On the Intrinsic Differentiability Theorem of Gromov-Schoen

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1 On the Intrinsic Differentiability Theorem of Gromov-Schoen Georgios Daskalopoulos Brown University Chikako Mese 2 Johns Hopkins University cmese@math.jhu.edu Abstract In this note, we generalize the key technical result of Gromov and Schoen [GS] Theorem 5. which states that a harmonic map from a smooth Riemannian domain to an F -connected complex can be well approximated near a point of order one by a homogeneous degree one map. The generalization provided here is necessary in order to estimate the codimension of the singular set of energy minimizing maps into a general class of Riemannian complexes and to prove superrigidity for targets (for example, hyperbolic buildings) that are not covered by the work of Gromov and Schoen. Introduction In their pioneering work, Gromov and Schoen [GS] developed the theory of harmonic maps when the target space is a non-positively curved Riemannian simplicial complex. This theory was substantially expanded in the subsequent papers of Korevaar and Schoen [KS] [KS2] to include general non-positively curved metric space targets. The main application of the harmonic map theory of Gromov and Schoen is to establish rank one superrigidity for representations into p-adic groups. The Archimedean rank one superigidity was previously established by Corlette [C] who used harmonic supported by research grant NSF DMS supported by research grant NSF DMS

2 map theory together with a new Bochner formula to prove the result. The proof of Gromov-Schoen s non-archimedean superrigidity theorem involves a delicate analysis of the local behavior of harmonic maps from smooth Riemannian manifolds into Euclidean buildings. In particular, they show that a harmonic map from a smooth Riemannian domain to a k-dimensional NPC (non-positively curved) F-connected complex locally maps into a Euclidean space outside a set of codimension at least 2. Here, a k-dimensional F- connected complex is a locally finite Riemannian complex such that any two adjacent simplices are contained in a k-flat, by which we mean a totally geodesic subcomplex isometric to a region in R k. In [DM] and [DM2], we consider a more general class of NPC complexes that are not necessarily Euclidean. Such complexes can have arbitrary Riemannian metrics but we assume that they are DM-connected in the sense that any two adjacent simplices are contained in a DM, an image of a differentiable Riemannian manifold isometrically embedded in Y. One of the technical points needed in the analysis of [DM] and [DM2] is a generalization of [GS] Theorem 5. and the purpose of this note is to provide such a generalization. The main ideas are all due to Gromov-Schoen. Our contribution consists of two observations. First, we replace the essential regularity assumption of the complex with an assumption of essential regularity of the tangent cone. More importantly, we loosen the assumption of closeness from C 0 -close to L -close and still draw the same conclusion. 2 Definitions A metric space (Y, d) is called an NPC space if (i) the space (Y, d) is a length space. That is, for any two points P and Q in Y, there exists a rectifiable curve c so that the length of c is equal to d(p, Q). We call such distance realizing curve a geodesic and (ii) for any three points P, R, Q Y, let c : [0, l] Y be the arclength parameterization of a geodesic from Q to R and let Q t = c(tl) for t [0, ]. Then d 2 (P, Q t ) ( t)d 2 (P, Q) + td 2 (P, R) t( t)d 2 (Q, R). Throughout this note, we will additionally assume that Y is locally compact and a Riemannian complex which we now define. Let E d be an affine space. A convex, bounded, piecewise linear polyhedron S with interior in 2

3 some E i E d is called a cell. We will use the notation S i to indicate the dimension of S. A complex Y in E d is a finite collection F = {S} of cells satisfying the following properties: (i) the boundary S of S i F is a union of T j F with j < i (called the faces of S) and (ii) if T j, S i F with j < i and S i T j, then T j S i. For example a simplicial complex is a cell complex whose cells are all simplices. We will denote by Y (i) the i-dimensional skeleton of Y, i.e. the union of all cells S j where j i. A complex Y is called k-dimensional or simply a k-complex if Y (k+) = but Y (k). Additionally, if each cell S of Y is equipped with a smooth Riemannian metric g S such that the component functions of g S extend smoothly all the way to the boundary of S and whenever S is a face of S then the restriction g S to S is equal to g S, then Y is called a Riemannian complex. We will furthermore assume that S is totally geodesic in S for any S and any of its faces S. We now review the notion of energy minimizing or harmonic maps in the singular context. Let Y be an NPC space. A map u : Ω Y is said to be an L 2 -map (or u L 2 (Ω, Y )) if there exists P Y such that Ω d 2 (u(x), P )dµ <. For u L 2 (Ω, Y ), we define the energy density u 2 as in [GS] (if Y is a locally compact Riemannian complex) or more generally for arbitrary NPC spaces as in [KS]. We set E(u) = X u 2 dµ and call a map u of Sobolev class W,2 (Ω, Y ) if E(u) <. If u W,2 (Ω, Y ), then there exists a well-defined notion of a trace of u, denoted T r(u), which is an element of L 2 ( Ω, Y ). Two maps u, v W,2 (Ω, Y ) have the same trace (i.e. T r(u) = T r(v)) if and only if d(u, v) W,2 0 (Ω). For details we refer to [KS]. A map u : Ω Y is said to be energy minimizing or harmonic if it is energy minimizing among all W,2 -maps with the same trace. The main regularity result of [GS] and [KS] is that harmonic maps are locally Lipschitz continuous. The key to Lipschitz regularity is the order function that we shall briefly review. Let u : Ω Y be a harmonic map. By Section.2 of [GS], given x Ω there exists a constant c > 0 depending only on the 3

4 C 2 norm of the metric on Ω such that σ Ord u (x, σ) := e cσ2 σ E x(σ) I x (σ) is non-decreasing for any x Ω. In the above E x (σ) := u 2 dµ and I x (σ) := B σ(x) As a non-increasing limit of continuous functions, B σ(x) Ord u (x) := lim σ 0 Ord u (x, σ) d 2 (u, u(x))dσ(x). is an upper semicontinuous function. By following the proof of Theorem 2.3 in [GS], we see that Ord u (x) (this is equivalent to the Lipschitz property of u). The value Ord u (x) is called the order of u at x. Before we proceed with the main Theorem 5, we recall some more terminology from [GS]. Definition Let T be an NPC space. A map l : B r0 (0) R n T is said to be intrinsically homogeneous of degree α if d(l(tx), l(0)) = t α d(l(x), l(0)) for t (0, ], x B r0 (0) and the curve t l(tx) is a constant speed parameterization of a geodesic. The number α is the degree of the map. Definition 2 Let T be an NPC space and T 0 be a subspace of T. We say that an instrically homogeneous degree map l 0 : B r0 (0) T is effectively contained in T 0 if l 0 (B r0 (0)) T 0 and for all P T 0 sufficiently close to P = l 0 (0), there exists an intrinsically homogeneous map l P : B r0 (0) R n T 0 of degree satisfying the following conditions: (a) l P (0) = P, (b) sup Br0 (x 0 ) d(l P, l 0 ) 0 uniformly as P P and (c) for any ɛ > 0, there exists δ = δ(ɛ) > 0 such that Vol{x B σ (0) : B T δσ(l P (x)) T 0 } < ɛσ n, σ (0, r 0 2 ]. 4

5 Note that Definition 2 is inspired by the definition of effectively contained in [GS]. However, even when T is an F -connected complex, our definition is slightly weaker than the one given in [GS]. Definition 3 Let Ω a smooth Riemannian domain and T 0 an NPC space. A map v : B r (0) R n T 0 is instrinsically differentiable if there exist ρ 0 > 0, c > 0 and β (0, ] and an intrinsically homogeneous degree map ˆl : Bρ0 (0) T 0 such that sup d(v, ˆl) cσ +β inf sup d(v, L), σ (0, ρ 0 B σ(0) L B ρ 02 (0) 2 ], where the infimum is taken over all intrinsically homogeneous maps L of degree. The constants c, ρ 0, and β may depend only on T 0 and the total energy of the map u. Definition 4 An NPC space T 0 is called essentially regular if for any harmonic map u : Ω Y with u(ω) T 0, the restriction of u to any compact subset of Ω is intrinsically differentiable. 3 Main Theorem Theorem 5 (Main theorem) Let Y be locally compact Riemannian complex and assume Y 0 is a totally geodesic subcomplex of Y. Fix P Y 0 and assume that T P Y 0 is essentially regular for P Y 0 close to P. Let l 0 : B r0 (0) R n T P Y be an instrinsically homogeneous degree map effectively contained in T P Y 0 and l 0 (0) = O P, the origin of T P Y. Let Ω be a smooth Riemannian domain, u : Ω Y a harmonic map, Ω 0 a compactly contained subset of Ω, x 0 Ω 0 such that u(x 0 ) = P and u is of order at x 0. Then there exist σ 0 > 0 and δ 0 > 0 such that if d(exp σ n (x), l 0 (x)) dµ < δ 0, σ (0, σ 0 ) B σ(0) then u(b σ (x 0 )) Y 0 for some σ > 0. The constants σ 0 and δ 0 depend only on Ω 0, Ω, Y, Y 0 and the total energy E u of u. 5

6 Before we proceed with the proof of Theorem 5, we prove the following two lemmas. Lemma 6 Suppose T is an NPC space and T 0 a totally geodesic subspace of T. Let ũ : R n T and define θ u(x) to be the point at a distance θ d(ũ(0), ũ(θx)) from ũ(0) such that the unique geodesic from ũ(0) to θ u(x) contains the point ũ(θx). Let l : R n T 0 be an instrisically homogeneous map of degree with l(0) = ũ(0). Assume the constants c >, β > 0, θ (0, 4 ), δ > 0 and ɛ > 0 satisfy the following conditions: (i) for all ρ [ 3, ], there exists a map v : B 4 ρ(0) R n T 0 with v = Π Bρ(0) T 0 ũ where Π Bρ(0) T 0 : T T 0 is the projection map such that sup d( θi u, v) c d( θi u, v). B 2 (0) 8 B ρ(0) (ii) For the map v : R n T 0 defined in (i), there exists an intrinsically homogeneous degree map ˆl : R n T 0 such that sup d(v, ˆl) cθ +β inf sup d(v, L) B θ (0) L B 2 (0) where the infimum is taken over all instrinsically homogeneous maps L of degree. (iii) The constants c, θ, δ and ɛ satisfy cθ β < 4, () cɛ θ + c 2 ɛ θ β < 4 (2) and V ol{x B σ (0) : B T δ σ(l(x)) T 0 } < ɛ σ n. (3) Let i l : R n T 0 be instrisically homogeneous maps of degree and iδ < ɛ δ, (4) 6

7 then we have the following implication: sup d( θi u, i l) =: D i B (0) d( θi u, θi l)dµ < i δ implies that there exists an instrinsically homogeneous degree map i+ l : T 0 so that D i+ := sup d( θi+ u, i+ l) < 2 D i (5) d( θi+ u, θi+ l)dµ < i+ δ := 2θ D i V 0 + i δ where V 0 is the volume of. Proof. We first give the proof of the first inequality of (5). The inequality of (3) with σ = θ i implies V ol{x : B T δ ( θi l(x)) T 0 } < ɛ. Since δ V ol{x : d( θi u(x), θi l(x)) > δ } inequality (4) implies Additionally, d( θi u, θi l)dµ < i δ, V ol{x : d( θi u(x), θi l(x)) > δ } < i δ δ < ɛ. V ol{x : θ i u(x) / T 0 } V ol{x : B T δ ( θi l(x)) T 0 } < 2ɛ, +V ol{x : d( θi u(x), θi l(x)) > δ } and thus there exists ρ [3/4, ] such that V ol{x B ρ (0) : θ i u(x) / T 0 } < 8ɛ. 7

8 Therefore, if Π T0 : T T 0 is the nearest point projection, then V ol{x B ρ (0) : θ i u(x) Π T0 θi u(x)} < 8ɛ. Let v : B ρ (0) T 0 be as given in (i). The fact that Π T0 is the nearest point projection and the above inequality imply that d( θi u, v)dσ = d( θi u, Π T0 θi u)dσ B ρ(0) and thus the inequality of (i) implies B ρ(0) 8ɛ sup d( θi u, Π T0 θi u) B ρ(0) 8ɛ sup d( θi u, i l) = 8ɛ D i, sup d( θi u, v) < cɛ D i. B 2 (0) By (ii), there exists a homogeneous degree harmonic map ˆl : T 0 so that sup d(v, ˆl) cθ +β sup d(v, i l). B θ (0) B 2 (0) The above two inequalities combine to give sup d( θi u, ˆl) sup d( θi u, v) + sup d(v, ˆl) B θ (0) B θ (0) B θ (0) < cɛ D i + cθ +β sup d(v, i l) B 2 (0) cɛ D i + cθ +β < cɛ D i + cθ +β (cɛ D i + D i ) = (cɛ + c 2 ɛ θ +β + cθ +β )D i. sup d(v, θi u) + sup d( θi u, i l) B 2 (0) B 2 (0) By () and (2), we have that (cɛ θ + c 2 ɛ θ β + cθ β ) < 2. Thus, if i+l : T 0 is defined by i+ l(x) = θˆl(x), then D i+ = sup d( θi+ u, i+ l) < 2 D i. 8

9 This completes the proof of the first inequality of (5). We now prove the second inequality of (5). Since θi l(0) = θi u(0), we have d( θi l(0), i l(0)) d( θi u(0), i l(0)) D i. Using the fact that θ i Y is NPC, for any x B (x), we obtain Thus, d( θi l(θx), i l(θx)) ( θ)d( θi l(0), i l(0)) + θd( θi l(x), i l(x)). which implies d( θi l(θx), i l(θx)) dµ ( θ)d i V 0 + θ ( θ)d i V 0 + θ ( θ)d i V 0 + θ( i δ + D i V 0 ) D i V 0 + θ i δ, d( θi l(x), i l(x)) dµ d( θi u(θx), θi l(θx)) dµ d( θi u(θx), i l(θx)) dµ + 2D i V 0 + θ i δ. d( θi l(x), θi u(x)) + d( θi u(x), i l(x)) dµ d( θi l(θx), i l(θx)) dµ Therefore, d( θi+ u(x), θi+ l(x)) dµ 2θ D i V 0 + i δ. This proves the second inequality of (5) and complete the proof. q.e.d. Lemma 7 Let Y be an NPC space, Ω be a smooth Riemannian domain, and Ω 0 a compactly contained subset of Ω. There exist σ > 0 and c > 0 9

10 depending only on Ω 0, Ω and Y such that if u : Ω Y is a harmonic map with only points of order in Ω 0, u(x 0 ) = P and E u C, then sup x B 2 (0) d(û(x), ˆv(x)) c d(û(x), ˆv(x)) dσ + r (6) for any x B σ (0) T x0 Ω R n, r (0, σ ] and any harmonic map ˆv : R n T P Y 0 with E ˆv C. In the above, we define û : R n T P Y 0 by setting û(x) = exp (x + rx). Proof. If the assertion of the lemma is not true, then there exist x 0i Ω 0, x i T x0i Ω R n with x i 0, r i 0, c i and harmonic 2 maps u i : Ω Y and i v : R n T P Y 0 such that sup x B 2 (0) d( i u, i v) c i d( i u, i v) dσ + where i u(x) = r i exp P u i exp x0i (x i + r i x). Here, we use the notation P to represent the point in T r P Y at a distance d(o r P, P ) such that the unique geodesic from O P to P contains the point P. Because of the uniform r bound on the energy and the smoothness of the metrics, the sequences i u, i v converge uniformly on every compact subset to harmonic maps u : B 2 (0) R n T P Y and v : R n T P Y 0. Thus, there exists c > 0 such that sup d(u, v ) c d(u, v ) dσ, x B 2 (0) B ρ(0) and this contradicts the fact that d( i u, i v) converge uniformly to d(u, v ). q.e.d. Proof of Theorem 5. For δ 0 > 0 and σ 0 > 0 to be chosen later, assume d(exp (x), l 0 (x)) dµ < δ 0. σ n 0 B σ0 (0) Since x B σ 0 2 (0) implies B σ 0 2 (x ) B σ0 (0), we have that B σ 0 2 (x ) d(exp (x), l 0 (x)) dµ < δ 0 σ n 0. 0

11 In particular, there exists r 0 [ σ 0 4, σ 0 2 ] such that d(exp (x), l 0 (x)) dµ < 4 δ 0 σ0 n = 4δ 0 σ0 n. σ 0 B r0 (x ) Therefore, we have and d(exp (x + r 0 x)), l 0 (x + r 0 x)) dµ d(exp (x), l 0 (x)) dµ = r n 0 r n 0 < δ 0 σ n 0 r n 0 B r0 (x ) B σ 0 2 (x ) = 4 n δ 0 d(exp (x), l 0 (x)) dµ d(exp (x + r 0 x)), l 0 (x + r 0 x))) dµ = r0 n d(exp (x), l 0 (x)) dµ B r0 (x ) σ0 n < 4δ 0 r0 n = 4 n δ 0. Let x B σ 0 2 (0) T x0 Ω be such that P := exp (x ) T P Y 0 and let û : T P Y be defined by setting Define l : T P Y 0 by setting û(x) = exp (x + r 0 x). l (x) = l P (r 0 x) where l P is as in Definition 2 with T = T P Y, T 0 = T P Y 0 and P = O P. Thus, û(0) = P = l P (0) by property (a) of Definition 2. Furthermore, by property (b) of Definition 2, if x is sufficiently close to 0 then d(û, l ) dµ < 4 n δ 0 and d(û, l ) dσ < 4 n δ 0.

12 Let Π TP Y 0 : T P Y T P Y 0 be the nearest point projection and ˆv : T P Y 0 be a harmonic map with ˆv = Π TP Y 0 û on. By (6), we have sup d(û, ˆv) c d(û, ˆv)dΣ + r 0 B 2 (0) = c d(û, T P Y 0 )dσ + r 0 c d(û, l )dσ + r 0 < 4 n c δ 0 + r 0. Without the loss of generality, we can assume r 0 < 4 n c δ 0 and hence sup d(û, ˆv) 2 4 n c δ 0. B 2 (0) Since T P Y 0 is essentially regular, there exist constants ρ 0 (0, ), ĉ > 0, β (0, ] and a homogeneous degree harmonic map ˆl : B ρ0 (x 0 ) T P Y 0 such that sup d(ˆv, ˆl) ĉr +β, r (0, ρ 0 B σ(0) 2 ]. Thus, for r (0, ρ 0 ] to be chosen later, we have 2 sup d(û, ˆl) sup d(û, ˆv) + sup d(ˆv, ˆl) < 2 4 n c δ 0 + ĉr +β. B r(0) B 2 (0) B r(0) Define ũ, l, 0 l : T P Y by setting ũ(x) = r û(rx), l(x) = r l (rx) and 0 l(x) = r ˆl(rx). Here, as before we use the notation P to represent the point in T r P Y at a distance d(o r P, P ) such that the unique geodesic from O P to P contains r the point P. Then and d(ũ, l) dµ = d(û, l )dµ < 4n δ 0 r n+ B r(0) r n+ sup d(ũ, 0 l) = r sup d(û, ˆl) < 2 4n c δ 0 + ĉr β. B r(0) r 2

13 With P = u(exp x0 (x )) Y 0, let c > and β > 0 (7) be constants such that for any harmonic map v exists a linear map ˆl : T P Y 0 such that : T P Y 0, there sup d(v, ˆl) c B r(0) 8 r+β inf sup d(v, L), r (0, L B 2 (0) 2 ) (8) The constants c > and β > 0 can be chosen independently of x sufficiently close to 0. Fix θ (0, 4 ) and ɛ satisfying cθ β < 4, and cɛ θ + c 2 ɛ θ β < 4. (9) By property (c) of Definition 2, we can now choose δ > 0 such that V ol{x B σ (0) : B T P Y δ σ (l(x)) T P Y 0 } < ɛ σ n. (0) This shows assumption (iii) of Lemma 6 is satisfied. We need the following claim to verify that assumptions (i) and (ii) are also satisfied. Claim With ũ(x) = r exp (x + r 0 rx) as defined above, set ϑ u(x) to be the point at a distance ϑ d(ũ(0), ũ(ϑx)) from ũ(0) such that the unique geodesic from ũ(0) to ϑ u(x) contains the point ũ(ϑx). There exists σ > 0 such that for any x T x0 Ω with x < σ, there exists r 0 > 0 such that for any r (0, r 0 ), we have the following: For every ϑ (0, ) and ρ [ 3, ], there exist 4 v : B ρ (0) T P Y 0 with v Bρ(0) = Π T P Y 0 ϑ u Bρ(0) and a linear map such that ˆl : Bρ (0) T P Y 0 sup d( ϑ u, v) c d( ϑ u, v)dσ () B 2 (0) 8 B ρ(0) 3

14 and sup d(v, ˆl) cθ +β inf sup d(v, L) (2) B θ (0) L B 2 (0) where the infimum is taken over all instrinsically homogeneous maps L of degree. Proof of Claim. For r > 0 and x T x0 Ω as in the assumption of the claim, fix ϑ (0, ) and ρ [ 3 4, ]. Set R = r 0rϑ and x = exp x0 (x ). Let w : B ρr (x ) Ω Y 0 be a harmonic map. Since d(u, w) is subharmonic, we can assume without the loss of generality that c > 0 defined in (7) is such that sup d(u, w) c d(u, w)dσ. (3) B R2 (x ) 64 B ρr (x ) For P = u(x ) as before, define ˆv : B ρ (0) T x Ω R n T P Y by ˆv(x) = rϑ exp P w exp x (Rx). If x is close to 0, then there exist maps Ψ : T x0 Ω T x Ω and Φ : T P Y Φ(T P Y ) T P Y which are close to isometries. Define v : B ρ (0) T x0 Ω R n T P Y by v(x) = Φ ˆv Ψ(x). If we assume that r > 0 is small and x is close to 0, then for x B ρr (x ) 2 d( ϑ u( x ), v( x )) R R 2. d(u(x), w(x)) rϑ Lastly choose w to have the appropriate boundary value on B ρr (x ) to ensure v = Π Bρ(0) T P Y 0 ϑ u Bρ(0). Thus (3) implies (). Next we prove (2). Choose σ > 0 sufficiently small such that Ψ and Φ are close to isometries. Fix x T x0 Ω such that x < σ. Suppose there exist r i 0, ϑ i (0, ) and ρ i [ 3, ] such that with u 4 i = ϑ i u, w i = w, ˆv i = ˆv and v i = v where r, ϑ, ρ are replaced by r i, ϑ i, ρ i and sup d(v i, l i ) cθ +β inf sup d(v, L) B θ (0) L B 2 (0) 4

15 for any instrinsically homogeneous degree map l i. Using the fact that Ψ and Φ are almost isometries, we have sup d(ˆv i, ˆl i ) c B θ (0) 2 θ+β inf sup d(v, L) L B 2 (0) for any instrinsically homogeneous degree map ˆl i and i sufficiently large. The modulus of continuity of the harmonic maps w i is uniformly bounded by that of u and hence so the modulus of continuity of v i is also uniformly bounded. Thus, by Arzela-Ascoli, there is a subsequence of v i that converges uniformly to a harmonic map v : B 3 (0) T P 4 Y. By the uniform convergence, we have sup d(v, ˆl) c B θ (0) 4 θ+β inf sup d(v, L) L B 2 (0) for any instrinsically homogeneous degree map ˆl. This contradicts (8) and proves the claim. The inequalities (9), (0), () and (2) show that for x T x0 Ω sufficiently close to 0 and r > 0 sufficiently small, the assumptions (i), (ii) and (iii) of Lemma 6 are satisfied with T = T P Y and T 0 = T P Y 0. Furthermore, we can require that ɛ σ n < 8 V ol(b σ(x )) (4) and V ol{x B σ (0) : B Y δ σ(l P (x)) T P Y 0 } < 8 V ol(b σ(0)). (5) Finally, we choose r > 0 and δ 0 > 0 sufficiently small such that ( 2 4 2θ n ) c δ 0 + ĉr β V 0 < ɛ δ r 4 and If we let (6) 4 n δ 0 r n+ < ɛ δ 2. (7) 0δ := 4n δ 0 r n+ and D 0 := sup d(ũ, 0 l) < 2 4n c δ 0 + ĉr β, r 5

16 we can write D 0 = sup d(ũ, 0 l) d(ũ, l) dµ < 0 δ < ɛ δ. Applying Lemma 6 with i = 0, we see that there exists essentially homogeneous degree map l : θ Y so that D := sup d θ ( θ u, l) < 2 D 0 d θ ( θ u, θ l) dµ < δ = 2θ D 0 V δ. By (6) and (7), we have 2θ D 0 V 0 < 2θ ( 2 4 n c δ 0 r ) + ĉr β V 0 < ɛ δ 4 and which then imply Assuming 0δ < ɛ δ 2, δ < ɛ δ 2 n= 2 n. D i < D 0 2, and iδ < ɛ i δ and proceeding inductively, Lemma 6 implies that for any i = 0,, 2,..., i+ n= 2 n D i+ := sup d θ i+( θi+ u, i+ l) < 2 D i d θ i+( θi+ u, θi+ l) dµ < i+ δ < 2θ D i V 0 + i δ. We can then conclude D i+ < 2 (i+) D 0 6

17 and i+δ < 2θ D 0 2 V i 0 + i δ ɛ δ 4 < ɛ δ 2 + ɛ δ i+2 i+ n= 2 n = ɛ δ 2 + iδ i i+2 i= 2 n < ɛ δ. This completes the induction and we have shown d( θi u, θi l) dµ < ɛ δ, i = 0,, On the hand, d( θi u, θi l) dµ = θ i(n+) B θ i(0) = r n θ i(n+) d(ũ, l) dµ B rθ i(0) = r n 0 r n θ i(n+) Therefore, B rr0 θ i (x ) d(û(x), l (x)) dµ B rr0 θ i (x ) d(exp (x), l P (x x ))) dµ. d(exp (x), l P (x x ))) dµ < ɛ δ r n+ 0 r n+ θ i(n+). For σ (0, rr 0 ], choose an integer i so that σ (θ i+ rr 0, θ i rr 0 ] to conclude d(exp (x), l P (x x ))) dµ < ɛ δ σ n+. and B σ(x ) V ol{x B σ (x ) : d(exp (x), l P (x x )) > σδ } d(exp P u exp σδ x0 (x), l P (x x )) dµ B σ(x ) < ɛ δ σ n+ σδ < ɛ σ n. 7

18 The above inequality combined with (4) and (5) implies that for x sufficiently close to 0, V ol{x B σ (x ) : exp (x) / T P Y 0 } V ol{x B σ (x ) : d(exp (x), l P (x x )) > σδ } +V ol{x B σ (x ) : Bδ Y σ(l P (x x )) Y 0 } ɛ σ n + V ol{x B σ (0) : B Y δ σ(l P (x)) Y 0 } < 4 V ol(b σ(x )). (8) Now assume that for σ > 0 arbitrarily small, u(b σ (x 0 )) Y 0. Then exp (B σ (0)) T P Y 0 and we can find a ball B exp x 0 u exp P (T P Y \T P Y 0 ) B σ (x 0 ) such that at least one point x in B such that exp (x ) T P Y 0. By the choice of x, we have for every r > 0 sufficiently small, close to half of B r (x ) is mapped by exp into the closure of T P Y \T P Y 0. If σ > 0 is sufficiently small (which implies x close 0), this contradicts (8). This shows that for some σ > 0, we have u(b σ (0)) Y 0. q.e.d. References [C] K. Corlette. Archimedean superrigidity and hyperbolic geometry. Ann. of Math. 35 (992) [DM] G. Daskalopoulos and C. Mese. On the singular set of harmonic maps into DM-complexes. Submitted for publication. [DM2] G. Daskalopoulos, C. Mese and A. Vdovina. Superrigidity of hyperbolic buildings. Submitted for publication. [GS] [KS] M. Gromov and R. Schoen. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ. Math. IHES 76 (992) N. Korevaar and R. Schoen. Sobolev spaces and harmonic maps into metric space targets. Comm. Anal. Geom. (993)

19 [KS2] N. Korevaar and R. Schoen. Global existence theorem for harmonic maps to non-locally compact spaces. Comm. Anal. Geom. 5 (997)