a regularity theorem for alexandrov spaces

Math. Nachr. 164 (1993) 91-102

A Regularity Theorem for Alexandrov Spaces

By ZHONGMIN SHEN of Ann Arbor

(Received December 7, 1992)

1. Introduction

Since the notion of Gromov-Hausdorff distance between metric spaces was introduced by GROMOV [GLP], many important results in Riemannian geometry have been obtained. In some applications, one needs to understand the limit space of Riemannian manifolds with curvature equal to or greater than - k 2 . It has been proved that the limit space, which inherits several properties of the Riemannian manifolds, is an Alex- androv space with curvature equal to or greater than - k2. Roughly seaking, an Alex- androv space X of curvature equal to or greater than - k 2 is a complete, connected length space of finite Hausdorff dimension, in which the Toponogov comparison theorem for triangles holds, comparing with those in the space form M ( - k 2 ) of curvature - k 2 . Further it is proved in [BGP] that the Hausdorff dimension of X is an integer, which is called the dimension of X . Since there are examples of Alexandrov spaces with curvature equal to or greater than 1 which are not limits of manifolds with sectional

1 curvature > 6 > - [PWZ], the class of Alexandrov spaces with curvature equal to or

greater than - k2 strictly contains that of limit spaces of Riemannian n-manifolds with sectional curvature equal to or greater than - k2 . Recently, the class of n-dimensional spaces with curvature bounded below has become a subject of interest in its own right. Also refer to [P2] for some interesting work in this direction. The purpose of this paper is to detect the singularities of Alexandrov spaces.

Let X be an n-dimensional Alexandrov spaces with curvature equal to or greater than - k Z . Let bE(r) denote the volume of the r-ball in M(- k2) . For a subset A in X we denote by Vn(A) the n-dimensional Hausdorff measure of A. Let p E X . We define the n-dimensional density at p by

4

92 Math. Nachr. 164 (1993)

where B ( p , r ) denotes the open metric ball at p with radius r. It is clear that O,,(p) is independent of k whenever it is defined. By the relative volume comparison theorem of Bishop-Gromov type [Y], one can show that the above limit must exist and that 0 < O,,(x) < 1 for all x E X . Put O,,(X) = inf O,(x). We call O J X ) the density of X . Notice that if X is a Riemannian n-manifold, then O,,(X) = 1. According to a result in [OS] and the arguments in the following sections, we will see that if an Alexandrov space X has density 1, then X is a complete C o Riemannian manifold.

Let 0 < a < O,(p). The a-density radius r,(p) at p is defined as the largest number r such that

X€,Y

UWP, 4) 3 4 ( r ) ' We have the following:

Main Theorem. Let X be an n-dimensional Alexandrov space (n 2 2) with curvature 1

equal to or greater than - k 2 . Suppose that O,(p) > - at some point p E X . Then fov any 1 - < a < O,,(p), there is a number r , defined by 2

2

ro = 100(n - 1) ( a - i r + min ( r , [ p ) , $), such that any open metric ball B(p, v), v < yo, is homeomorphic to IR".

A point p E X is called a topologically regular point if there is an open neighborhood U p of p, which is homeomorphic to an open set in IR". Otherwise p is called a topologi-

cally singular point. The main theorem tells us that if O,,(p) > -, then p is a topologically regular point. Hence we have:

1 2

Corollary 1. Let X be an n-dimensional Alexandrov space with curvature bounded below. 1 2

Suppose that O,,(x) > - everywhere. Tlzen X is a topological n-manifold.

1 2

Example. Let C = lRP"-' of constant curvature 1. Then V,- ,(C) = - V,- (Sn-'(l)).

Let K ( C ) denote the cone over C with the standard metric. Clearly, the vertex o is 1

a topologically singular point and O,,(o) = -. 2

For a point p e X we define the contractibility radius c ( p ) at p as the largest number r such that any open metric ball B ( p , v), v < Y, is contractible. We say that X has positive contractibility radius on compact sets if for any point p e X there is an open neighborhood Up of p such that c ( q ) : = lim c(x) > 0. When X is compact it is equivalent

to that c ( X ) = inf c(x) > 0. As pointed in [LS], there are examples of (compact) Alex- androv spaces which do not have positive contractibility radius on compact sets. Now

suppose that O,(p) > - at some point p e X . We will prove that O,,(x) is lower semi-

X E F

x o x

1 2

Shen, Alexandrov Spaces 93

continuous on X . Thus there is a small neighborhood Up at p and a number CI

such that O,(x) > a >- for all X E ~ . We will prove that the a-density radius

rJx) is also lower semi-continuous on Up. Thus ?=:= inf ra(x) > O . Let

1 2 -

x e U p

ro = ( C I - k ) ! ! min (fa, i). It follows from the main theorem that any

open metric ball B(x, v) , v < i,, x ~ q , is homeomorphic to IR". Therefore one obtains 100(n - 1)

the following corollary.

Corollary 2. Let X be an n-dimensional Alexandrov space with curvature bounded

below. Suppose that O,(x) > - everywhere. Then X has positive contractibility radius on compact sets.

Let X be an n-dimensional Alexandrov space with curvature equal to or greater than - k Z . For a pair ofpoints x, y E X , we denote by lxyl the distance between x and y . For any three points x, y, Z E X we denote by AZjZa triangle in M ( - k 2 ) with jxyl= 12j71, /yz( = /jZ?l and 12x1 = (221, and denote by e yxz the angle at 2 in AZjE. In [BGP] BURAGO, GROMOV and PERELMAN introduced a notion of (n, 6)-strained points in X . A point p E X is called an (n, +strained point if there exist points pi, qi E X , i = 1, . . . , n, such that the following holds:

1 2

C p i p q i > n - 6 , l < i < n , 71 n 71 < p i p p j > - - 6, e p ipq j > - - 6 , qipqj > - - 6, i # j . 2 2 2

(p i , qi);= is called an (n, &strainer of p and p:= min {Ippil, Ipqil} is called its length. BURAGO, GROMOV and PERELMAN proved that there exists a small number 6, > 0 such that if p has an (n, 6)-strainer with length equal to or greater than po, 6 < 6,, then the map f : X -+ IR" defined by f ( x ) : = ( I p l x ( , . . ., Ip,xl) is a z(6, r)-almost isometry of a metric ball B(p, r ) onto an open subset of IR", for sufficiently small r e ,uo, where lim z(6, r) = 0. In particular, this implies that pis a topologically regular point. Let Xn,d denote the set of all (n, &strained points in X . Then Xn,d is open dense in X and that Xn,a is a Lipschitz n-manifold [BGP]. In [OS], OTSU and SHIOYA introduce a notion of singular points in X. A point p is called a regular point if p is (n, &strained for all 6 > 0. Otherwise it is called a singular point. Topologically singular points must be singular points. Let S, denote the set ofsingular points in X . It is proved by OTSU and SHIOYA that S, is of Hausdorff dimension equal to or less than n - 1 and X\S, is a natural Co-Riemannian n-manifold.

1 I l I f l

6 , r - 0

Acknowledgement. The author would like to thank J. CHEEGER for bringing Yamaguchi's paper to his attention.

2. Critical points of distance functions

Since the notion of critical points of distance functions on Riemannian manifolds was introduced by GROVE and SHIOHAMA [GS], many important results have been obtained. In [PI PERELMAN has extended this concept to Alexandrov spaces and proved

94 Math. Nachr. 164 (1993)

that any point in an Alexandrov space has a spherical neighborhood which is homeomorphic to the tangent cone at this point. In this section we will recall some basic properties in the critical point theory for distance functions on Alexandrov spaces.

Let X be an Alexandrov space with curvature equal to or greater than - k 2 . Let

p E X and E E 0, ~ . A point x is called an E-critical point of d,(x):= 1~x1, x EX, if for

any point y E X , L ;>

7c e p x y d - + E .

2

Otherwise x is called an &-regular point of d,. Let p e X . We denote by X, the space of equivalence classes of minimal geodesics

emanating from p with a metric defined by the angle between minimal geodesics. Let C, denote the completion of ,EL. We call C, the space of directions. For a subset A c X we will denote by A’ c C, the set of all directions of minimal geodesics emanating from p to a point q in A with Ipq1 = IpAl.

Remark. By the first variation theorem [OS, Theorem 3.51, one can show that the above definition is equivalent to the following one: A point x is called an &-critical point of d , if for any direction E C,, there is a direction q E p’ c E x such that 1[ql< - + E.

This definition coincides with that given in [GS]. Let EX. It is proved in [BGP] that C, is an (n - 1)-dimensional Alexandrov space

with curvature equal to or greater than 1 and diameter equal to or less than n. The tangent cone at p is defined as the topological cone over C, with the following standard metric

n 2

I(<, s)(q, t)l = I/s2 + t2 - 2st cos Itql,

for any 5, q EL, and s, t 3 0. We denote the tangent cone at p by K ( Z J . Still denote by p the vertex of K(C,). For r > 0, let K,(C,) denote the open metric r-ball in K(C,) around p .

We need the following Isotopy Lemma for distance functions [PI (see also [LS]).

Jsotopy Lemma. Given E E 0, - . Let P E X and 0 < r1 < r2. Suppose that

B ( p , rJ\B(p, r , ) is free of &-critical point of d,. Then B ( p , r2)\B(p, r , ) is homeomorphic to S ( p , r l ) x [ r , . r z ] . Further if B ( p , r)\{p} is free of E-critical point of d,, then ( B ( p , r), S ( p , r ) ) is homeomorphic to (K,(C,,), C,), where S(p, r ) denotes the metric sphere with radius r around p.

One can show that for any point EX, there is a number r such that B ( p , r ) is free of critical point of d, ([PI [LS]). Therefore by the Isotopy Lemma, B ( p , v) , v < r, is homeomorphic to K,(C,). Notice that K,(C,) is contractible, so is B ( p , v) for all v < r. Thus the contractibility radius at p is positive, i.e. c ( p ) > 0.

For E E 0, - we denote by C,(E) the set of all E-critical points of d,. By [OS; Lemma

2.11 one can give an upper bound on the (n - 1)-dimensional Hausdorff measure of

( 3

( 3


C p ( ~ ) . More precisely, one has

3. The density

Let X be an n-dimensional Alexandrov space with curvature equal to or greater than - k2. We have the following relative volume comparison theorem of Bishop- Gromov type:

We outline the proof here since we need it. First by Toponogov comparison theorem one has

V , - , ( S ( p , R ) ) sinhfl-l kR V , - , ( S ( p , r)) sinh"-' kr'

< r < R . (3)

Then Proposition 1 follows from the co-area formula r

(4) j K- l ( S h t)) dt = K P h r ) ) . 0

Now it is easy to see that OJp) is well-defined and for all r > 0

The following proposition is the main result of this section.

Proposition 2. For any point p E X

where w , - ~ denotes the volume of the unit sphere S"-'(l) in R". Proof . First denote by I?(C,) the topological cone over C, with the following metric

I(<, s ) ( r , t)l = - cosh-'(cosh ks cosh kt -cos I<ql sinh ks sinh kt). 1 k (7)

Define a map p: X -, R(Z,) as follows. For every XEX, put p(x) = (5 , Ipxl), where 5 E X ' c Zp. Clearly, p maps B(p , r ) into Kr(CP), the open metric r-ball in E(C,) around the vertex. By Toponogov theorem, one has

IXYl d IP(x)P(Y)l*

96 Math. Nachr. 164 (1993)

Thus p is an expanding map and

V,(B(p , r)) G V , ( K ( q ) Therefore

Remark. According to the Russian version of [BGP], one can actually see that the equality (6) holds.

Proposition 3. The density function @,,(x) is lower semi-continuous on X

Proof . This proposition follows from the relative volume comparison theorem (I). Let EX and E < On@). There is a number r > 0 such that

KP(P> r ) ) 2- (@,(P) - 4 b m '

( @ , , ( P ) - 4 b;:(r) = (@Ad - 24 bnkk + 9) .

Take q > 0 which satisfies

This completes the proof. The above proposition immediately implies that if @,,(p) > a > 0 at some point p ,

then there is an open neighborhood Up at p such that @,(x) > a for all X E ~ . In order to prove Corollary 2 one also needs the following:

Proposition 4. Let a ~ ( 0 , 1). Suppose that @,,(x) > c1> 0 on some open subset tr in X . Then r,(x) is lower semi-continuous on U , i.e. at any point P E U .

> a. If not, there is a number ro < r,( p) T/,(B(P? r) ) We claim that for all r < r,(p), f ( r ) := such that b;:(r)

(8) T/,(B(p, r ) ) = a b W , r, < r < r,(p) From (3) one can see that K - , ( S ( p , r)) is a locally Lipschitz function. It follows from (4) and (8) that


By (3) one obtains

sinh kt "- Since K ( B ( p , r,)) = ctb;(r,), by (4) one has K-l(S(p, t)) = awn- (' ___ ) for all

This implies that ra(x) 3 r,(p) - E.

Now we suppose that r a ( p ) = + co. The same argument as above show that

4. A sphere theorem

Recall that a point p in an n-dimensional Alexandrov space X is said to be topologically regular ifp has an open neighborhood which is homeomorphic onto an open set in IR". By a result of PERELMAN [PI, this is equivalent to that C, is homeomorphic to S"- '( 1). Thus if all points on X are topologically regular, then X is a topological manifold. It is therefore natural to ask the following question: What geometric restrictions on the spaces of directions are sufficient to guarantee that they are homeomorphic to Sn-l(l)?

In [W], WILHELM proved that if C is an n-dimensional Alexandrov space with curvature equal to or greater than l, then C is homeomorphic to Sn(l) provided there are n + 2 points xI, ..., x , + ~ on C such that

71 l X i X j l > 2' i # j .

See [Pl] for related results. In this section we will prove the following:

7 Math. Nachr., Bd. 164

98 Math. Nachr. 164 (1993)

Theorem 1. If C is an n-dimensionul Alexandrov space with curvature equal to or greater than 1, then C is homeomorphic to S"(1) provided that

1 (9) Vfl(C) > 5 @ f l ?

where on denotes the volume of the unit sphere S"(1) in IR"+l. P r o o f . To prove Theorem 1 we use induction on n. Since 1-dimensional Alexandrov

spaces with curvature equal to or greater than 1 are circles with circumference equal to or less than 271 and intervals with length equal to or less than 71. Thus if

Vl(C) > 5 Vl(S1(l)) = 71, then the second case is excluded. Now we assume that Theorem

1 holds for (n - 1)-dimensional Alexandrov spaces. Let C be an n-dimensional Alex- androv space with curvature equal to or greater than 1 satisfying (9). Let p E C be an arbitrary point. By the relative volume comparison theorem, one has

1

It follows from Proposition 2 that

Since C, has curvature equal to or greater than 1, by hypotheses, C, is homeomorphic IL

to Sn-'( l ) . Take two points p , q E C such that Ipq1= diam (C). Clearly, diam (C) > -. 2

By the same argument as in [GS] one can show that for any x # p , q pxq > - + E for 2

some E. By [P; Theorem 4.51, C is homeomorphic to S(C,), the suspension on C,. Since C, is homeomorphic to S"- '(1) as we have proved, we conclude that I; is homeomorphic

71

to S"(1). rn We have essentially proved the following:

Theorem 2. Let C be an n-dimensional Alexandrov space with curvature equal to or greater than 1. Suppose that

71 1 @(C) > -.

2 2 diam (C) > -,

Then C is homeomorphic to S"(1).

Remark. Let X be an Alexandrov space. For any point p E X , define

rad ( X ) = inf sup 1~x1. ,EX xeX

We call Rad ( X ) the radius of X . Suppose C has curvature equal to or greater than

1 and satisfies (9). Then the volume comparison theorem implies that Rad ( X ) > -. 2

Thus one may expect that Theorem 1 still holds under a weaker assumption that

71

Rad (C) > c. Recently, GROVE-PETERSEN have claimed this is true. May be one can also

extand some results in [GP] to Alexandrov spaces. 2


YAMAGUCHI generalizes the differentiable sphere theorem in [OSY] to Alexandrov

Proposition 5 ([Y]). There is a number E(n) such that for any E < E(n) i f an n-dimensional

spaces.

Alexandrov spaces C of curvature equal to or greater than 1 satisfies v,(C) 3 0, - E ,

then C is .r,(E)-almost isometric to S"(l), where T,(E) depends only on n and E and lim z,(E) = 0. E+o

Proposition 2 and 5 immediately implies the following:

Proposition 6. There is a small number E(n) such that i f an n-dimensional Alexandrov space X satisfies.

O ( X ) > 1 - E(n), then X is a Lipschitz manifold.

5. The contractibility of metric balls

Let X be an n-dimensional Alexandrov space with curvature equal to or greater than - kZ, k > 0. Let p EX. Recall that the contractibility radius c(p) at p as the largest number r such that for any v < r, the metric ball B ( p , v) is contractible. As we have seen in [LS] X does not have positive contractibility radius on compact sets in general. In other words, there might exist a compact subset A in X such that c(A) = inf c(x) = 0, although c(x) > 0 for any x EX. This makes a difference in topology between Alexandrov spaces and Riemannian manifolds.

neA

Proposition 7 ([LS]). Let A > 1 and E E 0, - . Let p , p l , p z € X satisfy ( 3 . I

IPPZI G l - ' l P P I l , p z is an &-critical point of d,.

Then cos q p l p p z < {A-' + (1 + A-') sin E } eZklppll.

9 20

Lemma 1. Given 9 E . Suppose that B( p , v ) contains a --critical point x of d,.

7-c Then for any point y E B p , - with < xpy < - - 9, ( ':k) 2

10 9 9 20

Proof . Take d = - and E = -. Then

7'

100 Math. Nachr. 164 (1993)

Thus by Proposition 7, one obtains

10 IPYI < 21~x1 G 9 r .

In order to estimate the contractibility radius we need the following:

1 Y 4k 20 Lemma 2. Let Q E and r, p < -. Suppose that B(p, r) contins a --criticul point

of d,. Then

n P r o o f . Let A,(x) denote the set of points y # p such that xpy > - - 9. Clearly,

A,(x) is an open set in X. By Toponogov comparison theorem [BGP], one knows that A,(x)U ( p } contains any minimal segment joining p and y provided y E A,(x). Fix a di-

2

11 rection < E X ' c C,. Let r,(<) denote the set of directions ~ E C , such that I r ] ( l > - - 9.

Let I?(r,(()) denote the topological cone over &(t) with the metric defined as in (7). Define a map p : A,(x) 4 K(r,(<)) as follows. For every pointy E A,(x), put p ( y ) = (q , Ipyl), where r] E y' c r,(<). Clearly, p maps A , ( x ) n B(p , p) into KG(rs(()), the open metric p-ball in K(cq([)) around the vertex. By Toponogov comparison theorem,

2

lyzl d M Y ) P tzll . Thus p is an expanding map and

Applying the relative volume comparison theorem to r,([), one obtains

L J 0

By Lemma 1 one has

Thus

wn- 1


Proof of the main theorem. Take p = min rN(p), - . One still has ( :J 1 9

2(v - 1) 20 Take 9 = ~ ( a - i) and E = -. Suppose that B ( p , r) contains an &-critical point

of d,. Then by Lemma 2 one has that

o n - I

This implies that

r > ro:= 100(n - 1 )

Thus for all v less than r,, B ( p , v) is free of &-critical points of d,. By the Isotopy Lemma, one knows that B ( p , v) is homeomorphic to K,,(C,), the open metric v-ball in K(C,) around the vertex. It follows from Proposition 2 and (11) that

1 K-l(C,)> 2 wn-1.

By the sphere theorem 1 one concludes that C, is homeomorphic to Sn-l(l). Therefore KJCJ is homeomorphic to IR" for all v < r,. This completes the proof. 1

Finally we state the following theorem.

Theorem 3. Let X be an n-dimensional Alexandrov space with curvature equal to or greater than 0. Suppose that

Then X is homeomorphic to IR", where an denotes the volume of the unit ball in IR". 1 2

P roof . First one can see that On(p) > -. Thus C, is homeomorphic to Sn- I ( l ) . We

claim that if x is an &-critical point of d,, then t: 2 - (a, - i). Thus if we take

E, = f. ( a, - i), then X\{p> is free of &,-critical points of d,. Then it follows from the

Isotopy Lemma that X is homeomorphic to the tangent cone K(C,) which is homeomorphic to IR", since C, is homeomorphic to S"- ' ( l ) .

n - 1

n

102 Math. Nachr. 164 (1993)

Now we suppose that x ~ B ( p , r ) is an &-critical point of d,. By the same argument as in Proposition 7 one can show that for any y e X ,

71 Take an arbitrary ; > $ > E . The above inequality implies that if

r ) , then 71 xpy > - - 9. Fix a direction 5 EX‘ c C,. Define A,(x)

sin 9 - sin E 2 and I‘’(5) as in the proof of Lemma 2. Then for a sufficiently large number p,

Similarly, one has

Thus letting p+ + co, by (10) one obtains

Since 9 > E is arbitrary, one obtains that E > ~

n - 1

References

[BGP] Y. BURAGO, M. GROMOV and G. PERELMAN, A. D. Alexandrov’s spaces with curvature bounded

[GLP] M. GROMOV, J . LAFONTAINE, P. PANSU, Structures metriques pour les varietes riemanniennes,

[GP] [GS] [LS] [OS] [OSY] Y. OTSU and K. SHIOHAMA and T. YAMAGUCHI, A new version of differentiable sphere theorem,

[PI [Pl] [PZ] [PWZ] P. PETERSEN, F. WILHELM and S. ZHU, Spaces on and beyond the boundary of existence, preprint [W] [Y]

from below I, Uspekhi Mat. Nauk 47: 2 (1992). 3-51

Cedic\Fernand Nathan (1981) K. GROVE and P. PETERSEN, Volume comparison ii la Aleksandrov, Acta Math., to appear K. GROVE and K. Shiohama, A generalized sphere theorem, Ann. of Math. 106 (1977), 201-21 1 Z. LIU and Z. SHEN, On the Bctti numbers of Alexandrov spaces, preprint Y. OTSU and T. SHIOYA, The Riemannian structure of Alexandrov spaces, preprint

Invcnt. Math. 98 (1989), 219-228 G. PERELMAN, A. D. Alexandrov’s spaces with curvature bounded from below 11, preprint C. PLAUT, Two 2(n + 1)-point differential sphere theorem, preprint C. PLAUT, Spaces of Wald curvature bounded below I, 11, preprint

I;. WILHELM, A soft sphere theorem, preprint T. YAMAGUCHI, A convergence theorem in the geometry of Alexandrov spaces, preprint

Department of’ Mathematics The University of’ Michigan Ann Arbor, M I 48109-1003 V.S.A.

a regularity theorem for alexandrov spaces

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