Complete noncompact Alexandrov spaces of nonnegative. curvature.

Transcription

1 Arch. Math., Vol. 60, (1993) X/93/ $ 2.90/ Birkh/iuser Verlag, Basel Complete noncompact Alexandrov spaces of nonnegative curvature By KATSUHIRO SHIOHAMA Dedicated to Professor Hisao Nakagawa on his sixtieth birthday 1. Introduction. An Alexandrov space X with curvature bounded below is a length space with the property that the Alexandrov-Toponogov comparison theorem holds for all small geodesic triangles. It is interesting to investigate the properties of finite (Hausdorff) dimensional Alexandrov spaces of nonnegative curvature in connection with Riemannian geometry and Hausdorff convergence as well. In [1] it is known that a finite dimensional complete Alexandrov space X with curvature bounded below by I has the property that its diameter d(x) does not exceed zc. In particular it is compact. Thus the Myers-Toponogov compactness theorem holds for finite dimensional complete Alexandrov spaces whose curvature is bounded below by 1, (for details see w in [1]). Moreover if a complete n-dimensional Alexandrov space X with curvature bounded below by 1 has an (n + I, 6)-strainer for sufficiently small 6 > 0, then X is almost isometric to the standard unit n-sphere Sn(l), (e.g., there exists a bilipschitz homeomorphism f:x --, S ~ (1) whose Lipschitz constant is sufficiently close to 1, see Theorem 10.5; [1]). The basic idea of an (n + 1, cs)-strainer was first used in [13] to provide a diffeomorphism between S" (1) and a Riemannian n-manifold with curvature bounded below by 1 whose volume is sufficiently close to that of S n (1). The Riemannian version of the above result was established in [15], in which we see that the existence of an (n + 1, c~)-strainer for sufficiently small ~ is equivalent to the radius of a complete Riemannian n-manifold with curvature bounded below by I being sufficiently close to n. The Toponogov splitting theorem holds for complete noncompact Alexandrov spaces of nonnegative curvature. It was discussed in [10] for length spaces of nonnegative Toponogov curvature and in [18] for the Hausdorff limits of manifolds of almost nonnegative sectional curvature. It plays an important role for the proof of fibration theorem developped in [18]. However it is not certain if the soul and structure theorems hold for complete noncompact Alexandrov spaces of nonnegative curvature. The purpose of this article is to show that some of the Riemannian results can be extended to Alexandrov spaces. We first establish the end structure of complete noncompact Alexandrov spaces of nonnegative curvature. We next provide a natural extension of the Myers-Toponogov compactness theorem for complete Alexandrov spaces of nonnegative curvature.

2 284 K. SH1OHAMA ~.RCH. MATH. Theorem A. Let X be a complete noncompact Alexandrov space of finite Hausdorff dimension. If X has positive curvature, then X has exactly one end. If X has nonnegative curvature, then X has at most two ends. Theorem B. Let X be a complete noncompact Alexandrov space of dimension n, tf the curvature of X is nonnegative outside a compact set, then the n-dimensional Hausdorff volume of X is unbounded. Many attempts were made for complete noncompact Riemannian manifolds to have unbounded volume under certain restrictions for curvature, for instance, see [3], [6], [14], [20], [21], etc. As a direct consequence of Theorem B we have the Corollary to Theorem B. A complete Alexandrov space X of dimension n of nonnegative curvature is compact ~ and only if the n-dimensional Hausdorff volume of X is bounded. The proof of Theorem A is obtained by the direct application of the Alexandrov-Toponogov comparison theorem. It is proved that every Busemann function on a complete noncompact Alexandrov space of nonnegative curvature is convex. The proof of Theorem B is achieved by constructing an expanding map along rays which are asymptotic to a fixed ray. The original idea of it goes back to [14]. We refer the basic tools of Riemannian geometry to [4], [71, [i 1] and of Alexandrov spaces to [1], [9t, 2. Preliminaries. From now on let X be an n-dimensional complete noncompact Alexandrov space. A geodesic on X is by definition a curve whose length realizes the distance between its extremal points. Every geodesic is parametrized by arclength unless otherwise stated. A ray ;~ : [0, oo) -, X, (a line 7 : ff~ -* X respectively) is by definition a curve such that any subarc of it is a geodesic. It is elementary that through every opoint x e X there passes at least a ray. Let 7: [0, oo) -, X be a ray emanating from a point x e X. A ray o-: [0, ~) ~ X is by definition asymptotic to ~, iff there exists a sequence of geodesics {o): [0, 12] ~ X} such that {~rj(0)} and {aj(l)} converge to a(0) and a(1) respectively as j -~ ~ and such that {aj(lj)} is a monotone divergent sequence on 7' [0, ~). Thus the directions of o-j's at aj (0)'s converge to that of a at a (0). It is elementary that if o- is asymptotic to 7, then for every t > 0 the subray ~ 1 [t, oo) of ~r is a unique ray passing through a (t) and asymptotic to % A Busemann jhnction F;. for 7 is defined by F~(p):= lira [t-d(p,y(t))], t~00 pgx, where d is the distance function on X. If o- is asymptotic to 7, then F~o o-(t)= F~ o o- (0) + t holds for all t > 0. According to [I] the nonnegativity of curvature of X is defined as follows. First of all a geodesic triangle A = (7o, 71,72) in X is a tripple of geodesics, each 7i: [0, 1] --> X is parametrized proportionally to arc length such that Po := 7t (1) = ~2 (0), p~ := 7o(0) = Y2 (1), Pz := 70(1) = 71 (0). It is also expressed as the triple of points A = (Po, Pl, P2). Without loss of generality we only discuss the angle of A at the corner Po. For every s, t e [0, 1] let %t: [0, 1] --, X be a geodesic parametrized proportionally to arclength such

3 Vol. 60, 1993 Alexandrov spaces of nonnegative curvature 285 that Lr(0) = 72 (t) and Lt(1) = 71 (1 - s). Set A~t:= ( s, 1], 72110, t], z~t) and Ast: = (~7~ [ [1 - s, 1], 72 [ [0, t], Lt) the corresponding triangle sketched in R 2 having the same edge lengths as A~. X is by definition of nonnegative curvature (and henceforth denoted by K x > 0) iff the Alexandrov convexity property holds as follows: For every geodesic triangle A in X the angle 0" 0 (s, t) of A~t opposite Lt satisfies go (Sl, tl) => 0"0 (S2, t2) for 0 < s! < s2 =< I and for 0 < t 1 < t 2 < 1. The monotone property of 0o ensures the existence of the limit of it as s, t --, 0. Then the angle /P2 Po P~ at Po of A is defined as /- P2 Po Pl : = lim 0"o (s, t). S,t-~ O The Toponogov comparison theorem holds for every geodesic triangle A in X and its corresponding triangle A sketched in ]R 2 if and only if X has nonnegative curvature. The angles between two geodesics of X have the property that if p is an interior point of a geodesic joining x to y, then holds for all q e X. /-xpq + / qpy--- ~z 3. End Structure. The end structure of complete noncompact Riemannian manifolds of nonnegative curvature was investigated in [5] and [8]. The same conclusion holds for complete noncompact Alexandrov spaces of nonnegative and positive curvature. Lemma 3.1. If K x >= 0, then X admits at most two ends. P r o o f. Suppose that X has more than two ends. Then there exists a compact set C ~ X such that XkC has at least three unbounded components, say, U1, U 2 and U 3. Let 7 : ~ -~ X be a line such that 7 (0) ~ C, 7 [0, o9) ~ C c~ U 2 and 7 (- ~, 0] c C r~ U 1. Let {pj} be an unbounded sequence of points in U 3 and set tj:-- d(p~, C). Every geodesic joining pj to 7(_+ t~-) for large j passes through a point on C, and hence td(pj, 7(t;)) - 2@ =< 2d(C), Id(pj,~(-tj))- 2@ < 2d(C), where d (C) is by definition the diameter of C. Let Aj. be a geodesic triangle with vertices (Pj, 7 (tj), 7 (- tj)) and Aj the corresponding triangle sketched in R2. The above inequalities then imply that the sequence {A j} of triangles in ~2 converges to the right triangle by normalization. The Toponogov comparison theorem implies that for a given small positive number e there exists a number j (8) such that ifj > j (e), then all the angles of Aj are not less than ~/3 - e. On the other hand, a contradiction is derived by claiming that all the angles of ASs for sufficiently large j are bounded above by e. To show L T(-tj)7(tj)pj < for sufficiently large j, we use the property of angles stated at the end of Section 2. Take a point qj ~ C on the edge of Aj opposite 7 (- t j). Choose a large number j such that / qj 7 (t j) 7 (2 t j) > n - e. This is possible because d (q j, 7 (t j)) < t j- + d (C) and d (q j, ~/(2 t j)) > 2tj -- d(c). This proves Lemma 3.1.

4 286 K, NHIOHAMA ARCH. MATH. The following Lemma 3.2 is needed for the proof of uniqueness of end, The Toponogov splitting theorem is derived from this lemma, Lemma 3.2. Let K x > O. Assume that X admits a line 7 : ~ -" X. Then, through every point p e X\~ there passes a unique/ine 7p : ~ ~ X which is biasymptotic to 7. Moreover, for any points q, r e 7 O R) the geodesie triangle d = (p, q, r) has the property that the corresponding triangle zl = (/~, c~, f) in ~2 satisfies; /-~gtf= /-pqr, / Elf~= /-qrp, /-fpei= Z_rpq. P r o o f. Choose the parameter of 7 so as to satisfy d (p, 7 (0)) = d (p,? (N~)). It follows from triangle inequality that if F+ and F_ are Busemann functions for 7 + (t): =? (t) and 7- (t) : = 7 (- t) for t > 0, then F+ (p) + F_ (p) N 0. The Toponogov theorem for the limit of geodesic triangles (p, 7(0), 7(t)) as t -. ~ then implies that F+ (p) = F_ (p) = 0. Let a: [0,/] ~ X be a geodesic with a (0) =,~' (0),~ (l) = p, The above discussion means that F+ o a(u) = F_ o a(u) = 0 for all u e [0, I]. If 0(u, s) for (u, s) e [0, I] x [0, ~) is the angle at g(0) of the triangle,_~,~ := (~(~, ~(u), ~+ (s)) corresponding to A,,:= (7(0), or(u), 7+ (s)), then the above fact means that O(u, s) = zc/2 for all (u, s) e [0, l] x [0, oo). If7~' [0, oo) --, X for u e [0, l] is a ray obtained as the limit of geodesics joining ~r(u) to 7:~ (s) as s -r oo, then both 72 and?'.+ makes an angle 7z at a(u). If 7~(.): P- --* X is defined by 7~.)(t):= 7. + (t),,/~.) (- t): = 7~- (t) for t ~ 0, then triangle inequality implies that 7~(.) is a line which is biasymptotic to 7 and that /_ a (0) ~r (u) ~(,) (s) = Z_ a (0) cr (u) 7~,) (- s) = re/2. In particular, every angle of A,~ is equal to the corresponding angle of A,~ for all (u, s) e [0, I] x [0, ~). This proves Lemma 3.2. R e m a r k. It is not difficult to verify that for every (u, s) e [0, l] x [0,,~) the edge of A,~ opposite a(0) intersects 7~ for every v e [0, u] at the point?~+ (-u-~-s). The union of all lines U 7, (N) forms a convex set in X which is isometric to a flat strip [0,/] x ]R~ Also ue[o,/] we have a fiat strip consisting of all lines biasymptotic to 7 and passing through points on a geodesic joining any two points on F~71 (0), and they are all biasymptotic to each other. Thus F~- ~ (0) is isometric to F(g~(s) for all s e P,, and hence X is isometric to f~- 1 (0) x IR. Proof of Theorem A. It suffices for the proof of Theorem A to show that if Kx > 0, then X admits no line. Suppose that such an X admits a line 7 : 1R -, X. Lemma 3.2 implies that for any fixed point p e X\7 (IR) and for any two points q, r e 7(N) if = (/~, ~, f) is the triangle in N~ 2 corresponding to A = (p, q, r), then all the angles of A are equal to those of corresponding A. Since K x > 0, it has a positive lower bound on every compact set of X. Therefore the sum of all the angles of A exceeds z, a contradiction. This proves Theorem A, 4. The construction of expanding maps. Let X be an n-dimensional complete noncompact Alexandrov space of nonnegative curvature. The proof of Theorem B is achieved by constructing an expanding map outside a compact set. Rays asymptotic to a fixed ray are

5 Vot. 60, 1993 Alexandrov spaces of nonnegative curvature 287 employed for the construction. Let 7 : [0, Go) -, X be an arbitrary fixed ray. To each point x e X a ray 7x : [0, o~) -~ X emanating from x and asymptotic to 7 is assigned as follows. First of all fix a monotone divergent sequence {t~} of positive numbers. The 7~ is obtained as the limit of a converging subsequence of geodesics z j: [0, l j] ~ X with vj (0) = x, z~ (l~) = y (t~). With these notations we shall prove the Lemma 4.1. For an arbitrary fixed positive number a let (p~ : X ~ X be defined as Then ~o a is expanding. (Pa (x) : = Yx (a), x E X. P r o o f. For any points xl, x 2 ~ X let 71,72 be the rays assigned to these points. Let a: [0, l] -. X be a geodesic with a(0) = x i, a(1) = x2. For i = 1, 2 and forj = t, 2... let qj: [0, lq] ~ X be a geodesic with z~a(0) = x~, z~a(l~j) = 7(tj). Consider the triangles Aj with edges (a, zij, %i) and A)with corners (zlj(a), z2j(a), 7(t j)), and the corresponding triangles Aj with edges (#, flj, T2j) and A) with corners (flj (a), "Czj (a), ~ (@) sketched in IR 2. Then the Alexandrov convexity property implies that and in particular /- fij (a) ~ (t j) "~2j (a) > / flj (a) ~ (tj) z2j (a), d (zl2 (a), r22 (a)) = d (fij (a), "c2j (a)) > d (?ij (a),?2j (a)). Clearly ~% (xl) = lira rq (a) and j~ d(xi, x2)= dffa~(0), f22(0)) = lim diff,(a), ~z2(a)). j~ce This proves Lemma 4.1. From Lemma 4.1 we observe that if x; for i = 1, 2 is a point on ~,~[0, oe) intersecting F[~(t) for every t > max {F~(x0, Fr(x2)}, then /_x~ x~x'2 < ~/2 and /-x2x'2x~ < re~2. This fact suggests that every sublevel set of F~ is convex. Lemma 4.2. If K x > O, then F~ for every ray 7 is convex in the sense that it is convex along every geodesic. P r o o f. It is elementary that a continuous midconvex function is convex. The midconvexity of F~ along every geodesic a: [0, 1] ~ X is verified as follows. Let x 1 := a (0), x2:= a(1) and Xo:= cr(l/2). For a sufficiently large t we consider geodesic triangles A 0 (t):= (x i, x 2, ~ (t)), A 1 (t):= (x o, x 2, 7 (t)) and A 2 (t):= (xl, Xo, 7 (t)) and also their corresponding triangles Ao(t):=(21,2>~7(t)), ~i (t):= (20, 22, ~(t)) and z]2(t):= (xi, xo, "7(t)) respectively in IR 2. The edges ending at ~7(t) tend to parallel half lines as t ~ co. Let e be the unit vector parallel to these half lines. It follows from the definition of Busemann function that if (-,.) denotes the canonical dot product of N~ 2, then F,(x2) -- F,(x 0 = (e, 22-21), The Alexandrov convexity property then implies that )~ L2221~7(t) for all t>o. F,(xo) - F~(xl) = (e, Yo - 2i)-

6 288 K. SHIOHAMA ARCH. MATH. Letting t --, oe we see that This proves Lemma (e, 22-21) > (e, 20 2~) R e m a r k. It is welt-known that a complete Riemannian manifold admitting nontrivial convex function has unbounded volume. It is not certain if a complete Alexandrov n-space admitting nontrivial convex fupction has unbounded Hausdorff n-volume. Proof of Theorem B. Let C c X be a compact set such that K x > 0 on X\C. For a fixed ray 7: [0, oe) ~ X choose a positive number b such that b > max c F~. The basic property of n-dimensional Alexandrov spaces with curvature bounded below implies that for every point x e X there is a compact set B ~ X near x which is bilipschitz homeomorphic to a closed n-disk. For such a set B ~ F~ -1 [b, ~) choose a positive number a such that a > max~ F~ - min e F~. The sequence {Ps~ (B)} of sets has the properties that ~os~ (B) c~ (Ok, (B) = 0 forj 4: k and the Hausdorff n-volume of q0k~ (B) is not less than that of (ps, (B) for k >j. This proves Theorem B. References [1] Y. BURAGO, M. GROMOV and A. PERELMAN, A. D. Alexandrov's spaces with curvatures bounded from below I. Preprint. [2] H. BUSENANN, The Geometry of Geodesics. New York-London [3] E. CALABI, On manifolds with nonnegative Ricci curvature II. Notices Amer. Math. Soc. 22, A-205 (1975). [4] J. Chq?EGER and D. EBIN, Comparison Theorems in Riemannian Geometry. North-Holland Math. Library 9, Amsterdam-Oxford-New York [5] J. CEmEGER and D. GROMOLL, On the structure of complete manifolds ofnonnegative curvature. Ann. of Math. 96, (1972). [6] R. GREENE and H. C. Wu, Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27, (1974). [7] D. GROMOLL, W. KLINGENBERG und W. MEYER, Riemannsche Geometrie im Grossen. LNM 55, Berlin-Heidelberg-New York [8] D. GROMOLL and W. MEYER, On complete open manifolds of positive curvature. Ann. of Math. 95, (1969). [9] M. GROMOV, J. LAFONTAINE et P. PANSU, Structures m6triques pour les vari&es riemanniennes. Paris [10] K. GROVe and P. PETERSEN, On the excess of metric spaces and manifolds. Preprint. [11] W. KLINGEN~ERG, Riemannian Geometry. Studies in Mathematics 1, Berlin-New York [12] S. B. MYERS, Riemannian manifolds in the large. Duke Math. J. 1, (1935). [13] Y. OTSU, K. SHIOnAMA and T. YAMAGUCHI, A new version of differentiable sphere theorem. Invent. Math. 98, (1989). [14] K. SHIOHAMA, An extension of a theorem of Myers. J. Math. Soc. Japan 27, (1975). [15] K. SHIOHAMA and T. YAMAGUCHI, Positively curved manifolds with restricted diameter. Geometry of Manifolds, Perspectives in Mathematics, Vol. 8, , Boston-New York- Berkeley-Tokyo [16] V.A. TOVONOGOV, Riemannian spaces having their curvature bounded below by a positive number. (Uspehi Math. Nauk., 14 (1959)), Amer. Math. Soc. Transl. Ser. 37, (1964). [17] V. A. TOVONOGOV, Riemannian spaces which contain straight lines. Amer. Math. Soc~ Transl. Ser. 37, (1964). [18] T. YAMAGUCm, Collapsing and Pinching in lower curvature bound. Ann. of Math. 133, (1991).

7 Vol. 60, 1993 Alexandrov spaces of nonnegative curvature 289 [19] S.T. YAU, Nonexistence of continuous convex functions on certain Riemannian manifolds. Math. Ann. 207, (1974). [20] H. C. Wu, A structure theorem for complete noncompact hypersurfaces of nonnegative curvature. Bull. Amer. Math. Soc. 77, (1971). [21] H. C. Wu, On the volume of a noncompact manifold. Duke Math. J. 25, (1982). Anschrift des Autors: K. Shiohama Department of Mathematics Faculty of Science Kyushu University Fukuoka 812-Japan Eingegangen am Archiv der Mathematik 60 19