. (3.4)) is polar (in X). Indeed, P′ is covered by the sets P′_{i} ⊂ X_{i} obtained by replacing X by X_{i} on the right-hand side of (3.16), i ∈ {1, …, k}; and each P′_{i} is polar in X_{i} by (i) above, being polar in X̃_{i} = X_{i} ∪ τ_{i}(X_{i}) = R^{m} according to Case 1 because \sqrt{e(φ)}|_{X_{i}} ∈ L^{2}(X_{i})^{+} can be extended to a function of class L^{2}(X̃_{i})^{+} . Hence P′_{i} is polar in X, by (i), and so is therefore P′.

For any point a ∈ X \ P′, every fine neighbourhood U of a in X contains small segments issuing from a in almost all directions, and the restriction of φ to any of these segments has an essential limit at a. This follows at once from Case 1 if a ∉ X_{0} , so we may assume that a ∈ X_{0} . For i ∈ {1, …, k} write U ∩ X_{i} = U_{i}, and denote Ũ_{i} = U_{i} ∪ τ_{i}(U_{i}). Then X \ U is thin at a, and hence, by (ii) above, X_{i} \ U_{i} = X_{i} \ U is thin at a relative to X_{i} and therefore also relative to X̃_{i}. It follows by symmetry that τ_{i}(X_{i}) \ τ_{i}(U_{i}) = τ_{i}(X_{i} \ U_{i}) is thin at a relative to X̃_{i}, and so is therefore X̃_{i} \ Ũ_{i} ⊂ (X_{i} \ U_{i}) ∪ (τ_{i}(X_{i}) \ τ_{i}(U_{i})). This means that Ũ_{i} is a fine neighbourhood of a in X̃_{i} (= R^{m}). Writing φ|_{X_{i}} = φ_{i}, we have e_{ε}(φ_{i})(x) ⩽ e_{ε}(φ)(x) for every x ∈ X_{i} (because B_{X_{i}}(x, ε) ⊂ B_{X}(x, ε)). For ε → 0 it follows that e(φ_{i}) ⩽ e(φ) on X_{i}. Denote φ̃_{i}: X̃_{i} → Y the extension of φ_{i} to X̃_{i} by symmetry across X_{0} ; then e(φ̃_{i}) is the “even” extension of e(φ_{i}) from X_{i} to X̃_{i} (this clearly holds in X̃_{i} \ X_{0} , hence a.e. in X̃_{i}). It follows that E(φ̃_{i}) ⩽ 2E(φ_{i}) ⩽ 2E(φ), and (3.5) therefore holds with φ, U replaced by φ̃_{i}, Ũ_{i}, or by φ_{i}, U_{i} in particular.

The rest of the proof of Theorem 1 in Case 1 now carries over to the present Case 2, including the explicit description of a quasicontinuous version φ^{∗} of φ, whereby φ^{∗}(a) (even for a ∈ X_{0} ) is the essential radial limit of φ(x) as x → a in almost every direction (within each X_{i}), provided that a ∈ X \ P for a certain polar set P = P′ ∪ P′′ ⊂ X.

Case 3. Let X be any admissible m-dimensional polyhedron, embedded in some Euclidean space V , with each simplex affinely embedded, cf. [EF, Remark 4.1]. We give X the Euclidean Riemannian metric induced by that of V . The (m − 2)-skeleton X^{(m−2)} is a polar set [EF, Proposition 7.6], and may therefore be disregarded. The assertion of the theorem being local, we are left with the case of the star of an open (m − 1)-simplex of X; and that is covered by Case 2.

Case 4. This is the general case, where (X, g) is an arbitrary admissible Riemannian polyhedron (g simplexwise smooth). We may assume that X is compact; then g is equivalent with the Euclidean Riemannian metric g^{e} induced by that of a Euclidean space V in which X is embedded, as in Case 3. Simple estimates involving the ellipticity constant Λ of g allow us to reduce the theorem to Case 3 above, cf. [EF, Section 4 and Corollary 7.1, p. 120].

4. Proof of Proposition 1

Step 1. For any (ordered) quadruple P QRS in B = B_{Y} (q, R) (no restriction on the perimeter of the corresponding quadrilateral) with midpoints M and N of P S and QR, respectively, we show that

(4.1)  MN ⩽ \frac{c}{2}(P Q + RS) ⩽ c\sqrt{P Q^{2} + RS^{2}} ,  c = \frac{π}{\sqrt{2} cos R}.

Here and elsewhere we write briefly P Q in place of d_{Y} (P, Q) for points P, Q ∈ Y .

Consider first the case P Q, RS < ϱ := \frac{1}{2}π − R. Then [F2, Lemma 2(a)] applies and produces a (convex) comparison trapezoid P̃Q̃R̃S̃ in the unit sphere S^{2} in R^{3} , symmetric about a great circle γ̃ in S^{2} , and with side lengths

(4.2)  P̃S̃ = P S, Q̃R̃ = QR, P̃Q̃ = R̃S̃ = P Q ⋄ RS,

where the “cosine mean” a ⋄ b ∈ [0, \frac{1}{2}π] of two numbers a, b ∈ [0, \frac{1}{2}π] is defined by

(4.3)  cos(a ⋄ b) = \frac{1}{2}(cos a + cos b).

By [F2, Lemma 2(b)], MN ⩽M̃ Ñ (M̃ and Ñ denoting the midpoint of P̃S̃ and Q̃R̃, respectively). Let Õ denote the pole of γ̃ in S^{2} on the same side of γ̃ as R̃ and S̃. The cosine relation for the spherical triangle P̃Q̃Õ with angle θ at Õ may be written

sin^{2}(\frac{1}{2} P̃Q̃) = sin^{2}[\frac{1}{2}(ÕP̃ − ÕQ̃)] + sin ÕP̃ sin ÕQ̃ sin^{2}(\frac{1}{2}θ)

(also if the triangle degenerates). Because MN ⩽ M̃ Ñ = θ < π and sin ÕP̃ = cos(\frac{1}{2} P̃S̃) ⩾ cos R, etc., it follows by (4.2), (4.3) that

cos^{2} R sin^{2}(\frac{1}{2}MN) ⩽ sin^{2}(\frac{1}{2} P̃Q̃) = \frac{1}{2} sin^{2}(\frac{1}{2}P Q) + \frac{1}{2} sin^{2}(\frac{1}{2}RS)
⩽ \frac{1}{8}(P Q^{2} + RS^{2}) ⩽ \frac{1}{8}(P Q + RS)^{2},

and so indeed MN cos R ⩽ π cos R sin(\frac{1}{2}MN) ⩽ (π/\sqrt{8} )(P Q + RS).

For an arbitrary quadruple P QRS in B, (4.1) now follows by partitioning P Q, respectively SR, into n equal segments of length P Q/n, RS/n ⩽ 2R/n < ϱ. It remains to apply (4.1) to each of the n smaller quadrilaterals thus obtained, and to add up, using the triangle inequality. When applied to the quadrilateral

P QRS = φ_{0}(x)φ_{0}(x′)φ_{1}(x′)φ_{1}(x),

and hence M = φ_{1/2}(x), N = φ_{1/2}(x′), it follows from the latter inequality (4.1) in view of (1.2) that

(4.4)  E(φ_{1/2}) ⩽ c^{2}(E(φ_{0}) + E(φ_{1})).

Step 2. Suppose that u := d_{Y} (φ_{0}, φ_{1}) < ϱ. Note that u ∈ W^{1,2}(X), by (2.2). For any x ∈ X and x′ ∈ B_{X}(x, ε), [F2, Lemma 2] applies to the quadrilateral P QRS defined in the preceding paragraph. That produces a comparison trapezoid P̃Q̃R̃S̃ in S^{2} , symmetric about a great circle γ̃ in S^{2} , and with sidelengths as in (4.2). Write

\begin{aligned}φ_{κ}(x) & = (1 − κ(x))φ_{0}(x) + κ(x)φ_{1}(x), \\ φ_{κ′}(x′) & = (1 − κ(x′))φ_{0}(x′) + κ(x′)φ_{1}(x′); \end{aligned}

furthermore,

(4.5)  d_{Y} (φ_{i}(x), φ_{i}(x′)) = d_{i} (i = 0, 1),  d_{Y} (φ_{κ}(x), φ_{κ′}(x′)) = d_{κ},

and similarly with κ replaced by 1 − κ. Finally, write u(x) = u, u(x′) = u′, κ(x) = κ, κ(x′) = κ′, and consider in S^{2} the points

P̃_{κ} = (1 − κ)P̃ + κS̃, Q̃_{κ′} = (1 − κ′)Q̃ + κ′R̃.

According to [F2, Lemma 2(b)], d_{κ} ⋄ d_{1−κ} ⩽ P̃_{κ}Q̃_{κ′}, and since d_{0} ⋄ d_{1} = P̃Q̃ by (4.2), we obtain

(4.6)  \frac{1}{2}(cos d_{0} − cos d_{κ}) + \frac{1}{2}(cos d_{1} − cos d_{1−κ}) ⩽ cos P̃Q̃ − cos P̃_{κ}Q̃_{κ′}.

Let Õ denote the pole of γ̃ in S^{2} on the same side of γ̃ as R̃ and S̃, and write

(4.7)  d := d_{0} ⋄ d_{1} = P̃Q̃,

(4.8)  v := ÕP̃ = \frac{1}{2}(π + u), v′ := ÕQ̃ = \frac{1}{2}(π + u′).

Then v, v′ ∈ [\frac{1}{2}π, \frac{1}{2}π + R], and so 0 ⩽ − cot v, − cot v′ ⩽ tan R.

Define a function λ ∈ W^{1,2}(X), 0 ⩽ λ ⩽ 1, by λ = κu/v, and write λ(x) = λ, λ(x′) = λ′. Then P̃P̃_{κ} = κu = λv = λÕP̃ and Q̃Q̃_{κ′} = λ′ÕQ̃.

By the spherical cosine relation, applied to the triangles ÕP̃Q̃ and ÕP̃_{κ}Q̃_{κ′}, we therefore obtain (cf. [EF, p. 193]), after eliminating the common angle at Õ,

sin v sin v′ cos P̃_{κ}Q̃_{κ′} = sin(v − λv) sin(v′ − λ′v′) cos d
+ sin(v − λv) sin(λ′v′) cos v
+ sin v′ sin(λv) cos(v′ − λ′v′).

Insert this expression for cos P̃_{κ}Q̃_{κ′} in (4.6), together with cos P̃Q̃ = cos d = \frac{1}{2}(cos d_{0} + cos d_{1}) from (4.5), (4.7). After some manipulations serving to make (1.2) through (1.4) applicable this leads to

(4.9) \frac{1}{2}(cos d_{0} − cos d_{κ}) + \frac{1}{2}(cos d_{1} − cos d_{1−κ}) ⩽ R^{(1)} + R^{(2)} + R^{(3)} + R^{(4)},

cf. [EF, equation (10.17)], [F1, equation (8.2)]. Here

R^{(1)} := −2 sin^{2}(\frac{1}{2}d)(1 − \frac{sin(v − λv)}{sin v} \frac{sin(v′ − λ′v′)}{sin v′})

⩽ C_{1}(d_{0}^{2} + d_{1}^{2}); (4.10)

R^{(2)} := cos(λv) cos(λ′v′)(cos v − cos v′)(\frac{tan(λv)}{sin v} − \frac{tan(λ′v′)}{sin v′})

⩽ C_{2}[(cos v − cos v′)^{2} + (\frac{tan(λv)}{sin v} − \frac{tan(λ′v′)}{sin v′})^{2}];

R^{(3)} := (cos v − cos v′)^{2} \frac{sin(λv)}{sin v} \frac{sin(λ′v′)}{sin v′} ⩽ (cos v − cos v′)^{2};
R^{(4)} := 2 sin^{2} \frac{λv − λ′v′}{2} − 2 sin^{2} \frac{v − v′}{2} \frac{sin(λv)}{sin v} \frac{sin(λ′v′)}{sin v′} ⩽ \frac{1}{2}(λv − λ′v′)^{2},

where C_{1}, C_{2} and subsequent constants C_{3}, … depend on R and dim X = m only.

The power series of 1−cos t is alternating, with terms that decrease in absolute value t^{2n}/(2n)!, n ⩾ 1, when t^{2} < 4!/2! = 12. Since d_{0}, d_{κ} ⩽ 2R < π < \sqrt{12} , it follows that \frac{1}{2}d_{0}^{2} ⩾ 1 − cos d_{0} and \frac{1}{2}d_{κ}^{2} − \frac{1}{24}d_{κ}^{4} ⩽ 1 − cos d_{κ}. Inserting 1 − \frac{1}{12}d_{κ}^{2} ⩾ 1 − \frac{1}{12}π^{2} = C_{3} > 0, leads to

\frac{1}{2}C_{3}d_{κ}^{2} ⩽ \frac{1}{2}d_{0}^{2} + (cos d_{0} − cos d_{κ}).

Adding to this the corresponding inequality with κ replaced by 1−κ, and 0 by 1, we obtain for f ∈ C_{c}(X, [0, 1]) after dividing by 2ε^{m+2} and invoking (1.1), (1.3), (4.3), and (4.9):

C_{3} lim sup_{ε→0} ∫_{X}(e_{ε}(φ_{κ}) + e_{ε}(φ_{1−κ}))f dμ ⩽ lim sup_{ε→0} ∫_{X}(e_{ε}(φ_{0}) + e_{ε}(φ_{1}))f dμ

+ 4 lim sup_{ε→0} ∫_{X} f(x) dμ(x) ∫_{B_{X}(x,ε)} \frac{1}{ε^{m+2}} ∑_{j=1}^{4} R^{(j)} dμ(x′). (4.11)

Inserting the above estimates of R^{(1)} , R^{(2)} , R^{(3)} , and R^{(4)} in (4.11), we obtain by application of (1.2) through (1.4) and viewing C_{c}(X, [0, 1]) as an upper directed set:

C_{3}(E(φ_{κ}) + E(φ_{1−κ})) ⩽ (1 + 4C_{1})(E(φ_{0}) + E(φ_{1}))

+ 4C_{2} ∫_{X}(|∇ cos v|^{2} + |∇\frac{tan(λv)}{sin v}|^{2}) dμ

+ 4 ∫_{X} |∇ cos v|^{2} dμ + 2 ∫_{X} |∇(λv)|^{2} dμ

⩽ C_{4}(E(φ_{0}) + E(φ_{1}) + ∫_{X}(|∇u|^{2} + |∇κ|^{2}) dμ)

after an easy reduction, invoking (2.2), (4.8), and λ = κu/v. This leads to (2.3) in view of (2.2).

Step 3. In the general case we have, by (4.4), E(φ_{1/2}) ⩽ c^{2}(E(φ_{0}) + E(φ_{1})), hence E(φ_{1/4}), E(φ_{3/4}) ⩽ (c^{2} +c^{4})(E(φ_{0})+E(φ_{1})), etc. Choose an integer n ⩾ 1 so that 3·2^{−n}2R < ϱ. For any number α ∈ [0, 1] such that 2^{n}α is an integer it follows that

(4.12)  E(φ_{α}) ⩽ C_{5}(E(φ_{0}) + E(φ_{1})), C_{5} := c^{2} + ⋯ + c^{2n}.

For integers i ∈ [−1, 2^{n} + 1] write 2^{−n}i = α_{i}. For i ∈ [0, 2^{n} − 1] and x ∈ X define κ_{i}(x) ∈ [0, 1] as the number in the interval [α_{i−1}, α_{i+2}] nearest to κ(x); then κ_{i} ∈ W^{1,2}(X), by [EF, Proposition 5.1(c)]. Suppose κ ∈ Lip(X) (cf. Step 4 below). Let φ_{κ}^{(i)} denote the restriction of φ_{κ} to the open set

X^{(i)} := κ^{−1}( ]α_{i−1}, α_{i+2}[ )

in which κ = κ_{i} and hence φ_{κ} = φ_{κ_{i}}. Every connectivity component of X^{(i)} is an admissible Riemannian polyhedron, like X; and (in case i ∈ [1, 2^{n} − 2])

φ_{κ}^{(i)} = φ_{κ_{i}} = (1 − η)φ_{α_{i−1}} + ηφ_{α_{i+2}} in X^{(i)}

with η(x) = \frac{1}{3}2^{n}(κ(x) − α_{i−1}) ∈ ]0, 1[ for x ∈ X^{(i)} ; furthermore, η has finite Dirichlet integral, like κ; and

u^{(i)} := d_{Y} (φ_{α_{i−1}}, φ_{α_{i+2}}) = (α_{i+2} − α_{i−1})d_{Y} (φ_{0}, φ_{1}) = 3·2^{−n}u < ϱ,

by choice of n (since u := d_{Y} (φ_{0}, φ_{1}) ⩽ 2R). We therefore obtain from the case of (2.3) established in Step 2, but now applied with X, φ_{0}, φ_{1}, κ replaced by X^{(i)}, φ_{α_{i−1}}, φ_{α_{i+2}}, η:

E(φ_{κ}^{(i)}) ⩽ C_{6}(E(φ_{α_{i−1}}) + E(φ_{α_{i+2}}) + ∫_{X^{(i)}} |∇η|^{2} dμ).

Here |∇η| = \frac{1}{3}2^{n}|∇κ|. In view of (4.12) applied to α = α_{i−1}, α_{i+2} , this establishes (2.3) restricted to X^{(i)} with i ∈ [1, 2^{n}−2]; the remaining cases i = 0 and i = 2^{n}−1 are handled similarly. Because the X^{(i)} , i ∈ [0, 2^{n} − 1], cover X we have now proved (2.3) for the whole of X.

Step 4. Suppose first that X is compact, hence κ ∈ W^{1,2}(X). To remove the extra hypothesis κ ∈ Lip(X) made in Step 3, we approximate κ in W^{1,2} -norm by a sequence of functions κ_{j} ∈ Lip(X), 0 ⩽ κ_{j} ⩽ 1; then φ_{κ_{j}} → φ_{κ} in L^{2}(X, B) because d_{Y} (φ_{κ_{j}}, φ_{κ}) = |κ_{j} − κ|d_{Y} (φ_{0}, φ_{1}) with d_{Y} (φ_{0}, φ_{1}) ⩽ 2R. Since (2.2) holds for each κ_{j} in place of κ, and since ∫_{X} |∇κ_{j}|^{2} dμ → ∫_{X} |∇κ|^{2} dμ < ∞, we have sup_{j} E(φ_{κ_{j}}) < ∞. Consequently, (2.2) follows for j → ∞ (and with the same constant C) by lower semicontinuity of energy [EF, Lemma 9.1].

Finally, if X is noncompact, exhaust it by an increasing sequence of finite subpolyhedra X_{j}. From (2.2) applied to the restrictions of φ_{0} , φ_{1} , and κ to each X_{j} follows that φ_{κ} ∈ W_{ψ}^{1,2}(X, B), and in the limit as j → ∞ that (2.2) holds as it stands.

5. Proof of Proposition 2

In our presentation of Serbinowski’s proof of his inequality (2.7) in Proposition 2 above we bring in Lemma 4 below a variant of the proof of [Se, Lemma 2.9]; and in the proof of Lemma 3 we bring some details underlying [Se, proof of Lemma 1.10], drawing for that purpose on [F2, Lemma 2]. Furthermore Proposition 1 above is indispensable.

In this section the domain of maps will be a Riemannian domain (Ω, g) in the sense of Korevaar and Schoen [KS, Section 1.1], i.e., a connected open subset of a Riemannian manifold (M, g) such that the metric completion \overline{Ω} is a compact subset of M. The volume measure on M is denoted by μ = μ_{g}. We refer to [KS, Section 1.7] for the class Γ(T Ω) of all Lipschitz vector fields ω on \overline{Ω}, and the associated directional energies ^{ω}E(φ) and directional energy densities ωe(φ) = |φ_{∗}(ω)|^{2} of maps φ ∈ W^{1,2}(Ω, Y), i.e., maps Ω → Y of finite 2-energy. For any open set U ⊂ Ω write U_{ε} = {x ∈ U : d_{Ω}(x, ∂U) > ε}, ε > 0. We sometimes abbreviate d_{Y} (P, Q) = P Q for points P, Q ∈ Y . In the following lemma, Y may be any complete metric space.

Lemma 2 ([Se, Lemmas 2.5, 2.6, 2.7]). Let φ ∈ W^{1,2}(Ω, Y), let ω ∈ Γ(T Ω) be a unit vector field, and denote by (x̄(x, t))_{t} the flow generated by ω. Then, for ε > 0,

(a) d_{Y} (φ(x), φ(x̄(x, ε))) ⩽ ∫_{0}^{ε} |φ_{∗}(ω)|(x̄(x, t)) dt for μ-a.e. x ∈ Ω_{2ε}.

(b) The functions x ↦ (1/ε)d_{Y} (φ(x), φ(x̄(x, ε))) are locally uniformly square μ-integrable for small ε, and

(5.1)  \frac{1}{ε}d_{Y} (φ(x), φ(x̄(x, ε))) → |φ_{∗}(ω)|  in L_{loc}^{2}(Ω) as ε → 0.

(c) Defining for α > 0

(5.2)  A_{α}^{ε} = A_{α}^{ε}(φ) = {x ∈ Ω_{2ε} : d_{Y} (φ(x), φ(x̄(x, ε))) > α}

we have for any f ∈ C_{c}(Ω)

(5.3) ^{ω}E^{φ}(f) := ∫_{Ω} |φ_{∗}(ω)|^{2}f dμ = lim_{ε→0} ∫_{Ω\A_{α}^{ε}} \frac{d_{Y}^{2} (φ(x), φ(x̄(x, ε)))}{ε^{2}}f(x) dμ(x).

Proof. Ad (a). Let U ⊂ Ω_{ε} be a coordinate patch such that, in coordinates x^{i} on U, we have ω = ∂/∂x^{1} . Then dμ(x) = \sqrt{det g(x)} dx, and for f ∈ Cc^{+}(U_{ε}) and n = 1, 2, …, writing f̃(x) = f(x)\sqrt{det g(x)} ,

∫_{Ω} \frac{d_{Y} (φ(x), φ(x̄(x, ε)))}{ε}f(x) dμ(x) = ∫_{U} \frac{d_{Y} (φ(x), φ(x + εω))}{ε}f̃(x) dx

⩽ ∫_{U} f̃(x)\frac{1}{ε} ∑_{k=0}^{n−1} d_{Y} (φ(x + \frac{k}{n}εω), φ(x + \frac{k + 1}{n}εω)) dx

= ∫_{U} \frac{1}{n} ∑_{k=0}^{n−1} f̃(x − \frac{k}{n}εω)\frac{d_{Y} (φ(x), φ(x + εω/n))}{(ε/n)} dx.

By [KS, Theorem 1.8.1],

\frac{d_{Y} (φ(x), φ(x + εω/n))}{(ε/n)} → |φ_{∗}(ω)|(x)

weakly as measures for n → ∞. Furthermore,

\frac{1}{n} ∑_{k=0}^{n−1} f̃(x − \frac{k}{n}εω) → \frac{1}{ε} ∫_{0}^{ε} f̃(x − tω) dt

uniformly for x ∈ U as n → ∞. Thus

\begin{matrix}∫_{Ω} d_{Y} (φ(x), φ(x̄(x, ε)))f(x) dμ(x) ⩽ ∫_{U} ∫_{0}^{ε} f̃(x − tω)|φ_{∗}(ω)|(x) dt dx \\ ⩽ ∫_{Ω} f(x) ∫_{0}^{ε} |φ_{∗}(ω)|(x̄(x, t)) dt dμ(x), \end{matrix}

and Part (a) of the lemma follows by varying f and U.

Ad (b). For δ > 0 define

(5.4) V (δ) = sup{∫_{A} |φ_{∗}(ω)|^{2} dμ : A ⊂ Ω, A is μ-measurable, μ(A) ⩽ δ}.

Since |φ_{∗}(ω)| ∈ L^{2}(Ω) it follows that V (δ) ↘ 0 as δ ↘ 0.

Let K ⋐ Ω be μ-measurable, and choose ε > 0 so that K ⋐ Ω_{2ε}. For |t| ⩽ ε define

K^{t} = {x̄(x, t) : x ∈ K} (⋐ Ω_{ε}).

Then we have μ(K^{t}) ⩽ cμ(K) with c independent of ε and t. By Cauchy’s inequality, applied to the integral in (a), we obtain after integrating over K

∫_{K} \frac{d_{Y}^{2} (φ(x), φ(x̄(x, ε)))}{ε^{2}} dμ(x) ⩽ \frac{1}{ε} ∫_{K} ∫_{0}^{ε} |φ_{∗}(ω)|^{2}(x̄(x, t)) dt dμ(x).

When combined with

∫_{K} |φ_{∗}(ω)|^{2}(x̄(x, t)) dμ(x) ⩽ C ∫_{K^{−t}} |φ_{∗}(ω)|^{2} dμ,

with C independent of ε and t for |t| ⩽ ε, this leads to

(5.5) ∫_{K} \frac{d_{Y}^{2} (φ(x), φ(x̄(x, ε)))}{ε^{2}} dμ(x) ⩽ \frac{1}{ε} ∫_{0}^{ε} C V (μ(K^{−t})) dt ⩽ C V (cμ(K)),

and the uniform (in small ε) square integrability over K follows.

For brevity, write (1/ε)d_{Y} (φ(x), φ(x̄(x, ε))) = u_{ε}(x), x ∈ K. According to [KS, Theorem 1.9.6], u_{ε} → |φ_{∗}(ω)| pointwise μ-a.e. in K as ε → 0. By Egoroff’s theorem there exists for any δ > 0 a μ-measurable set E ⊂ K such that μ(K \ E) < δ and that u_{ε} → |φ_{∗}(ω)| uniformly on E as ε → 0. Then

\begin{aligned}∫_{K}(u_{ε} − |φ_{∗}(ω)|)^{2} dμ & ⩽ μ(K) sup_{E}(u_{ε} − |φ_{∗}(ω)|)^{2} \\ & + 2 ∫_{K\E} u_{ε}^{2} dμ + 2 ∫_{K\E} |φ_{∗}(ω)|^{2} dμ, \end{aligned}

and here the last two integrals are majorized by CV (cδ) and V (δ), respectively, by (5.5) (applied with K replaced by K \ E, whereby we may keep C) and (5.4). Making first ε → 0 and then δ → 0 leads to (5.1).

Ad (c). We may assume that φ has been chosen within its equivalence class so that the map [0, ε] ∋ t ↦ φ(x̄(x, t)) is (Hölder) continuous for every x ∈ Ω_{ε} off some μ-nullset N, [KS, Lemma 1.9.2]. Define for α > 0

Ã_{α}^{ε} = {x ∈ Ω_{2ε} : d_{Y} (φ(x), φ(x̄(x, t))) > α for some t ∈ [0, ε]}.

Then A_{α}^{ε} ⊂ Ã_{α}^{ε}, and both sets are μ-measurable, the latter because it suffices to consider rational t when x ∉ N. For ε → 0 through a decreasing sequence, Ã_{α}^{ε} decreases to ⋂_{ε} Ã_{α}^{ε} ⊂ N, and hence μ(Ã_{α}^{ε}) → 0 as ε → 0; in particular,

(5.6)  μ(A_{α}^{ε}) → 0  as ε → 0.

In proving (5.3) we may assume that f ⩾ 0. With integration over all of Ω, this limit relation is contained in [KS, Theorem 1.8.1]. As it stands, (c) therefore follows from (5.5) applied to K = K_{ε} := A_{α}^{ε} ∩ supp f, taking ε small enough so that supp f ⊂ Ω_{2ε}:

∫_{A_{α}^{ε}} \frac{d_{Y}^{2} (φ(x), φ(x̄(x, ε)))}{ε^{2}}f(x) dμ(x) ⩽ C∥f∥_{L^{∞}}V (cμ(K_{ε})).

By (5.6), μ(K_{ε}) → 0, hence V (cμ(K_{ε})) → 0 as ε → 0.

It is not known to the author whether (b) and (c) in Lemma 2 have analogues for the “full” energy functional E^{φ}(f) = ∫_{Ω} e(φ)f dμ in place of the directional energy functional ^{ω}E^{φ}(f). Actually, rather than using (5.3), we shall in the sequel employ (5.6) combined with

(5.7)  ess lim_{ε→0} \frac{d_{Y}^{2} (φ(x), φ(x̄(x, ε)))}{ε^{2}} = |φ^{∗}(ω)|^{2}(x),

valid for μ-a.e. x ∈ Ω and for any φ ∈ W^{1,2}(Ω, Y), according to [KS, Theorem 1.9.6]. For a Sobolev function u ∈ W^{1,2}(Ω) this reads

(5.8)  ess lim_{ε→0} \frac{|u(x) − u(x̄(x, ε))|^{2}}{ε^{2}} = |∇_{ω}u|^{2}(x)

for μ-a.e. x ∈ Ω, as is well known.

Henceforth we consider maps φ: Ω → B with (Y, d_{Y}) and B = B_{Y} (q, R) as in the paragraph containing (2.1); in particular R < \frac{1}{2}π.

Lemma 3 ([Se, Lemma 2.8]). Let φ_{0}, φ_{1} ∈ W^{1,2}(Ω, B). For any unit vector field ω ∈ Γ(T Ω), generating a flow (x̄(x, t))_{t}, we have

(5.9)  |(φ_{1/2})_{∗}(ω)|^{2} cos^{2}(\frac{1}{2}u) ⩽ \frac{1}{2}|(φ_{0})_{∗}(ω)|^{2} + \frac{1}{2}|(φ_{1})_{∗}(ω)|^{2} − \frac{1}{4}|∇_{ω}u|^{2}

μ-a.e. in Ω, whereby u(x) = d_{Y} (φ_{0}(x), φ_{1}(x)), while ∇_{ω}u = ⟨ω, ∇u⟩ is the derivative of u ∈ W^{1,2}(Ω) in the direction ω = ω(x).

It is the factor cos^{2}(\frac{1}{2}u) ⩽ 1 in (5.9) which causes the need for considering the map φ̂_{1/2} instead of φ_{1/2} in Serbinowski’s inequality (2.6).

Proof. Proposition 1 shows that φ_{1/2} ∈ W^{1,2}(Ω, B), B being convex. Let 0 < α ⩽ \frac{1}{2}π − R. For ε > 0 and x ∈ Ω_{2ε} \ (A_{α}^{ε}(φ_{0}) ∪ A_{α}^{ε}(φ_{1}) ∪ A_{α}^{ε}(φ_{1/2})), cf. (5.2), hence

(5.10)  d_{i} := d_{Y} (φ_{i}(x), φ_{i}(x̄(x, ε))) ⩽ α,  i = 0, 1, \frac{1}{2},

the ordered quadruple

P = φ_{0}(x), Q = φ_{0}(x̄(x, ε)), R = φ_{1}(x̄(x, ε)), S = φ_{1}(x)

defines a quadrilateral P QRS in Y , and 2 max{P Q, RS} + QR + SP < 2π. According to [F2, Lemma 2(a)], P QRS has a comparison trapezoid P̃Q̃R̃S̃ in the unit sphere S^{2} in R^{3} , i.e., a convex spherical quadrilateral, symmetric about a great circle γ̃ in S^{2} , and with side lengths

P̃S̃ = P S = u(x),  Q̃R̃ = QR = u(x̄(x, ε)),
P̃Q̃ = R̃S̃ = P Q ⋄ RS = d_{0} ⋄ d_{1},

where the cosine mean d_{0} ⋄ d_{1} ∈ [0, \frac{1}{2}π] of numbers d_{0}, d_{1} ∈ [0, \frac{1}{2}π] is defined by cos(d_{0} ⋄ d_{1}) = \frac{1}{2}(cos d_{0} + cos d_{1}), i.e., 2 sin^{2}[\frac{1}{2}(d_{0} ⋄ d_{1})] = sin^{2}(\frac{1}{2}d_{0}) + sin^{2}(\frac{1}{2}d_{1}).

If Õ ∈ S^{2} denotes the pole of γ̃ on the same side of γ̃ as R̃ and S̃, the spherical cosine relation applied to P̃Q̃Õ yields (also if P̃Q̃R̃S̃ degenerates)

(5.11)  2 cos \frac{u}{2} cos \frac{ū}{2} sin^{2} \frac{θ}{2} = sin^{2} \frac{d_{0}}{2} + sin^{2} \frac{d_{1}}{2} − 2 sin^{2} \frac{u − ū}{4}.

Here u = u(x), ū = u(x̄(x, ε)), while θ denotes the angle at Õ, i.e., the spherical distance between the midpoints M̃ , Ñ ∈ γ̃ of P̃S̃ and Q̃R̃, respectively. According to [F2, Lemma 2(b)], the distance d_{1/2} = d_{Y} (φ_{1/2}(x), φ_{1/2}(x̄)) = MN between the midpoints M and N of P S and QR, respectively, satisfies d_{1/2} ⩽M̃ Ñ = θ. By the triangle inequality, |u − ū| ⩽ d_{0} + d_{1} ⩽ 2α, cf. (5.10).

Becausee 2 sin^{2}(\frac{1}{2}t) = \frac{1}{2}t^{2}(1 − O(t^{2})) for t = d_{0}, d_{1}, d_{1/2} , and \frac{1}{2}|u − ū| (all ⩽ α), we obtain from (5.11) after division by \frac{1}{2}ε^{2} cos(\frac{1}{2}u) cos(\frac{1}{2} ū):

\frac{d_{1/2}^{2}}{ε^{2}} ⩽ \frac{1 + O(α)}{cos^{2}(\frac{1}{2}u)}(\frac{d_{0}^{2} + d_{1}^{2}}{2ε^{2}} − \frac{(u − ū)^{2}}{4ε^{2}} (1 − O(α^{2})))

μ-a.e. in Ω_{2ε}. Making first ε → 0 and then α → 0 leads in view of [KS, Theorem 1.9.6] to (5.9). For this, apply (5.6), (5.7) to φ_{0} , φ_{1} , φ_{1/2} ; and (5.8) to u ∈ W^{1,2}(Ω), cf. (2.2), whilst noting that 1/ cos^{2}(\frac{1}{2}u) is bounded and measurable.

Lemma 4 ([Se, Lemma 2.9]). Let φ ∈ W^{1,2}(Ω, B) (B = B_{Y} (q, R)), and write d_{Y} (φ(x), q) = ϱ(x), x ∈ Ω. For any function η ∈ W^{1,2}(Ω), 0 ⩽ η < 1, the map φ̂:= (1 − η)φ + ηq ∈ W^{1,2}(Ω, B) satisfies

(5.12)  |φ̂_{∗}(ω)|^{2} ⩽ \frac{sin^{2}(ϱ − ηϱ)}{sin^{2} ϱ}(|φ_{∗}(ω)|^{2} − |∇_{ω}ϱ|^{2}) + |∇_{ω}(ϱ − ηϱ)|^{2}

μ-a.e. in Ω for any unit vector field ω ∈ Γ(T Ω) generating a flow (x̄(x, t))_{t}.

Proof. Proposition 1 shows that φ̂ ∈ W^{1,2}(Ω, B), B being convex; and (2.2) shows that ϱ ∈ W^{1,2}(Ω). Given ε > 0 and 0 < α ⩽ \frac{1}{2}π − R, let x ∈ Ω_{2ε} \ (A_{α}^{ε}(φ) ∪ A_{α}^{ε}(φ̂)), cf. (5.2). Write η(x) = η, ϱ(x) = ϱ, x̄(x, ε) = x̄, and

d_{Y} (φ(x), φ(x̄)) = d, d_{Y} (φ̂(x), φ̂(x̄)) = d̂, η(x̄) = η̄, ϱ(x̄) = ϱ̄.

Then d, d̂⩽ α. At points x where d̂⩾ d we find that

(5.13)  \frac{1}{2!}(d̂^{2} − d^{2}) − \frac{1}{4!}(d̂^{4} − d^{4}) ⩽ cos d − cos d̂

because d ⩽ d̂⩽ π < \sqrt{15} . Indeed, in the series (with terms of alternating signs)

∑_{n=2}^{∞}(−1)^{n} \frac{d̂^{2n} − d^{2n}}{(2n)!} = cos d̂− cos d + \frac{d̂^{2} − d^{2}}{2!}

the terms do not increase in absolute value because

d̂^{2n+2} − d^{2n+2} ⩽ (d̂^{2n} − d^{2n})(d̂^{2} + d^{2})

and d̂^{2} + d^{2} ⩽ 2π^{2} < 30 ⩽ (2n + 1)(2n + 2) for n ⩾ 2. The sum of the series is therefore no bigger than its first term \frac{1}{4!}(d̂^{4} − d^{4}), as claimed in (5.13).

By triangle comparison for the triangle φ(x)φ(x̄)q we obtain d̂⩽ d+ \frac{1}{2}π|η−η̄|, cf. [F1, equation (8.1)], and hence d̂^{4} − d^{4} ⩽ (d^{2} + |η̄ − η|^{2})O(|η − η̄|). Inserting this in (5.13) yields

(5.14)  \frac{1}{2} d̂^{2} ⩽ \frac{1}{2}d^{2} + (cos d − cos d̂) + (d^{2} + |η̄ − η|^{2})O(|η − η̄|).

The same holds for d̂⩽ d, even without the remainder term, as shown by applying the meanvalue theorem to the function − cos \sqrt{t} with derivative ⩽ \frac{1}{2} for t > 0.^{3}

3 Arguments and estimates corresponding to those of the paragraphs containing (5.13) and (5.14) should have been used in the remainder term in [F1, equation (8.3)] and in the former inequality [EF, equation (10.17)], both of which therefore should be replaced (without further consequences) by

\frac{1}{2} (d_{λ}^{2} − d^{2}) ⩽ (cos d − cos d_{λ}) + (d^{2} + |λ − λ′|^{2})∥λ∥_{Lip}O(ε).

Now combine (5.14) with the following analogue of (4.9) (cf. [EF, equation (10.17)] or [F1, equation (8.2)]):

(5.15)  cos d − cos d̂⩽ R^{(1)} + R^{(2)} + R^{(3)} + R^{(4)},

where the R^{(j)} are defined by the respective former expression in (4.10), etc. (and likewise defined in [EF, p. 193]), now with v, λ replaced by ϱ, η. Note that 2 sin^{2}(\frac{1}{2}d) = \frac{1}{2}d^{2}(1−O(α)), and that the function [sin(ϱ−ηϱ)]/ sin ϱ of ϱ ∈ ]0, R[ and η ∈ [0, 1[, extends smoothly to [0, R]×[0, 1], and therefore is of class W^{1,2}(Ω) as a function of x ∈ Ω, along with ϱ and η, by the chain rule [EF, Lemma 5.2]. Inserting these expressions for the R^{(j)} in (5.15) combined with (5.14), dividing by \frac{1}{2}ε^{2} , and making first ε → 0 whilst using (5.6), (5.7), (5.8), and then α → 0, we obtain μ-a.e. in Ω (cf. end of proof of the preceding lemma):

|φ̂_{∗}(ω)|^{2} ⩽ \frac{sin^{2}(ϱ − ηϱ)}{sin^{2} ϱ}|φ_{∗}(ω)|^{2} + 2 cos^{2}(ηϱ)⟨∇_{ω} cos ϱ, ∇_{ω} \frac{tan(ηϱ)}{sin ϱ}⟩

+ 2\frac{sin^{2}(ηϱ)}{sin^{2} ϱ}|∇_{ω} cos ϱ|^{2} + |∇_{ω}(ηϱ)|^{2} − \frac{sin^{2}(ηϱ)}{sin^{2} ϱ}|∇_{ω}ϱ|^{2}.

After further applications of the chain rule this reduces to (5.12) because

sin^{2}(ϱ − ηϱ) + 2 sin ϱ sin(ηϱ) cos(ϱ − ηϱ) ≡ sin^{2} ϱ + sin^{2}(ηϱ).

Proof of Proposition 2 (cf. [Se]). In proving (2.6) and hence (2.7), it suffices to consider each m-simplex of X separately, and so we may assume that X is an open simplex and hence a Riemannian domain Ω, cf. footnote 2 to (2.7). In view of Lemma 4 (to be applied to φ_{1/2} ∈ W^{1,2}(Ω), cf. Proposition 1) we define the function η on Ω, 0 ⩽ η < 1, by (2.5):

(5.16)  \frac{sin[(1 − η)ϱ]}{sin ϱ} = cos \frac{u}{2},

understood so that η(x) = 1 − cos(\frac{1}{2}u(x)) when ϱ(x) = 0. Then

1 − η = \frac{1}{ϱ} Arcsin(cos(\frac{1}{2}u) sin ϱ)

extended to a smooth function of (ϱ, u) ∈ [0, R]×[0, 2R]. Since ϱ, u ∈ W^{1,2}(Ω), cf. (2.2), it follows by the chain rule [EF, Lemma 5.2] that η ∈ W^{1,2}(Ω). Moreover, (1 − η)ϱ ∈ W^{1,2}(Ω), and

|∇_{ω}[(1 − η)ϱ]|^{2} = \frac{|∇_{ω}(cos(\frac{1}{2}u) sin ϱ)|^{2}}{1 − cos^{2}(\frac{1}{2}u) sin^{2} ϱ} .

By Lemma 4, applied to φ_{1/2} , and Lemma 3 we obtain, after inserting (5.16),

(5.17)

|(φ̂_{1/2})_{∗}(ω)|^{2} ⩽ \frac{1}{2}|(φ_{0})_{∗}(ω)|^{2} + \frac{1}{2}|(φ_{1})_{∗}(ω)|^{2} − \frac{1}{4}|∇_{ω}u|^{2}

− cos^{2} \frac{u}{2}|∇_{ω}ϱ|^{2} + \frac{|∇_{ω}(cos(\frac{1}{2}u) sin ϱ)|^{2}}{1 − cos^{2}(\frac{1}{2}u) sin^{2} ϱ} .

Simple computations show that

\frac{1}{4}|∇_{ω}u|^{2} + cos^{2} \frac{u}{2}|∇_{ω}ϱ|^{2} − \frac{|∇_{ω}(cos(\frac{1}{2}u) sin ϱ)|^{2}}{1 − cos^{2}(\frac{1}{2}u) sin^{2} ϱ}

= \frac{cos^{4} ϱ cos^{4}(\frac{1}{2}u)}{1 − sin^{2} ϱ cos^{2}(\frac{1}{2}u)}|∇_{ω} \frac{tan(\frac{1}{2}u)}{cos ϱ}|^{2}

⩾ cos^{8} R|∇_{ω} \frac{tan(\frac{1}{2}u)}{cos ϱ}|^{2}.

Inserting this in (5.17) leads to

|(φ̂_{1/2})_{∗}(ω)|^{2} ⩽ \frac{1}{2}|(φ_{0})_{∗}(ω)|^{2} + \frac{1}{2}|(φ_{1})_{∗}(ω)|^{2} − cos^{8} R|∇_{ω} \frac{tan(\frac{1}{2}u)}{cos ϱ}|^{2}

μ-a.e. in Ω. And integrating that with respect to ω over the unit sphere S^{m−1} in R^{m} establishes (2.6), by [KS, Theorem 1.8.1]:

(5.18)  e(φ̂_{1/2}) ⩽ \frac{1}{2}e(φ_{0}) + \frac{1}{2}e(φ_{1}) − c_{m} cos^{8} R|∇\frac{tan(\frac{1}{2}u)}{cos ϱ}|^{2}

μ-a.e. in Ω. Writing ∇[(tan(\frac{1}{2}u))/ cos ϱ] = ξ ∈ T (Ω), we have in fact

∫_{S^{m−1}}⟨ω, ξ⟩^{2} dσ(ω) = |ξ|^{2} ∫_{S^{m−1}} ω_{1}^{2} dσ(ω) = \frac{|ξ|^{2}}{m} |S^{m−1}| = |ξ|^{2}ω_{m},

where ω_{m} denotes the volume of the unit ball in R^{m}. This leads to (5.18) with c_{m} = ω_{m}; and that is the same as (2.6). ^{4} 458 Bent Fuglede

4 Actually, we have defined c_{m} = ω_{m}/(m + 2) in [EF] because it was chosen, in the definition of the approximate energy density e_{ε}(φ) in [EF, (9.2)], (cf. (1.1) above) to suppress the factor m+2 (occurring as n+p with p = 2 in [KS, (1.2vii)]). In [EF, Corollary 9.2], c_{m} is inadvertently written as ω_{m}/(m + 1), while the correct expression c_{m} = ω_{m}/(m + 2) is written elsewhere in [EF], in particular on [EF, pp. 168, 171].

Finally suppose that u ∈ W_{0}^{1,2}(X), and let us establish that σ := ηϱ = d_{Y} (φ_{1/2}, φ̂_{1/2}) ∈ W_{0}^{1,2}(X). From sin(ϱ − σ) = sin ϱ cos(\frac{1}{2}u), by (5.16), we easily obtain

(5.19)  sin(\frac{1}{2}σ) cos(ϱ − \frac{1}{2}σ) = sin ϱ sin^{2}(\frac{1}{4}u),

where ϱ − \frac{1}{2}σ = (1 − \frac{1}{2}η)ϱ < R < \frac{1}{2}π. It follows that 1/ cos(ϱ − \frac{1}{2}σ) ∈ W^{1,2}(X) and hence sin(\frac{1}{2}σ) ∈ W_{0}^{1,2}(X), so that indeed σ ∈ W_{0}^{1,2}(X). Finally, by (5.19),

η = \frac{σ}{ϱ} = \frac{σ}{sin(\frac{1}{2}σ)} \frac{sin ϱ}{ϱ} \frac{sin^{2}(\frac{1}{4}u)}{cos(ϱ − \frac{1}{2}σ)} ∈ W_{0}^{1,2}(X)

after removing the singularities of σ/ sin(\frac{1}{2}σ) and (sin ϱ)/ϱ.

References

[Br1] Brelot, M.: Introduction axiomatique de l’effilement. - Ann. Mat. Pura Appl. 57, 1962, 77–96.

[Br2] Brelot, M.: On Topologies and Boundaries in Potential Theory. - Lecture Notes in Math. 175, Springer-Verlag, Berlin, 1971.

[De1] Deny, J.: Un théorème sur les ensembles effilés . - Ann. Univ. Grenoble 23, 1948, 139–142.

[De2] Deny, J.: Sur la convergence de certaines intégrales de la théorie du potentiel. - Arch. Math. 5, 1954, 367–370.

[EF] Eells, J., and B. Fuglede: Harmonic Maps between Riemannian Polyhedra. - Cambridge Tracts in Math. 142, Cambridge University Press, 2001.

[F1] Fuglede, B.: Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature. - Calc. Var. 16, 2003, 375–403.

[F2] Fuglede, B.: The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature. - Trans. Amer. Math. Soc. (to appear).

[KS] Korevaar, N. J., and R. M. Schoen: Sobolev spaces and harmonic maps for metric space targets. - Comm. Anal. Geom. 1, 1993, 561–659. Reprinted as Chapter X in Lectures on Harmonic Maps by R. Schoen and S. T. Yau, Conf. Proc. and Lecture Notes in Geometry and Topology, Vol. II, 1997, 204–310.

[Se] Serbinowski, T.: Harmonic maps into metric spaces with curvature bounded above. - Thesis, Univ. Utah, 1995.

Received 12 February 2003