- Email: [email protected]
Quasiflats in Hadamard spaces
Ann. sciwt. 152 Norm. Sup., 4’ serie, t. 30, 1997, p, 339 %t 352.
QUASIF’LATS
BY
URS
IN HADAMARD
LANG
AND
VIKTOR
SPACES
SCHROEDER
ABSTRACT. - Let 9 be a simply connected, complete geodesic metric space which is nonpositively curved in the sense of Alexandrov. We assume that S contains a k-flat E’ of maximal dimension and consider quasiisometric embeddings f: IF;p” - X whose distance function from F satisfies a certain asymptotic growth condition. We prove that if X is locally compact and cocompact, then the Hausdorff distance between ,f(@) and F is uniformly bounded. This generalizes a well-known lemma of Mostow on quasiflats in symmetric spaces of noncompact type.
RBsut&. - Soit X un espace metrique geodesique complet et simplement connexe courbe de maniere nonpositive au sens d’Alexandrov. Noits supposons que S contient un k-p]& F de dimension maximale et considtrons des quasi-plats f : R” -+ *Y teh que les fonctions mesurant la distance a partir de F veritient une certaine condition de croissance lineaire. Nous demontrons que la distance de Hausdorff entre f( Iw’. ) et F est uniformement born&e lorsque X est localement compact et cocompact. Ceci generalise un lemme bien connu de Mostow sur les quasi-plats des espaces symetriqu,:s de type non-compact.
Introduction Quasiflats, i.e. quasiisometrically embedded euclidean spaces,play an important role in the theory of nonpositively curved manifolds. A striking example is Mostow’s proof of the rigidity theorem for locally symmetric spaceswhich, in the caseof higher rank, largely relies on the interplay between flats and quasiflats. More precisely, let X’: X be two symmetric spaces of noncompact type and rank k, and let p: r’ + I‘ be an isomorphism between discrete groups of isometries acting cocompactly on S’ and X respectively. A &flat F’ in X’ is called I”-compact if its stabilizer I?>, in r’ acts cocompactly on F’. Given such F’, there exists a unique r-compact X--flat F in X with stabilizer I’F = p(l$), $ [MO, 13.11. A key step in Mostow’s argument is the following uniform estimate, c$ [MO, 13.21. THEOREM A (Mostow). -- Let f: X’ --+ X be a quasiisometric map equivariant with respect to p. Then there exists a constant D 2 0 such that the following holds. If F’ is a II’-compact k-Jiat in X’, aEd if F is its unique companion in X as described above, then the Hausdorff distance between f (F’) and F satisfies Hd( f (F’), F) < D. This result then gives rise to a homeomorphism between the Furstenberg boundaries of X’ and X, which by a theorem of Tits is induced, after suitable renormalization, by a p-equivariant isometry of X’ and X. AUNALES
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In Theorem A, the assumptions on F’ and F clearly imply that Hd(f(F’), F) < cc. One may ask the following question: If a single quasillat f: W” -+ X lies within finite distance of a k-flat F in X, does this already imply that Hd(f(R”): F) is bounded above by some constant depending only on X and the quasiisometry constants of f? In this paper we show that this holds true in an even more general context. We consider quasiflats f: R” ---f X, where X is a Hadamard space in the sense of Alexandrov. i.e. a simply connected, complete geodesic space all of whose geodesic triangles satisfy the CAT(O) inequality, c$ Section 1. Moreover, X is assumed to be locally compact and cocompact. We prove: THEOREM B. - Let (X, d) be a locally compact and cocompact Hadamard space containing a k-flat but no (k + l)-JEat, where k > 1. Then for all L > 0 and C > 0 there exists D > 0 such that the following holds. Let FcX be a k-flat, f: Rk -+ X an (L> C)quasiisometric map (cf. l.l), and for T > 0 define u(r) := sup{d(f(z), F) : (z( <_ r}. rf lim s~p,,~ u(Y)/~ < L-l, then Hd(f(R’),F) 5 D. The constant L-’ bounding the linear growth of a(r) is optimal, as is shown by the following simple example. Let F be a geodesic line in the hyperbolic plane H2, and let f:lR t H2 be a geodesic of speed L-l with f(R) # F (and hence Hd(f(R),F) = m>. Then f is an (L, 0)-quasiisometric map as defined in 1.1, and it is easily seen that in this case, a(r)/ T + 1L-l for T -+ ‘cc. By taking products H2 x RkrP1, similar examples are obtained for all ic > 1.
In the case where J*: R” -+ X is an isometric map, and hence F’ := f(R”) is a k-flat in X, the upper limit considered in Theorem B is actually a limit and can be interpreted in terms of the angle metric i on the boundary at infinity X(m) of X, cj 5.1. Let HdL denote the Hausdorff #distance induced by i, and let F(m) and F’(m) be the boundaries at infinity of F and F’ respectively. Then Theorem B yields the following dichotomy. THEOREM C. - Let X and k be given as in Theorem B, and let F, F’ be two k-juts in X. F’(m)) = 0 (and F, F’ lie within Then either HdL(F(oc), F’(m)) 2 7r/2, or HdL(F(ec), uniformly bounded distance from each other).
The paper is divided into five and prove a basic approximation between quasiflats and flats. In discuss the special case where
short sections. In the first section we fix the terminology lemma. Sections 2 and 3 contain several results on maps Section 4 we prove Theorem B. In the last section we F’ is itself a hat in X.
After finishing this paper we learned that Bruce Kleiner had announced a result very similar to Theorem B in summer 1995, before the completion of our work. The proofs rely on different methods. Another result in the same spirit is proved in [KL, Section 41.
1. Preliminaries We briefly recall some basic notions and results from the theory of nonpositively curved metric spaces in the sense of Alexandrov. The general references are [Ba], [BGS], and the forthcoming book. [BH]. 4e
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Let (X, d) be a metric space. A continuous map 0: I - X from a connected subset I of R into X is said to be a nzinimi,-ing geodesic if it is isometric up to a constant factor, i.e. if there exists u > 0 such that d(a(r).g(s)) = /:lr - s/ for all ‘r,.s E 1. Then I! is called the speed of 0. More generally. the curve (T is said to be a geodesic if there is G > 0 such that 0 is 1ocall:y minimizing and of speed U. A geodesic .segmenf is a geodesic defined on a compact interval. A geodesic triangle A in X is a triple (al. ~2. m3) of geodesic segments cr,: 1, --$ X, the sides of a. whose endpoints match in the usual way. A euclidean comparison triangle A for a is a correspondin g triple of geodesic segments -5i : I, --f R” such that ??, has the same length as cr;, ,i = 1.2.3, and such that the endpoints of F1 : a2, 53 match in the same way as those of nl, g2. ‘T:~, Then a is said to satisfy the CAT(O) inequali4 if 3 as above exists and if d(aL@). gJ(s)) 5 Ia, - rr,(.s)I whenever ‘r E Ii, s E I,, and a.,j cz {1.2:3}. A metric space (X, $) is called a geodesic space if for all .I:. y E X there exists a minimizing geodesic 0: [O, l] - X with a(O) = s and m(l) = y. Note that for a geodesic triangle a in X with minimizing sides, a euclidean comparison triangle ,4 exists (and is unique up to isometry). Throughout the paper, (X, d) is assumed to be a simply connected, complete geodesic space all of whose geodesic triangles satisfy the CAT(O) inequality. We will refer to X simply ~1sa Hadamard space. A fundamental feature of these spaces is that for all Z, y E X there is a unique geodesic from :c to ;y (whose image we denote by [:1:,y]), and hence all geodesics in X are minimizing. Moreover, for all pairs of geodesics crl, g2: I -+ X, the function mapping t E I to ~i(a~(t), cam) is convex. We will often use the nearest point projection r: X -+ F from X onto a closed convex subset E’ of X. This map is well-defined and Lpschitz with constant 1, 0 and C 2 0. A map ,f: X’ metric spaces (X’,d’) am! (X, d) is called (L,C)-q uasiisometric I;-ld’(z, y) - c < d(f(zj: f(y)) 2 Ld’(z. y) + c.
4 X between two if for all :r. y E X’,
It is often necessary to approximate general quasiisometric maps by continuous ones. The argument proving the followin, 0 lemma is well-known, but an appropriate reference does not seem to be available in the literature. LEMMA 1.2. - Let C 2 0, and f: I3 -+ Then for ever?j X > &(L + X-lC) and (L, C)-quasiisometric,
B be a closed convex subset of R”, X a Hadamard space, L > 0, X a map satisfying d(f(i7:): f(y)) 5 Ljx - yl + C for all :r. y E B. 0 there exists a map f’: B -+ X which is Lipschitz with constant satisjies d(f’(z): f(z)) 5 &XL + Cfor all z E B. ZfmoreoL>er f is then d(f’(:r), f’(y)) > LP1/z-y - (2v’%XL+SC)forall .I:, y E B.
Proof. - Let 7r: Iw’ + B denote the nearest point projection. We f o 7r: R” - X and call this map f again. Then f satisfies ~j(f(.r)~ f(y)) < for all z,y E R”. For 2’ = 0: 1,. . . , Ic let 5; denote the closed i-skeleton of the canonical R” into cubes of edge length X. On So = (AZ)“, put f’ := f. Then for assuming that f’ is already defined on SP1, we extend f’ to S; as follows. ANNALES
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replace f by Ll.1: - yl + C subdivision of 1. = 1,. . . 3k:, Let (:I, . . . . ek
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be the canonical basis of R”, and let s E S,,\S,-1. Let .j = j(x) E (1,. . . ~k} be the maximal index such that there exists z = Z(X) E S,-i with :I: E [z, z + Xcj]. Then f/(.x) is determined by defining .t”][z. z + Xe,,] to be a geodesic. Consider the following assertion -4(i): for all pairs (j,~) = (j(s),z(z)) with .): E &\.!?-I, d(f’(~:~,f’(~ + Xc,)) i XL + C, and for each %-plane PicSt, f’]P; is Lipschitz with constant &(L + A-‘C). Clearly A(1 ) holds. We show that for i = 2. . . . : A:, A(i- 1) implies A(1:). (Given (3,~) = (j(x). Z(X)) with .r E S;\S,-i, there exist p, Q E S,-2 such that z E [n. (I] ard both f’][2J> 91 and f’][Tj + Xc,, 4 + Aej] are geodesics. Moreover, A(i - 1) implies that n(f’(p). j’(n + Xej)), d(f’(q). f’(q + A:,,)) < XL + C. Thus by convexity, d(.f’(s),
f’(Z
+ Xi:,))
< XL + C.
It suffices to prove the claimed Lipschitz property for ,f’]Q, where Qi N [O, Xli is an i-cell of the given cubical subdivision of R”. Let :I:. z’ E QZ. and let [z, z + XeJ] and [z’, z’ + Xc,] be th e segments in Q1 containing :I: and .I:’ respectively, where j is maximal. Let y E [z! z + Xcj] be the point closest to 2. Since ,f’][~, z + Xe,r] is a geodesic of length at most XL + C,
d(,f’(:c), f’(!/)) I (L + PC&: On the other hand, Alla - 1) asserts that d(f’(z). m(L + X-lC)/z - 2’1, hence d(f’(y). by convexity.
Combining
f’(d))
2 &-T(L
these estimates
- yl.
.f’( .?)), d(~f’(r: + Xcj), f’(2
+ x-lC)I!/
+ Xe, )) <
- :r’j
we see that
d(f’(.~). f’(~‘j)~ 5 [d(f’(+ .f’(:vj) + d(f’j?l). f’(:#))12 < I;d(f’(:x), f’(y))’ + i(i - I)-‘d(,f’(yy): f’(2’))2 < 6jL + )\-‘q21:r.
- :cy
proving A(i). In particular, we have shown that f’ is Lipschitz with constant &(L+X-‘C). Given IL’ E R’;, let k'c (X7)k be the set of vertices of a cube of edge length X containing Z. Then f maps I/ into the metric ball with center f(:~) and radius &XL + C. By the above construction, f’(~) is contained in the convex hull of f(V). Since metric balls in a Hadamard space are convex, it follows that d(f’(z): f(x)) < &XL + C. Finally, if f is (L, C)-quasiisometric on J3, we see that L-ll.~
- ?/I - c L W(x),
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TOME
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5 d(f’(x),
f’(y))
+ q&XL
+ C) 0
for all x,?/ E B. ae S6RfE
f(y))
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QUASIFLATS IN HADAMARD SPACES
2. Continuous
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quasiflats
In this section we prove two independent results applying to continuous quasiflats f: W” -+ X, where X is a general Hadamard space containing a k-flat F for some k 2 1. We emphasize that here k, need not be maximal as in Theorem B. We start with a simple topological lemma. For r > 0 and z E Rk’ we define B’(z) r) := {r E Rk- : 1~1. - ~1 5 I.} and S”-l(z,r) := {.r E R” : 1.1.- ~1 = r}. Instead of B’(f). rj and S’“-l(O,r) we write B”(r) and Skpl(,rj respectively. LEMMA 2.1. - Let k 2 I, r > 0, and let f: B”(, r ) - R” be a continuous mup suti@ing f(r) # f(y) for all 3:. y E Bk( r ) with 1.r + /yI = I:/. - yl = 7’. Then flSk.-‘(;l,) i.s not contractible (i.e. homotopic to a constant map) in R”\{ f’(0)).
Proof. - Assume the contrary. Then there exists a continuous map f: n’(r) R’\{f(O)} with flSP1(r~) = dlS-i(~). Define /a,l:ll’(r,) + S”-‘(l) by
For z E
Sk-l(r)
+
and 0 <: t 5 1, the denominator of
h,(z) :=
f((l
- tP):r)
- f((-tPb-)
If((i
- t/z):r)
- f((-t/2)3-)1
is nonzero. The map ki: S”-‘(r) -+ Sk-l(l) defined this way is homotopic to /lalS”-l(r) and can thus be extended to a continuous map ILL: B”(r) - S”-i (1). Finally, since h,(-z) = -h,(x) for :c f S“-l(r). h,l gives rise to a continuous map from S”‘(1) into S’+l( 1) preserving antipc’des. in contradiction to the Borsuk-Ulam theorem, c$ [Ma]. 0 For the statement of the next lemma the following definition is useful. DEFINITION 2.3. - Let k: > 1 and d; > 0. We call u continuous map f: W” --f Rk n-expanding {ff.for all 0 < b < -d, the following condition is satis$ed for 7‘ > 0 sufficiently large (depending on 6): If(x) - f(O)1 > ST ,for :I: E Sk’-l(r.), and .flS”‘-l(r) is not contractible in W”\{f(O)}.
The definition is independent of the choice of basepoints. Clearly 2.1 implies that every continuous map f: W” i UP satisfying If(.r) - .f(,f/)l 2 J!-~~:I: - y// - c7 for some constants L > 0. C 2 Clis L-‘-exp.mding. In particular, it follows that f is surjective, a fact which was proved in (MO, 10.1:. More generally, the following holds. LEMMA 2.3. - Let X be a Hadumard space containing a k-jlat F for some X: 2 1, and let T: X + E’ be the nearest point projection. Let L > 0, C > 0, and let ,f: k%” - X be a continuous map sati@ng d(f(:r). f(y)) > L,-‘lz - yI - C ,fin all :I:. y E Wk. For r > 0 define u(,r) := sup-(d(f(.x), F) : 1x15 ,I,}. 1f~ := limsup,.,, a(,r)/r. < L-l, then h := T o f: R” --+ F N U;P”is (I,-’ - cr.)-expanding. ANNALES
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Prooj: - For all -c :> 0 there exists T, > 0 such that a,(‘r)/r for .r:.?/ E B”(r) with ].K(+ 1~~1 = ]:I: - :r/j = 7‘. Il(f(.l*,l.
F) + d(f(;l/).
F) + d(h,(s).
h(y))
5 7i + E for r 2 r,. Then
> d(,f(x): >_ L-l1
f(y)) - c.
In case 1.x.1. ]yJ > ‘I’~, we have ~i(f(:~:), F) + ~1(f’(;//). F) < (CL+ E)(]J] + ];v]) = (E + E)T. In case ]:I./ < rE or ](I] < TV, we see that d(.f’(.K). F) + d(f(y).
F) < (ii + E)(?. + rE) 2 (E + E)(l + E)T
for I’ > Y~/E. Thus ri( r?(.l.). It,(l/)) > [C’ from 2.1.
- ((I + ~)(l + z)]r - C, and the claim follows 0 The definition of the Hausdorff distance of two subsetsA, A’ of X reflects the fact that there are two ways of measuring distances between these sets: Hd(A, A’) := sup{n, a’}. where (I := SI~I)~,~-~~ d(:r:‘. A) and (1,’:= SUI),,~,~d(n:, A’). The following lemma shows, in particular, that if Iz = F and A’ = f(R”j are (quasi-)flats in X1 then a is bounded above in terms of (I’. More generally. the statement is adapted to the hypotheses of Theorem B. LEMMA 2.4. - Let .I7 be u Hadmnurd space containing a k-&t F ,for same X: > 1, and let z E F be some busepoint. Let L, L’ > 0, C > 0, and let ,f: Rk’ + X be a map .satisfiirzg L-l I:/: - I// -~ C 5 d(.f(.r). j(y)) < L’1.r - ?//I for all .r, y E R”. For T > o dejine n(r) := sup{d(f(:r). f’) : 1.1.1< ,I,} afzd c/(7.) := SllI,{d(?J, f(@)) : y E F, d(y,z) < r}. Jf i? := lim sitp,.,,~ a( V)/T < L-l, then,for all 0 < $ < (1 - iZ)/L’ there exists r;j > 0 such that (I(/%) 5 I,a’:r) + T;‘,for I’ > r ,,. where z := 3LL’ + 1 and c := LL’C. Proof. - Let n: X -7‘ F denote the projection. and let h, := r o f. Without loss of generality we may assume that z = /l,(O). For fixed ‘I’ > 0 we define a map 9: Bk’(,l) - R” as follows. First, by the properties of f, we can associate to each :I: E Bk(?-) a point YEE R” with
fl(f(Fj. Note that h(x) O(:C) 5 o’(L’r).
5
/L(T)) = d(h(.r). f(R”))
and. since d(l~(:c),z)
o(r)
5
=: [J(T) . d(S(z),%(O))
5 L’]z]
< L’r,
+ C]) c 1osestto Z: then ]g(:‘;.) -F]
< Lh(.c).
Moreover. 17 - XI < L[d(j(S). < L[d(f(T). 2 L[b(.r)
f(X))
+ C]
h(3.))
+ d(h(.r)!
L[n(r)
l,y(:I’)
!/(!/)I
+ C]
+ a(7.) + C].
We define !](,I,) to be the point in Bk(.r, Hence, for .I:. y E Bk:7.), -
f(X))
5
L[b(X)
+
h(y)]
<
L[h(:C)
+
b(y)
<
L[t?h(.K)
5
L[h’(L’7.)
+
+ +
2b(y) +
1-r d(f(;r;), +
L’lX
-/I f(g))
d(h(T), -
z/l +
+ h(?/))
c].
c] +
c]
QUASIFLATS
Let X > 0. The procedure carried g’: B”(r) + R” which satisfies
IN HADAMARD
out in the proof
lg’(.c) - g(z)\ for all 3’ E B”(r)
and coincides
5 L[vGAL’
with g on (AZ)”
345
SPACES
of 1.2 yields
+ 4u’(L’r)
a continuous
map
+ C]
n B”(r).
We claim that the image of g’ contains B”(r - L[a(r) + C] - 2&X) =: B’. Let y E B”(r - &A), and let V c (AZ)’ n B”(r) denote the set of vertices of a cube of edge length X containing y. For L: E V, lg(z) - 1~15 1,9(z) - :cI + Iz - ~1 5 L[u(T) + C] + &A. Hence, by the definition of g’?
Id(Y) - 1Jl5 a+-) + Cl + km for all y E B”(r - fix). N ow the claim follows easily from the fact that there is no retraction B”(1) -+ Sk-l(l). Let y E B’. Then there exists .2: E B”(r) with g’(z) = y, hence W(Y),
E‘) L W(g), f(3) + Wm. N:c)) 5 L’[lg’(z) - g(.r)l + lg(:r) - %I] + b(n:) 5 L’L[dLw
+ Sa’(L’r)
Since this holds for all X > 0, using the continuity
+ C] + n’(L’r) of f we obtain
d(f(:r), F) < ,o,‘(L’r) + c for all 2 E B”(r--L[a(r)t-Cl), w h ere z := 5L’L+l and c := L’LC. Let E > 0. Then for T sufficiently large: a(r) 5 (E+ t-) T and C < ET. Thus a(~ - L[z + 2&]7-) 5 %‘(L’r) + ??, and replacing 1^by T/L’ we get the desired result. 0
3. A basic area estimate In [MO, 6.41, Mostow proved an estimate for the k-dimensional Jacobian of the normal projection K: X -+ F onto a &flat FcX, where X is a nonpositively curved symmetric space of rank Ic. Our aim is to deduce a non-infinitesimal analogue of this result applying to flats of maximal dimension in a Hadamard space X. Here now we assume X to be locally compact and cocompact, where the latter condition means that there exists a compact subset K of X such that ihe translates of K by the isometry group of X cover X. First we establish a diameter estimate for certain subsets of X. LEMMA 3.1. - Let X be a locally compact and cocompact Hadamard space containing a k-flat but no (k + l)-$at, where k > 1. Then there exist s,m > 0 such that the following holds. Let Fc.Y be a k-flat, r: X 4 F the projection, and let QC F be a closed k-dimensional euclidean cube of edge length 3s. Let ql> . . . , q,U be the vertices of Q, where 2: := 2’“. For i = 1) . . , u let QiCQ denote the k-cube of edge length s containing qi.
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If V is a subset of X such that r(V) meets each Qi, and if d(V, F) > s, then the diameter of V satisfies diamV 2 s + md( V! F). Proof. - By the assumptions on S there exists s > 0 such that S does not contain a closed convex set isometric to a (k + I)-dimensional euclidean cube of edge length S, cJ: [Br, 3.11. For i = 1,. . . 12~ pick zi E V with x(x~) E Q;, and let 2:: E [:I:~,T(x;)] be the point with d(:r:‘, F) = s. First we show that there are %,,j E { 1. . ,,v} such that rl(~:: z$) > cI(~(.r;): ~(2:~)). If not, then d(~::, x$) = d(7r(ri). n(.r.j)j for all i,j since 7r is Lipschitz with constant 1. The convexity of the distance function on X x X then implies that the segments [.t.:. rr(~;)] and [:rS. .ir(.r~,~)] are at constant distance from each other. As in [Ba, IS.91 it follows that the convex hull of lJ:1,[:1:. X(X,)] is isometric to the product of the convex hull of {r(.rt), . . . : ~(:r,;)} and [(I, .s]. Thus X contains an isometric copy of a (k: + l)-dimensional euclidean cube of edge length s, contrary to the choice of s. Using the cocompactness of X again, we see that in fact d(s::. .I$) > Xd(~(:r~), 7r( :r,i)) for some 1-j E {l..... ,I;} and for some X > 1 depending only on X and s (but not on the particular choice of F. Q and V). Hence by convexity, d(z,: .cj) 2 [l + (X - l)s-‘d(l~~. > s + (A - l)d(V.
F)]d(~(z~),
n(x,
j)
F) .
Thus the lemma holds for m = X - 1.
cl
Together with the coarea formula, 3.1 leads to the desired analogue of [MO, 6.41. For 0 < r: < k we denote by XFli the i-dimensional Hausdorff measure on R”. PROPOSITION3.2. - Let ,Y, k, s, and vx be given as in 3.1. Let FcX be a k-flat, K: X + F the projection, and let QcF be a closed k-dimensional cube of edie length 3s. Let f be an L-Lipschitz map from a compact ball BCR” into X, arid let h = n; o f. Assume that h(8B) f’ Q = v) and that h13B is not contractible in F\Q. [f d := d(f[t’(Q)], 8’) > s, then ?f’(h-l(Q)) 2 s’-l(s + rnd)/LC”. Proof. - Let v = 2k. Using induction on X: one shows that the vertices ql, . . , ql, of Q can always be ordered in such a way that for % = 1. . . ,7! -- 1, [q?, ql+l] is an edge of Q. Then let P = u:z,‘[qi, q,+l]. and let R denote the union of all k--cubes Q’cQ of edge length s meeting P. Define the cubes QicR as in 3.1, and denote by Z the common (X: - 1)-face of R and Q1 isometric to [O: s]‘.-l. For h: = 3 the situation is illustrated in Figure. We define a 1-Lipschitz retraction 4: R + 2 such that each fiber I’, := qV1{z} is a connected polygonal arc lying at constant distance from P. Let jr = rr o f; then 4 o /I, is Lipschitz with constant L. Now the coarea formula [Fe, 3.2.221 yields
I’
.z
3-t1(h-l(P,))
d’FI”-‘(z)
I L”-“Ft”(h-l(Rj).
We claim that for each :: in the relative interior 2’ of 2 there exists a connected component C, of h-l ( Pz) such that h( CZ) meets each QT. With this at hand one can proceed as follows. By 3.1, dianif(CZ) > :: + ,rnd. Since CZ is connected, and since f is L-Lipschitz, we get IFtl(/l,-‘(Pz))
> X’(C,)
2 diamCZ
1 (s +
7724/L.
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IN HADAMARD
The
proof
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of 3.2
Together with the above integral estimate and the fact that IKQ this yields the desired result. It remains to show that for each z E 2’ there exists a connected component C,? of h-‘(Pz) such that h(Cz) meets each Q,. Suppose this is false for some z E 2’. Every connected component C of /L-‘(I’~) is mapped by 11 to a connected subarc of I’,. Denote by G the union of those components C with h(C) fl Qi # Cn.Then /i(G) n Qtz= 0. Note also that both G and P,\G are compact. For E > 0 let CT, := {X E h-‘(R) : n(:r, G) < E}. We can choose E such that 17, II h-l(P;) = G and, since h is L-Lipschitz. such that h( CT=) intersects the boundary of 12 (relative to F) only in 2. Our aim is to construct a continuous map &: B --+ F that coincides with h. on B\U, and maps G to {z}. For :y E Z and 5 E P,y let (Y(X) denote the length of the subarc of Pv with endpoints y/ and 2. Then for t 2 0 let - t}. For z E 157~define q!+(z) be the point on Py with N(&(x)) = max{O,tr(z) t(r)
:= I(1 - E-%(X,
G)) .
where I is the length of P,. Finally, set k(z) := &cX,(/l(z)) for 3: E Cr, and A(X) := h(z) for z E B\U,. Clearly k i:i a continuous map with h[iJB = hldB, and it is not difficult to see that /A omits the set (Pz\{x}) t7 Qr . It follows that /rliJB is contractible in F\Q, 0 contrary to the assumption.
4. Main
result
In this section we prove our main result. We use the following
elementary
fact.
a(r)/r < cc. Then LEMMA 4.1. - Let a: (0,03) --f [O; OS) be afunction with limsup,,,,, fiw all 0 < E’ < E < 1 there exists an unbounded sequence 0 < 7’1 < 7’2 < . , . such that E’U(T-,) < U(ET~) for all i. ANNALES
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Proof - Assume first that for all TO > 0 there exists T > r-a with u,(r)/r > 5 := lim SUF),,~ u(r)/r. Then one easily finds 0 < r1 < r2 < . . . such that a(ri)/ri < u,(u~)/(E~.~) for all i. Thus &a(~,) 5 ECU < u(eri). On the other hand, if u(r)/r 5 Z for r >_ r-0 > 0 say, then one may pick rt, 5 r1 < ‘7.2< . . . such that ~(ET,)/(ET~) > (E‘/ E)-a f or i > 1, hence e’a(ri) 5 E’ZT; 5 CL(ET,). We restate Theorem B for convenience. THEOREM 4.2. - Let (X, d) be a locally, compact and cocompact Hadamard space containing a k-$at but 110(k + l)-$at, where k 2 1. Then for all L > 0 and C 2 0 there exists D > 0 such that the following holds. Let FcX be a k-flat, f: R” + X an (L, C)-quasiisometric map, and for r > 0 define a(r) := sup{d(f(z): F) : (51 <_ r). Zf lim sup,,, a(r)/r < L-l, then Hd(f(R”),F) < D.
Proof. - According to 1.2 there exists a map f’: R” + X satisfying
I;-112 - y1 - C’ F d(f’(z): f’(Y)) I L’lz - Yl and d(f’(rc), f(:r)) < C” for all 2: y E [w”, where L’ := &(L + C), C’ := 2v%L + SC, and C” := AL + C. Redefining u(r) := sup{d(f’(z),F) : (z( 5 r}, we still have E := lim sup a(r)/r T’X
< L-l .
Let 7r: X --+ F denote the projection, h := 7ro f’, and F’ := f’(lw”). For z E F and T > 0 define BF(z, r) := {y E F : d(y, 2) < r} and a,‘(r) := sup{d(y, F’) : y E BF(h.(0),,r)}. Fix 0 < 6 < L-l - zi, and let /3 := SL/L’. From 2.4 we know that for T sufficiently large, a(fir) 5 En’(r) + C, where z := 5LL’ + 1 and ?? := LL’C’. Replacing T by Sr we see that (L(U) < &‘(nr) f ?? for E := S2L/L’. Finally, by 4.1 and 2.3, we get the existence of an unbounded sequence 0 < r1 < r2 < . . such that for all i, u(c) < 2E-1[7A(s7-i) + C] (say) and h(Bk(ri)) contains BF(h(0), Sri). We pick y; E BF(h(0), 6~~) with d(yi, F’) = a’(6ri). Then we choose zi E B” (T,) such that yi E Dj := BF (W!
+w)
cB,(h(O),
(since a’(Sri) 5 Sri -t d(h(0): F’) we may assume that a’(6ri) II: E K, := h-‘(Di) ,I Bk(ri) we have
67-i) 5 4Sr, for all i). For
L-l 12- ZiI - c’ < d(“f’(z), f’(G)) < d(h(s), h(G)) + 247-i)
5 (; + 4i-‘L)u’(hri)+ dE-lc. 4e SeRlE
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Hence, denoting by ok the volume of the unit ball in I?“, we see that
On the other hand, for z E K,, qf’(4,
F)
2
4S’(.7:),
>
n’(6r,)
=
;o’ihr,i.
Y2) -
-
d(h(.c),
!I%)
$uf(S7$
Let n(i) be the maximal number of pairwise disjoint k-cubes of edge length 3s fitting into 11,. where s > 0 is the constant given by 3.1. We may assume that ia’(?5ri) > 3&s, then clearly 77,(i) > cxk[$u'(hr;) - 3&s]"/(39)". A ccording to 2.3 we may apply 3.2 to each of these cubes, thus we deduce that
k
I(
1 + +z(Srr)
>
/L'".
Combining the two estimai:es for Xk(Ki), we obtain an upper barrier for a’(&,). Since I 5 2~-‘[%‘(Sri) +??I, it follows that Hd(F’, F) is bounded above by some constant which, however, still depends on the choice of 6 E (0, L-l - a) and hence on 5. But now the argument can be repeated with Z = 0, which then yields a estimate in terms of L, C, k, s, and m. Finally, since d(f’(~),f(~)) 5 C” = &I, + C for all z E RK, the theorem follows. 0 As mentioned in the introduction, the constant L-l given in the theorem is optimal. Moreover, if for a concrete Hadamard space X a pair of numbers s, m satisfying 3.1 is known. then the present method allows, in principle, to determine D explicitly as a function of the quasiisometry constants L and C.
5. Pairs of flats In this section we specialize the statement of Theorem 4.2 to the case where f: R” + X is an isometric map, i.e. F’ := f( W”) IS . a k-flat in X. In this case, the upper limit U := lim sup,,, U(T)/T considered in the theorem is actually a limit and can be interpreted in terms of the angle metric i on the boundary at infinity X(m) of X, as explained below. (See [Ba, Ch. II] for the definition of the metric space (X(x’), i).) For illustration, consider two intersecting straight lines F: F’ in R ‘. Then clearly a correspondingly defined E would just be equal to the sine of the angle between F and F’. More generally, for two flats F, F’ of possibly distinct dimension in a Hadamard space X, define an asymptotic angle by
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where F(m) and F’( xc) denote the boundaries at infinity of F and F’, respectively. By compactness, the supremum is attained at some point < E F’(K ). The Hausdorff distance between F(x) and E“( cc ) with respect to i is defined by
HdL(Fb):F'(~))
:= rnax{~~&!~F'}.
Now Theorem C in the introduction is a direct consequenceof Theorem B and the following proposition. Note that everything stated below remains true if F is replaced by a locally compact, closed convex subset of X containing a geodesic ray. PROPOSITION 5.1. - Let (X, d) he a Hudamurd spac,econtaining twojluts F. F’ (of possibl) distinct dimensions). Pick z f F’, and for r > 0 define (l,(r) := sup{d(z. F) : 1‘ E F’, d(z, z) 5 r}. Then the limit Z := lim,,cx. u(,r)/r exists (and is independent of the choice of 2). Moreover E E [O. 11, and Z = 1 {f cmd only if i,lF > ~12. If L p F 5 ~12, then sin(iFrF) = 5. We need several lemmas. LEMMA 5.2. - Thefu.action dejined by U{)(T) := [a,@)-a(O)]/rfi>r
and 0 5 a0 5 1. In particular the limit U := lim,,,
u(r)/r
r > 0 is nondecreasing, exists and z E [(I, I].
Proof. - Clearly a(0) 5 u(r) 5 r’ + a(O). thus 0 < cl,o < 1. Let 0 < ~1 < ~2. Since II: H d(z, F) is a convex function on F’? a(rl) = d(zl, F) for some ~‘1 E F’ with d(zr:l. 2) = ‘r1. Let (T: [O.m) + F’ be the unit speed ray with (~(0) = z and O(Q) = x1, and let b: [&co) --i R be the convex function defined by b(t*) := d(cT(r), F). Then b(0) = IL(O), In = a(ri), and b(r) < a(,r) for all r 2 0. By convexity, [b(~) - ~(O)]/Q > [tf(~r) - b(O)]/ri. which implies the result. 5.3. -Let CJ:[O! x) - X be a unit speedrax \vith o( WI) =: <, andfor r > 0 deji”n_ b(r) := d(a(r), F). Then 6 := lim,.+x, ?/(T)/T exists, b E [O: I], atzd i(<. P(x#)) > arcsin b. LEMMA
Proof. - The existence of 5 E [0, l] follows as in 5.2. Assume first that 8 < 1. We may replace g by an asymptotic ray and therefore assume without loss of generality that (~(0) E F. Let 11E F(~G) be an arbitrary point, and let p: [0, co) -+ F be the ray of speed (1 - -* b ) I/* from a(0) to 71. Consider the convex function defined by (‘(7.) := d(a(r),p(r)) for *r > 0. Clearly b(r) 5 c(r) 5 2r, thus C := lim,,, c(r)/.r exists and C 2 $. By [Ba, 11.4.41,i([, 77)equals the angle opposite to the side of length ;? in a euclidean triangle with sides of length 1, (1 - &2)1/2, and C. Since C 2 6 it follows that i(<. 7) > arcsiu 5. cl An obvious limit argument yields the claim for 6 = 1. LEMMA
q E F(x#)
5.4. - Let CT, I, b, and 6 be given as in 5.3. If 8 < 1 then there exists a point with i(<, q) = arcsinb < 7r/2.
Proof. - As in the proof of 5.2 we may assumethat a(0) E F. For r > 0 we consider the geodesic triangle with vertices g(O), g(r). Q(T). where q(r) is the point in F closest to cr(r). We may further assumethat b(r) = d(a(s),g(r)) > 0, and since b(r) < r6 < r, we 4e
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also have d(,r) := d(cr(O),q(r)) > 0. Then it is easily seen that iy(r.)(~(0)!~(~)) 2 T/Z, thus $(r) := i cr(T)(m. (1(r)) I n/2. We claim furthermore that cos @cr.) < 5 for all r’ > 0. To see this, consider for small .5 > 0 the geodesic triangle a, in X with vertices a(r), (T(T - s2),ps, where ps is the point on [U(T), q(r)] with d(ps, n(,r)) = s. Since 6 is convex, 6(r) - 6(~ - s2) 5 s2$, thus
d((T(T - .S’),J&) >
lJ(7’
-
S2)
-
6(7’)
+
S >
S -
S2,.
Let h,s be the euclidean comparison triangle for a,. The angle at the vertex of As corresponding to n(r) is al: least as large as the angle subtended by two sides of length 1 and .$-l in a euclidean triangle with third side of length s-l - 5. For s --+ 0 this angle converges to arccos6. Thi:; proves the claim. By the cosine inequality [Ba. 1.5.2(ii)] d(r)2 > r2 + 6(rj2 - 2r61
Since linl,.,ix:
b(r)/r
it follows
> -d(7-)* > 1 + -QTl* -
,7.2
-
7-2
= $ we see that lim++,
that 1.’ > d(r)’
- 26(‘)r
+ 6(~)~
and
’
c~(T)/T = (1 - 1,‘)l’“.
with ~~(0) = ~(0) and p,.(r) = q(r). Let pT: [O. r] + F be the geodesic of speed d(r)/r Consider the convex function defined by I := n’(rr(t), pr(t)) for t E [O, r]. We have c,,(O) = 0 and C,(T) = 6( t*). thus c,.(t) < th(r)/,r for all t E [O:r]. Let /I: [0, a) -+ F be an accumulation ray of the family (P~)~>O. This is a ray of speed (1 - b2)l/‘, and n’(~(t).p(t)) 5 th for all t > (1. Let c := lim+,= nj(~~(f),p(t))/f and 7~ := p(jz~). As in the proof of 5.3 we apply [Ba, 11.4.41 which, since C < 6, this time yields L:(<: TJ) 2 arcsiuz < 7r/2. Together with 5.3 this gives the result. 0 Now
we are ready to prove Proposition
5.1.
Proof
qf PI-oposirion 5.1. - Lemma 5.2 shows that E E [O. 11 exists.
It is not difficult to set: that there exists a sequence of rays g’,: [0, x) -+ F’ with lim,,, 5; = U, where $; :.= lim,,, d(uI(,r). F)/r. By 5.3, ipF > i(o+(~),F(cm)) 2 arcsin$,, hence iF, F 2 arcsinK In particular, 5 = 1 implies that i~t F > r/2. Now assume that Z < 1. Pick < E F’(x) with i([. F(x)) = iF’ F, and let 0: [0, CC) ----* F’ be the unit speed ray from z to [. Then clearly 6 := lin17,x d(a(r), F)/r < Z < 1. By 5.4 there exists 71 E F@) with i(<;q) = arcsing, thus IF/F = i((!F(xj) < a,rcsinI; 5 arcsinx < 7r/2. 0
These arguments also show that sin( iF( F) = ?i if LF~ F 5 7r/2.
ACKNOWLEDGEMENT
We would like to thank Pierre Pansu for initiating stated after Theorem A. 4NNALES
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REFERENCES [Ba] [BGS] [Br] [BH]
Fe1 [KL] [Ma] [MO]
W. B.~LLMANN. Lectures on spaces of nonpositive curwture, DMV Seminar Band. No. 25, Base]: Birkhluser 1995. W. BALLMANN, M. GXOMOV and V. SCHROEDER: Munifolds of nonpositive curvuture, Base]. Birkhauser 1985. M. R. BRIDSON. On t,he existence ofjlat planes in spaces of non-positive curvature (Proc. Amer. Muth. Sot., Vol. 123, 1995, pp. 223-235). M. R. BRIDSON and A. HAEFLIGER, Metric spacrs @‘non-positive curvature, in preparation. H. FEDERER. Geomrt;ic measure theory, Berlin, Springer 1969. M. KAPOVICH and B. LEER, Quasi-i.sometries presewr the geometric decomposition of Haken manifolds, preprint, December 29, 1995. W. S. MASSEY. A ha:ic conrxe in algebraic topology, New York. Springer 1991. G. D. MOSTOW, Strorlg rigidit?; of locally symmetric spaces (Ann. of Math. Studies, No. 78, Princeton Univ. Press 1973). (Manuscript u. LANG Forschungsinstitut fur Mathematik, ETH Zentrum, Rlmistr. 101, CH-8092 Zurich, S;witzerland. E-mail: [email protected] V. SCHROEIER Institut fiir Mathematik, Universitlt Zurich-Irchel, Winterthurer Str. 190, CH-8057 Zurich, E,witzerland. E-mail: [email protected]
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QUASIF’LATS
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URS
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VIKTOR
SPACES
SCHROEDER
ABSTRACT. - Let 9 be a simply connected, complete geodesic metric space which is nonpositively curved in the sense of Alexandrov. We assume that S contains a k-flat E’ of maximal dimension and consider quasiisometric embeddings f: IF;p” - X whose distance function from F satisfies a certain asymptotic growth condition. We prove that if X is locally compact and cocompact, then the Hausdorff distance between ,f(@) and F is uniformly bounded. This generalizes a well-known lemma of Mostow on quasiflats in symmetric spaces of noncompact type.
RBsut&. - Soit X un espace metrique geodesique complet et simplement connexe courbe de maniere nonpositive au sens d’Alexandrov. Noits supposons que S contient un k-p]& F de dimension maximale et considtrons des quasi-plats f : R” -+ *Y teh que les fonctions mesurant la distance a partir de F veritient une certaine condition de croissance lineaire. Nous demontrons que la distance de Hausdorff entre f( Iw’. ) et F est uniformement born&e lorsque X est localement compact et cocompact. Ceci generalise un lemme bien connu de Mostow sur les quasi-plats des espaces symetriqu,:s de type non-compact.
Introduction Quasiflats, i.e. quasiisometrically embedded euclidean spaces,play an important role in the theory of nonpositively curved manifolds. A striking example is Mostow’s proof of the rigidity theorem for locally symmetric spaceswhich, in the caseof higher rank, largely relies on the interplay between flats and quasiflats. More precisely, let X’: X be two symmetric spaces of noncompact type and rank k, and let p: r’ + I‘ be an isomorphism between discrete groups of isometries acting cocompactly on S’ and X respectively. A &flat F’ in X’ is called I”-compact if its stabilizer I?>, in r’ acts cocompactly on F’. Given such F’, there exists a unique r-compact X--flat F in X with stabilizer I’F = p(l$), $ [MO, 13.11. A key step in Mostow’s argument is the following uniform estimate, c$ [MO, 13.21. THEOREM A (Mostow). -- Let f: X’ --+ X be a quasiisometric map equivariant with respect to p. Then there exists a constant D 2 0 such that the following holds. If F’ is a II’-compact k-Jiat in X’, aEd if F is its unique companion in X as described above, then the Hausdorff distance between f (F’) and F satisfies Hd( f (F’), F) < D. This result then gives rise to a homeomorphism between the Furstenberg boundaries of X’ and X, which by a theorem of Tits is induced, after suitable renormalization, by a p-equivariant isometry of X’ and X. AUNALES
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In Theorem A, the assumptions on F’ and F clearly imply that Hd(f(F’), F) < cc. One may ask the following question: If a single quasillat f: W” -+ X lies within finite distance of a k-flat F in X, does this already imply that Hd(f(R”): F) is bounded above by some constant depending only on X and the quasiisometry constants of f? In this paper we show that this holds true in an even more general context. We consider quasiflats f: R” ---f X, where X is a Hadamard space in the sense of Alexandrov. i.e. a simply connected, complete geodesic space all of whose geodesic triangles satisfy the CAT(O) inequality, c$ Section 1. Moreover, X is assumed to be locally compact and cocompact. We prove: THEOREM B. - Let (X, d) be a locally compact and cocompact Hadamard space containing a k-flat but no (k + l)-JEat, where k > 1. Then for all L > 0 and C > 0 there exists D > 0 such that the following holds. Let FcX be a k-flat, f: Rk -+ X an (L> C)quasiisometric map (cf. l.l), and for T > 0 define u(r) := sup{d(f(z), F) : (z( <_ r}. rf lim s~p,,~ u(Y)/~ < L-l, then Hd(f(R’),F) 5 D. The constant L-’ bounding the linear growth of a(r) is optimal, as is shown by the following simple example. Let F be a geodesic line in the hyperbolic plane H2, and let f:lR t H2 be a geodesic of speed L-l with f(R) # F (and hence Hd(f(R),F) = m>. Then f is an (L, 0)-quasiisometric map as defined in 1.1, and it is easily seen that in this case, a(r)/ T + 1L-l for T -+ ‘cc. By taking products H2 x RkrP1, similar examples are obtained for all ic > 1.
In the case where J*: R” -+ X is an isometric map, and hence F’ := f(R”) is a k-flat in X, the upper limit considered in Theorem B is actually a limit and can be interpreted in terms of the angle metric i on the boundary at infinity X(m) of X, cj 5.1. Let HdL denote the Hausdorff #distance induced by i, and let F(m) and F’(m) be the boundaries at infinity of F and F’ respectively. Then Theorem B yields the following dichotomy. THEOREM C. - Let X and k be given as in Theorem B, and let F, F’ be two k-juts in X. F’(m)) = 0 (and F, F’ lie within Then either HdL(F(oc), F’(m)) 2 7r/2, or HdL(F(ec), uniformly bounded distance from each other).
The paper is divided into five and prove a basic approximation between quasiflats and flats. In discuss the special case where
short sections. In the first section we fix the terminology lemma. Sections 2 and 3 contain several results on maps Section 4 we prove Theorem B. In the last section we F’ is itself a hat in X.
After finishing this paper we learned that Bruce Kleiner had announced a result very similar to Theorem B in summer 1995, before the completion of our work. The proofs rely on different methods. Another result in the same spirit is proved in [KL, Section 41.
1. Preliminaries We briefly recall some basic notions and results from the theory of nonpositively curved metric spaces in the sense of Alexandrov. The general references are [Ba], [BGS], and the forthcoming book. [BH]. 4e
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Let (X, d) be a metric space. A continuous map 0: I - X from a connected subset I of R into X is said to be a nzinimi,-ing geodesic if it is isometric up to a constant factor, i.e. if there exists u > 0 such that d(a(r).g(s)) = /:lr - s/ for all ‘r,.s E 1. Then I! is called the speed of 0. More generally. the curve (T is said to be a geodesic if there is G > 0 such that 0 is 1ocall:y minimizing and of speed U. A geodesic .segmenf is a geodesic defined on a compact interval. A geodesic triangle A in X is a triple (al. ~2. m3) of geodesic segments cr,: 1, --$ X, the sides of a. whose endpoints match in the usual way. A euclidean comparison triangle A for a is a correspondin g triple of geodesic segments -5i : I, --f R” such that ??, has the same length as cr;, ,i = 1.2.3, and such that the endpoints of F1 : a2, 53 match in the same way as those of nl, g2. ‘T:~, Then a is said to satisfy the CAT(O) inequali4 if 3 as above exists and if d(aL@). gJ(s)) 5 Ia, - rr,(.s)I whenever ‘r E Ii, s E I,, and a.,j cz {1.2:3}. A metric space (X, $) is called a geodesic space if for all .I:. y E X there exists a minimizing geodesic 0: [O, l] - X with a(O) = s and m(l) = y. Note that for a geodesic triangle a in X with minimizing sides, a euclidean comparison triangle ,4 exists (and is unique up to isometry). Throughout the paper, (X, d) is assumed to be a simply connected, complete geodesic space all of whose geodesic triangles satisfy the CAT(O) inequality. We will refer to X simply ~1sa Hadamard space. A fundamental feature of these spaces is that for all Z, y E X there is a unique geodesic from :c to ;y (whose image we denote by [:1:,y]), and hence all geodesics in X are minimizing. Moreover, for all pairs of geodesics crl, g2: I -+ X, the function mapping t E I to ~i(a~(t), cam) is convex. We will often use the nearest point projection r: X -+ F from X onto a closed convex subset E’ of X. This map is well-defined and Lpschitz with constant 1,
4 X between two if for all :r. y E X’,
It is often necessary to approximate general quasiisometric maps by continuous ones. The argument proving the followin, 0 lemma is well-known, but an appropriate reference does not seem to be available in the literature. LEMMA 1.2. - Let C 2 0, and f: I3 -+ Then for ever?j X > &(L + X-lC) and (L, C)-quasiisometric,
B be a closed convex subset of R”, X a Hadamard space, L > 0, X a map satisfying d(f(i7:): f(y)) 5 Ljx - yl + C for all :r. y E B. 0 there exists a map f’: B -+ X which is Lipschitz with constant satisjies d(f’(z): f(z)) 5 &XL + Cfor all z E B. ZfmoreoL>er f is then d(f’(:r), f’(y)) > LP1/z-y - (2v’%XL+SC)forall .I:, y E B.
Proof. - Let 7r: Iw’ + B denote the nearest point projection. We f o 7r: R” - X and call this map f again. Then f satisfies ~j(f(.r)~ f(y)) < for all z,y E R”. For 2’ = 0: 1,. . . , Ic let 5; denote the closed i-skeleton of the canonical R” into cubes of edge length X. On So = (AZ)“, put f’ := f. Then for assuming that f’ is already defined on SP1, we extend f’ to S; as follows. ANNALES
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be the canonical basis of R”, and let s E S,,\S,-1. Let .j = j(x) E (1,. . . ~k} be the maximal index such that there exists z = Z(X) E S,-i with :I: E [z, z + Xcj]. Then f/(.x) is determined by defining .t”][z. z + Xe,,] to be a geodesic. Consider the following assertion -4(i): for all pairs (j,~) = (j(s),z(z)) with .): E &\.!?-I, d(f’(~:~,f’(~ + Xc,)) i XL + C, and for each %-plane PicSt, f’]P; is Lipschitz with constant &(L + A-‘C). Clearly A(1 ) holds. We show that for i = 2. . . . : A:, A(i- 1) implies A(1:). (Given (3,~) = (j(x). Z(X)) with .r E S;\S,-i, there exist p, Q E S,-2 such that z E [n. (I] ard both f’][2J> 91 and f’][Tj + Xc,, 4 + Aej] are geodesics. Moreover, A(i - 1) implies that n(f’(p). j’(n + Xej)), d(f’(q). f’(q + A:,,)) < XL + C. Thus by convexity, d(.f’(s),
f’(Z
+ Xi:,))
< XL + C.
It suffices to prove the claimed Lipschitz property for ,f’]Q, where Qi N [O, Xli is an i-cell of the given cubical subdivision of R”. Let :I:. z’ E QZ. and let [z, z + XeJ] and [z’, z’ + Xc,] be th e segments in Q1 containing :I: and .I:’ respectively, where j is maximal. Let y E [z! z + Xcj] be the point closest to 2. Since ,f’][~, z + Xe,r] is a geodesic of length at most XL + C,
d(,f’(:c), f’(!/)) I (L + PC&: On the other hand, Alla - 1) asserts that d(f’(z). m(L + X-lC)/z - 2’1, hence d(f’(y). by convexity.
Combining
f’(d))
2 &-T(L
these estimates
- yl.
.f’( .?)), d(~f’(r: + Xcj), f’(2
+ x-lC)I!/
+ Xe, )) <
- :r’j
we see that
d(f’(.~). f’(~‘j)~ 5 [d(f’(+ .f’(:vj) + d(f’j?l). f’(:#))12 < I;d(f’(:x), f’(y))’ + i(i - I)-‘d(,f’(yy): f’(2’))2 < 6jL + )\-‘q21:r.
- :cy
proving A(i). In particular, we have shown that f’ is Lipschitz with constant &(L+X-‘C). Given IL’ E R’;, let k'c (X7)k be the set of vertices of a cube of edge length X containing Z. Then f maps I/ into the metric ball with center f(:~) and radius &XL + C. By the above construction, f’(~) is contained in the convex hull of f(V). Since metric balls in a Hadamard space are convex, it follows that d(f’(z): f(x)) < &XL + C. Finally, if f is (L, C)-quasiisometric on J3, we see that L-ll.~
- ?/I - c L W(x),
-
TOME
30 -
5 d(f’(x),
f’(y))
+ q&XL
+ C) 0
for all x,?/ E B. ae S6RfE
f(y))
1997
- U”
3
QUASIFLATS IN HADAMARD SPACES
2. Continuous
343
quasiflats
In this section we prove two independent results applying to continuous quasiflats f: W” -+ X, where X is a general Hadamard space containing a k-flat F for some k 2 1. We emphasize that here k, need not be maximal as in Theorem B. We start with a simple topological lemma. For r > 0 and z E Rk’ we define B’(z) r) := {r E Rk- : 1~1. - ~1 5 I.} and S”-l(z,r) := {.r E R” : 1.1.- ~1 = r}. Instead of B’(f). rj and S’“-l(O,r) we write B”(r) and Skpl(,rj respectively. LEMMA 2.1. - Let k 2 I, r > 0, and let f: B”(, r ) - R” be a continuous mup suti@ing f(r) # f(y) for all 3:. y E Bk( r ) with 1.r + /yI = I:/. - yl = 7’. Then flSk.-‘(;l,) i.s not contractible (i.e. homotopic to a constant map) in R”\{ f’(0)).
Proof. - Assume the contrary. Then there exists a continuous map f: n’(r) R’\{f(O)} with flSP1(r~) = dlS-i(~). Define /a,l:ll’(r,) + S”-‘(l) by
For z E
Sk-l(r)
+
and 0 <: t 5 1, the denominator of
h,(z) :=
f((l
- tP):r)
- f((-tPb-)
If((i
- t/z):r)
- f((-t/2)3-)1
is nonzero. The map ki: S”-‘(r) -+ Sk-l(l) defined this way is homotopic to /lalS”-l(r) and can thus be extended to a continuous map ILL: B”(r) - S”-i (1). Finally, since h,(-z) = -h,(x) for :c f S“-l(r). h,l gives rise to a continuous map from S”‘(1) into S’+l( 1) preserving antipc’des. in contradiction to the Borsuk-Ulam theorem, c$ [Ma]. 0 For the statement of the next lemma the following definition is useful. DEFINITION 2.3. - Let k: > 1 and d; > 0. We call u continuous map f: W” --f Rk n-expanding {ff.for all 0 < b < -d, the following condition is satis$ed for 7‘ > 0 sufficiently large (depending on 6): If(x) - f(O)1 > ST ,for :I: E Sk’-l(r.), and .flS”‘-l(r) is not contractible in W”\{f(O)}.
The definition is independent of the choice of basepoints. Clearly 2.1 implies that every continuous map f: W” i UP satisfying If(.r) - .f(,f/)l 2 J!-~~:I: - y// - c7 for some constants L > 0. C 2 Clis L-‘-exp.mding. In particular, it follows that f is surjective, a fact which was proved in (MO, 10.1:. More generally, the following holds. LEMMA 2.3. - Let X be a Hadumard space containing a k-jlat F for some X: 2 1, and let T: X + E’ be the nearest point projection. Let L > 0, C > 0, and let ,f: k%” - X be a continuous map sati@ng d(f(:r). f(y)) > L,-‘lz - yI - C ,fin all :I:. y E Wk. For r > 0 define u(,r) := sup-(d(f(.x), F) : 1x15 ,I,}. 1f~ := limsup,.,, a(,r)/r. < L-l, then h := T o f: R” --+ F N U;P”is (I,-’ - cr.)-expanding. ANNALES
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Prooj: - For all -c :> 0 there exists T, > 0 such that a,(‘r)/r for .r:.?/ E B”(r) with ].K(+ 1~~1 = ]:I: - :r/j = 7‘. Il(f(.l*,l.
F) + d(f(;l/).
F) + d(h,(s).
h(y))
5 7i + E for r 2 r,. Then
> d(,f(x): >_ L-l1
f(y)) - c.
In case 1.x.1. ]yJ > ‘I’~, we have ~i(f(:~:), F) + ~1(f’(;//). F) < (CL+ E)(]J] + ];v]) = (E + E)T. In case ]:I./ < rE or ](I] < TV, we see that d(.f’(.K). F) + d(f(y).
F) < (ii + E)(?. + rE) 2 (E + E)(l + E)T
for I’ > Y~/E. Thus ri( r?(.l.). It,(l/)) > [C’ from 2.1.
- ((I + ~)(l + z)]r - C, and the claim follows 0 The definition of the Hausdorff distance of two subsetsA, A’ of X reflects the fact that there are two ways of measuring distances between these sets: Hd(A, A’) := sup{n, a’}. where (I := SI~I)~,~-~~ d(:r:‘. A) and (1,’:= SUI),,~,~d(n:, A’). The following lemma shows, in particular, that if Iz = F and A’ = f(R”j are (quasi-)flats in X1 then a is bounded above in terms of (I’. More generally. the statement is adapted to the hypotheses of Theorem B. LEMMA 2.4. - Let .I7 be u Hadmnurd space containing a k-&t F ,for same X: > 1, and let z E F be some busepoint. Let L, L’ > 0, C > 0, and let ,f: Rk’ + X be a map .satisfiirzg L-l I:/: - I// -~ C 5 d(.f(.r). j(y)) < L’1.r - ?//I for all .r, y E R”. For T > o dejine n(r) := sup{d(f(:r). f’) : 1.1.1< ,I,} afzd c/(7.) := SllI,{d(?J, f(@)) : y E F, d(y,z) < r}. Jf i? := lim sitp,.,,~ a( V)/T < L-l, then,for all 0 < $ < (1 - iZ)/L’ there exists r;j > 0 such that (I(/%) 5 I,a’:r) + T;‘,for I’ > r ,,. where z := 3LL’ + 1 and c := LL’C. Proof. - Let n: X -7‘ F denote the projection. and let h, := r o f. Without loss of generality we may assume that z = /l,(O). For fixed ‘I’ > 0 we define a map 9: Bk’(,l) - R” as follows. First, by the properties of f, we can associate to each :I: E Bk(?-) a point YEE R” with
fl(f(Fj. Note that h(x) O(:C) 5 o’(L’r).
5
/L(T)) = d(h(.r). f(R”))
and. since d(l~(:c),z)
o(r)
5
=: [J(T) . d(S(z),%(O))
5 L’]z]
< L’r,
+ C]) c 1osestto Z: then ]g(:‘;.) -F]
< Lh(.c).
Moreover. 17 - XI < L[d(j(S). < L[d(f(T). 2 L[b(.r)
f(X))
+ C]
h(3.))
+ d(h(.r)!
L[n(r)
l,y(:I’)
!/(!/)I
+ C]
+ a(7.) + C].
We define !](,I,) to be the point in Bk(.r, Hence, for .I:. y E Bk:7.), -
f(X))
5
L[b(X)
+
h(y)]
<
L[h(:C)
+
b(y)
<
L[t?h(.K)
5
L[h’(L’7.)
+
+ +
2b(y) +
1-r d(f(;r;), +
L’lX
-/I f(g))
d(h(T), -
z/l +
+ h(?/))
c].
c] +
c]
QUASIFLATS
Let X > 0. The procedure carried g’: B”(r) + R” which satisfies
IN HADAMARD
out in the proof
lg’(.c) - g(z)\ for all 3’ E B”(r)
and coincides
5 L[vGAL’
with g on (AZ)”
345
SPACES
of 1.2 yields
+ 4u’(L’r)
a continuous
map
+ C]
n B”(r).
We claim that the image of g’ contains B”(r - L[a(r) + C] - 2&X) =: B’. Let y E B”(r - &A), and let V c (AZ)’ n B”(r) denote the set of vertices of a cube of edge length X containing y. For L: E V, lg(z) - 1~15 1,9(z) - :cI + Iz - ~1 5 L[u(T) + C] + &A. Hence, by the definition of g’?
Id(Y) - 1Jl5 a+-) + Cl + km for all y E B”(r - fix). N ow the claim follows easily from the fact that there is no retraction B”(1) -+ Sk-l(l). Let y E B’. Then there exists .2: E B”(r) with g’(z) = y, hence W(Y),
E‘) L W(g), f(3) + Wm. N:c)) 5 L’[lg’(z) - g(.r)l + lg(:r) - %I] + b(n:) 5 L’L[dLw
+ Sa’(L’r)
Since this holds for all X > 0, using the continuity
+ C] + n’(L’r) of f we obtain
d(f(:r), F) < ,o,‘(L’r) + c for all 2 E B”(r--L[a(r)t-Cl), w h ere z := 5L’L+l and c := L’LC. Let E > 0. Then for T sufficiently large: a(r) 5 (E+ t-) T and C < ET. Thus a(~ - L[z + 2&]7-) 5 %‘(L’r) + ??, and replacing 1^by T/L’ we get the desired result. 0
3. A basic area estimate In [MO, 6.41, Mostow proved an estimate for the k-dimensional Jacobian of the normal projection K: X -+ F onto a &flat FcX, where X is a nonpositively curved symmetric space of rank Ic. Our aim is to deduce a non-infinitesimal analogue of this result applying to flats of maximal dimension in a Hadamard space X. Here now we assume X to be locally compact and cocompact, where the latter condition means that there exists a compact subset K of X such that ihe translates of K by the isometry group of X cover X. First we establish a diameter estimate for certain subsets of X. LEMMA 3.1. - Let X be a locally compact and cocompact Hadamard space containing a k-flat but no (k + l)-$at, where k > 1. Then there exist s,m > 0 such that the following holds. Let Fc.Y be a k-flat, r: X 4 F the projection, and let QC F be a closed k-dimensional euclidean cube of edge length 3s. Let ql> . . . , q,U be the vertices of Q, where 2: := 2’“. For i = 1) . . , u let QiCQ denote the k-cube of edge length s containing qi.
ANNALES
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AND
V. SCHROEDER
If V is a subset of X such that r(V) meets each Qi, and if d(V, F) > s, then the diameter of V satisfies diamV 2 s + md( V! F). Proof. - By the assumptions on S there exists s > 0 such that S does not contain a closed convex set isometric to a (k + I)-dimensional euclidean cube of edge length S, cJ: [Br, 3.11. For i = 1,. . . 12~ pick zi E V with x(x~) E Q;, and let 2:: E [:I:~,T(x;)] be the point with d(:r:‘, F) = s. First we show that there are %,,j E { 1. . ,,v} such that rl(~:: z$) > cI(~(.r;): ~(2:~)). If not, then d(~::, x$) = d(7r(ri). n(.r.j)j for all i,j since 7r is Lipschitz with constant 1. The convexity of the distance function on X x X then implies that the segments [.t.:. rr(~;)] and [:rS. .ir(.r~,~)] are at constant distance from each other. As in [Ba, IS.91 it follows that the convex hull of lJ:1,[:1:. X(X,)] is isometric to the product of the convex hull of {r(.rt), . . . : ~(:r,;)} and [(I, .s]. Thus X contains an isometric copy of a (k: + l)-dimensional euclidean cube of edge length s, contrary to the choice of s. Using the cocompactness of X again, we see that in fact d(s::. .I$) > Xd(~(:r~), 7r( :r,i)) for some 1-j E {l..... ,I;} and for some X > 1 depending only on X and s (but not on the particular choice of F. Q and V). Hence by convexity, d(z,: .cj) 2 [l + (X - l)s-‘d(l~~. > s + (A - l)d(V.
F)]d(~(z~),
n(x,
j)
F) .
Thus the lemma holds for m = X - 1.
cl
Together with the coarea formula, 3.1 leads to the desired analogue of [MO, 6.41. For 0 < r: < k we denote by XFli the i-dimensional Hausdorff measure on R”. PROPOSITION3.2. - Let ,Y, k, s, and vx be given as in 3.1. Let FcX be a k-flat, K: X + F the projection, and let QcF be a closed k-dimensional cube of edie length 3s. Let f be an L-Lipschitz map from a compact ball BCR” into X, arid let h = n; o f. Assume that h(8B) f’ Q = v) and that h13B is not contractible in F\Q. [f d := d(f[t’(Q)], 8’) > s, then ?f’(h-l(Q)) 2 s’-l(s + rnd)/LC”. Proof. - Let v = 2k. Using induction on X: one shows that the vertices ql, . . , ql, of Q can always be ordered in such a way that for % = 1. . . ,7! -- 1, [q?, ql+l] is an edge of Q. Then let P = u:z,‘[qi, q,+l]. and let R denote the union of all k--cubes Q’cQ of edge length s meeting P. Define the cubes QicR as in 3.1, and denote by Z the common (X: - 1)-face of R and Q1 isometric to [O: s]‘.-l. For h: = 3 the situation is illustrated in Figure. We define a 1-Lipschitz retraction 4: R + 2 such that each fiber I’, := qV1{z} is a connected polygonal arc lying at constant distance from P. Let jr = rr o f; then 4 o /I, is Lipschitz with constant L. Now the coarea formula [Fe, 3.2.221 yields
I’
.z
3-t1(h-l(P,))
d’FI”-‘(z)
I L”-“Ft”(h-l(Rj).
We claim that for each :: in the relative interior 2’ of 2 there exists a connected component C, of h-l ( Pz) such that h( CZ) meets each QT. With this at hand one can proceed as follows. By 3.1, dianif(CZ) > :: + ,rnd. Since CZ is connected, and since f is L-Lipschitz, we get IFtl(/l,-‘(Pz))
> X’(C,)
2 diamCZ
1 (s +
7724/L.
QUASIFLATS
IN HADAMARD
The
proof
347
SPACES
of 3.2
Together with the above integral estimate and the fact that IKQ this yields the desired result. It remains to show that for each z E 2’ there exists a connected component C,? of h-‘(Pz) such that h(Cz) meets each Q,. Suppose this is false for some z E 2’. Every connected component C of /L-‘(I’~) is mapped by 11 to a connected subarc of I’,. Denote by G the union of those components C with h(C) fl Qi # Cn.Then /i(G) n Qtz= 0. Note also that both G and P,\G are compact. For E > 0 let CT, := {X E h-‘(R) : n(:r, G) < E}. We can choose E such that 17, II h-l(P;) = G and, since h is L-Lipschitz. such that h( CT=) intersects the boundary of 12 (relative to F) only in 2. Our aim is to construct a continuous map &: B --+ F that coincides with h. on B\U, and maps G to {z}. For :y E Z and 5 E P,y let (Y(X) denote the length of the subarc of Pv with endpoints y/ and 2. Then for t 2 0 let - t}. For z E 157~define q!+(z) be the point on Py with N(&(x)) = max{O,tr(z) t(r)
:= I(1 - E-%(X,
G)) .
where I is the length of P,. Finally, set k(z) := &cX,(/l(z)) for 3: E Cr, and A(X) := h(z) for z E B\U,. Clearly k i:i a continuous map with h[iJB = hldB, and it is not difficult to see that /A omits the set (Pz\{x}) t7 Qr . It follows that /rliJB is contractible in F\Q, 0 contrary to the assumption.
4. Main
result
In this section we prove our main result. We use the following
elementary
fact.
a(r)/r < cc. Then LEMMA 4.1. - Let a: (0,03) --f [O; OS) be afunction with limsup,,,,, fiw all 0 < E’ < E < 1 there exists an unbounded sequence 0 < 7’1 < 7’2 < . , . such that E’U(T-,) < U(ET~) for all i. ANNALES
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Proof - Assume first that for all TO > 0 there exists T > r-a with u,(r)/r > 5 := lim SUF),,~ u(r)/r. Then one easily finds 0 < r1 < r2 < . . . such that a(ri)/ri < u,(u~)/(E~.~) for all i. Thus &a(~,) 5 ECU < u(eri). On the other hand, if u(r)/r 5 Z for r >_ r-0 > 0 say, then one may pick rt, 5 r1 < ‘7.2< . . . such that ~(ET,)/(ET~) > (E‘/ E)-a f or i > 1, hence e’a(ri) 5 E’ZT; 5 CL(ET,). We restate Theorem B for convenience. THEOREM 4.2. - Let (X, d) be a locally, compact and cocompact Hadamard space containing a k-$at but 110(k + l)-$at, where k 2 1. Then for all L > 0 and C 2 0 there exists D > 0 such that the following holds. Let FcX be a k-flat, f: R” + X an (L, C)-quasiisometric map, and for r > 0 define a(r) := sup{d(f(z): F) : (51 <_ r). Zf lim sup,,, a(r)/r < L-l, then Hd(f(R”),F) < D.
Proof. - According to 1.2 there exists a map f’: R” + X satisfying
I;-112 - y1 - C’ F d(f’(z): f’(Y)) I L’lz - Yl and d(f’(rc), f(:r)) < C” for all 2: y E [w”, where L’ := &(L + C), C’ := 2v%L + SC, and C” := AL + C. Redefining u(r) := sup{d(f’(z),F) : (z( 5 r}, we still have E := lim sup a(r)/r T’X
< L-l .
Let 7r: X --+ F denote the projection, h := 7ro f’, and F’ := f’(lw”). For z E F and T > 0 define BF(z, r) := {y E F : d(y, 2) < r} and a,‘(r) := sup{d(y, F’) : y E BF(h.(0),,r)}. Fix 0 < 6 < L-l - zi, and let /3 := SL/L’. From 2.4 we know that for T sufficiently large, a(fir) 5 En’(r) + C, where z := 5LL’ + 1 and ?? := LL’C’. Replacing T by Sr we see that (L(U) < &‘(nr) f ?? for E := S2L/L’. Finally, by 4.1 and 2.3, we get the existence of an unbounded sequence 0 < r1 < r2 < . . such that for all i, u(c) < 2E-1[7A(s7-i) + C] (say) and h(Bk(ri)) contains BF(h(0), Sri). We pick y; E BF(h(0), 6~~) with d(yi, F’) = a’(6ri). Then we choose zi E B” (T,) such that yi E Dj := BF (W!
+w)
cB,(h(O),
(since a’(Sri) 5 Sri -t d(h(0): F’) we may assume that a’(6ri) II: E K, := h-‘(Di) ,I Bk(ri) we have
67-i) 5 4Sr, for all i). For
L-l 12- ZiI - c’ < d(“f’(z), f’(G)) < d(h(s), h(G)) + 247-i)
5 (; + 4i-‘L)u’(hri)+ dE-lc. 4e SeRlE
-
TOME
30 - 1997
- No 3
QUASIFLATS
IN HADAYARD
349
SPACES
Hence, denoting by ok the volume of the unit ball in I?“, we see that
On the other hand, for z E K,, qf’(4,
F)
2
4S’(.7:),
>
n’(6r,)
=
;o’ihr,i.
Y2) -
-
d(h(.c),
!I%)
$uf(S7$
Let n(i) be the maximal number of pairwise disjoint k-cubes of edge length 3s fitting into 11,. where s > 0 is the constant given by 3.1. We may assume that ia’(?5ri) > 3&s, then clearly 77,(i) > cxk[$u'(hr;) - 3&s]"/(39)". A ccording to 2.3 we may apply 3.2 to each of these cubes, thus we deduce that
k
I(
1 + +z(Srr)
>
/L'".
Combining the two estimai:es for Xk(Ki), we obtain an upper barrier for a’(&,). Since I 5 2~-‘[%‘(Sri) +??I, it follows that Hd(F’, F) is bounded above by some constant which, however, still depends on the choice of 6 E (0, L-l - a) and hence on 5. But now the argument can be repeated with Z = 0, which then yields a estimate in terms of L, C, k, s, and m. Finally, since d(f’(~),f(~)) 5 C” = &I, + C for all z E RK, the theorem follows. 0 As mentioned in the introduction, the constant L-l given in the theorem is optimal. Moreover, if for a concrete Hadamard space X a pair of numbers s, m satisfying 3.1 is known. then the present method allows, in principle, to determine D explicitly as a function of the quasiisometry constants L and C.
5. Pairs of flats In this section we specialize the statement of Theorem 4.2 to the case where f: R” + X is an isometric map, i.e. F’ := f( W”) IS . a k-flat in X. In this case, the upper limit U := lim sup,,, U(T)/T considered in the theorem is actually a limit and can be interpreted in terms of the angle metric i on the boundary at infinity X(m) of X, as explained below. (See [Ba, Ch. II] for the definition of the metric space (X(x’), i).) For illustration, consider two intersecting straight lines F: F’ in R ‘. Then clearly a correspondingly defined E would just be equal to the sine of the angle between F and F’. More generally, for two flats F, F’ of possibly distinct dimension in a Hadamard space X, define an asymptotic angle by
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where F(m) and F’( xc) denote the boundaries at infinity of F and F’, respectively. By compactness, the supremum is attained at some point < E F’(K ). The Hausdorff distance between F(x) and E“( cc ) with respect to i is defined by
HdL(Fb):F'(~))
:= rnax{~~&!~F'}.
Now Theorem C in the introduction is a direct consequenceof Theorem B and the following proposition. Note that everything stated below remains true if F is replaced by a locally compact, closed convex subset of X containing a geodesic ray. PROPOSITION 5.1. - Let (X, d) he a Hudamurd spac,econtaining twojluts F. F’ (of possibl) distinct dimensions). Pick z f F’, and for r > 0 define (l,(r) := sup{d(z. F) : 1‘ E F’, d(z, z) 5 r}. Then the limit Z := lim,,cx. u(,r)/r exists (and is independent of the choice of 2). Moreover E E [O. 11, and Z = 1 {f cmd only if i,lF > ~12. If L p F 5 ~12, then sin(iFrF) = 5. We need several lemmas. LEMMA 5.2. - Thefu.action dejined by U{)(T) := [a,@)-a(O)]/rfi>r
and 0 5 a0 5 1. In particular the limit U := lim,,,
u(r)/r
r > 0 is nondecreasing, exists and z E [(I, I].
Proof. - Clearly a(0) 5 u(r) 5 r’ + a(O). thus 0 < cl,o < 1. Let 0 < ~1 < ~2. Since II: H d(z, F) is a convex function on F’? a(rl) = d(zl, F) for some ~‘1 E F’ with d(zr:l. 2) = ‘r1. Let (T: [O.m) + F’ be the unit speed ray with (~(0) = z and O(Q) = x1, and let b: [&co) --i R be the convex function defined by b(t*) := d(cT(r), F). Then b(0) = IL(O), In = a(ri), and b(r) < a(,r) for all r 2 0. By convexity, [b(~) - ~(O)]/Q > [tf(~r) - b(O)]/ri. which implies the result. 5.3. -Let CJ:[O! x) - X be a unit speedrax \vith o( WI) =: <, andfor r > 0 deji”n_ b(r) := d(a(r), F). Then 6 := lim,.+x, ?/(T)/T exists, b E [O: I], atzd i(<. P(x#)) > arcsin b. LEMMA
Proof. - The existence of 5 E [0, l] follows as in 5.2. Assume first that 8 < 1. We may replace g by an asymptotic ray and therefore assume without loss of generality that (~(0) E F. Let 11E F(~G) be an arbitrary point, and let p: [0, co) -+ F be the ray of speed (1 - -* b ) I/* from a(0) to 71. Consider the convex function defined by (‘(7.) := d(a(r),p(r)) for *r > 0. Clearly b(r) 5 c(r) 5 2r, thus C := lim,,, c(r)/.r exists and C 2 $. By [Ba, 11.4.41,i([, 77)equals the angle opposite to the side of length ;? in a euclidean triangle with sides of length 1, (1 - &2)1/2, and C. Since C 2 6 it follows that i(<. 7) > arcsiu 5. cl An obvious limit argument yields the claim for 6 = 1. LEMMA
q E F(x#)
5.4. - Let CT, I, b, and 6 be given as in 5.3. If 8 < 1 then there exists a point with i(<, q) = arcsinb < 7r/2.
Proof. - As in the proof of 5.2 we may assumethat a(0) E F. For r > 0 we consider the geodesic triangle with vertices g(O), g(r). Q(T). where q(r) is the point in F closest to cr(r). We may further assumethat b(r) = d(a(s),g(r)) > 0, and since b(r) < r6 < r, we 4e
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30 - 1997 - No 3
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IN HADAMARD
351
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also have d(,r) := d(cr(O),q(r)) > 0. Then it is easily seen that iy(r.)(~(0)!~(~)) 2 T/Z, thus $(r) := i cr(T)(m. (1(r)) I n/2. We claim furthermore that cos @cr.) < 5 for all r’ > 0. To see this, consider for small .5 > 0 the geodesic triangle a, in X with vertices a(r), (T(T - s2),ps, where ps is the point on [U(T), q(r)] with d(ps, n(,r)) = s. Since 6 is convex, 6(r) - 6(~ - s2) 5 s2$, thus
d((T(T - .S’),J&) >
lJ(7’
-
S2)
-
6(7’)
+
S >
S -
S2,.
Let h,s be the euclidean comparison triangle for a,. The angle at the vertex of As corresponding to n(r) is al: least as large as the angle subtended by two sides of length 1 and .$-l in a euclidean triangle with third side of length s-l - 5. For s --+ 0 this angle converges to arccos6. Thi:; proves the claim. By the cosine inequality [Ba. 1.5.2(ii)] d(r)2 > r2 + 6(rj2 - 2r61
Since linl,.,ix:
b(r)/r
it follows
> -d(7-)* > 1 + -QTl* -
,7.2
-
7-2
= $ we see that lim++,
that 1.’ > d(r)’
- 26(‘)r
+ 6(~)~
and
’
c~(T)/T = (1 - 1,‘)l’“.
with ~~(0) = ~(0) and p,.(r) = q(r). Let pT: [O. r] + F be the geodesic of speed d(r)/r Consider the convex function defined by I := n’(rr(t), pr(t)) for t E [O, r]. We have c,,(O) = 0 and C,(T) = 6( t*). thus c,.(t) < th(r)/,r for all t E [O:r]. Let /I: [0, a) -+ F be an accumulation ray of the family (P~)~>O. This is a ray of speed (1 - b2)l/‘, and n’(~(t).p(t)) 5 th for all t > (1. Let c := lim+,= nj(~~(f),p(t))/f and 7~ := p(jz~). As in the proof of 5.3 we apply [Ba, 11.4.41 which, since C < 6, this time yields L:(<: TJ) 2 arcsiuz < 7r/2. Together with 5.3 this gives the result. 0 Now
we are ready to prove Proposition
5.1.
Proof
qf PI-oposirion 5.1. - Lemma 5.2 shows that E E [O. 11 exists.
It is not difficult to set: that there exists a sequence of rays g’,: [0, x) -+ F’ with lim,,, 5; = U, where $; :.= lim,,, d(uI(,r). F)/r. By 5.3, ipF > i(o+(~),F(cm)) 2 arcsin$,, hence iF, F 2 arcsinK In particular, 5 = 1 implies that i~t F > r/2. Now assume that Z < 1. Pick < E F’(x) with i([. F(x)) = iF’ F, and let 0: [0, CC) ----* F’ be the unit speed ray from z to [. Then clearly 6 := lin17,x d(a(r), F)/r < Z < 1. By 5.4 there exists 71 E F@) with i(<;q) = arcsing, thus IF/F = i((!F(xj) < a,rcsinI; 5 arcsinx < 7r/2. 0
These arguments also show that sin( iF( F) = ?i if LF~ F 5 7r/2.
ACKNOWLEDGEMENT
We would like to thank Pierre Pansu for initiating stated after Theorem A. 4NNALES
SCIENTIFIQUES
DE L’6COl.E
NORMALE
SUPkRIEURE
this work
by asking
the question
352
U. LANG
AND
V. SCHROEDER
REFERENCES [Ba] [BGS] [Br] [BH]
Fe1 [KL] [Ma] [MO]
W. B.~LLMANN. Lectures on spaces of nonpositive curwture, DMV Seminar Band. No. 25, Base]: Birkhluser 1995. W. BALLMANN, M. GXOMOV and V. SCHROEDER: Munifolds of nonpositive curvuture, Base]. Birkhauser 1985. M. R. BRIDSON. On t,he existence ofjlat planes in spaces of non-positive curvature (Proc. Amer. Muth. Sot., Vol. 123, 1995, pp. 223-235). M. R. BRIDSON and A. HAEFLIGER, Metric spacrs @‘non-positive curvature, in preparation. H. FEDERER. Geomrt;ic measure theory, Berlin, Springer 1969. M. KAPOVICH and B. LEER, Quasi-i.sometries presewr the geometric decomposition of Haken manifolds, preprint, December 29, 1995. W. S. MASSEY. A ha:ic conrxe in algebraic topology, New York. Springer 1991. G. D. MOSTOW, Strorlg rigidit?; of locally symmetric spaces (Ann. of Math. Studies, No. 78, Princeton Univ. Press 1973). (Manuscript u. LANG Forschungsinstitut fur Mathematik, ETH Zentrum, Rlmistr. 101, CH-8092 Zurich, S;witzerland. E-mail: [email protected] V. SCHROEIER Institut fiir Mathematik, Universitlt Zurich-Irchel, Winterthurer Str. 190, CH-8057 Zurich, E,witzerland. E-mail: [email protected]
4e SBRlE
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TOME
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1997 --
No
3
received
February
1. 1996.)