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Proceedings of the International Congress of Mathematicians Helsinki, 1978 Convex Sets and Convex Functions on Complete Manifolds Katsuhiro Shiohama The aim of this paper is to show how convex sets and functions give strong restrictions to the topology of a certain class of complete Riemannian manifolds without boundary. The idea of convexity plays an essential role for the proofs of "finiteness theorems", which give a priori estimates for the number of topological types of a certain class of compact Riemannian manifolds characterized by geometric quantities. Small convex sets such as strongly convex balls are used in the proofs of finiteness theorems. Weinstein's theorem [18], which is the first attempt in this direction, states that given n and (56(0, 1), there are only finitely many homotopy types of 2/7-dimensional simply connected <5-pinched manifolds, and the number of homotopy types depends on ô and n. Then it has been developed by Cheeger [3], Margulis [13] and Gromov. On the other hand, large convex sets (such as a closed hemisphere of standard sphere and hyperplanes at infinity on the projective space with standard metrics) are useful in the proof of "uniqueness theorem". Well known examples of such theorems are the sphere and rigidity theorems investigated by Berger [1], Klingenberg [12] and the author [17]. A sphere theorem states that a (5-pinched connected M is a topological sphere if ö = 1/4 and the diameter d(M) of M is greater than n. The rigidity theorem due to Berger states that an even dimensional (l/4)-pinched simply connected M is isometric to a compact symmetric space of rank 1 if its diameter d(M)=n. Under the assumptions of both the sphere andrigiditytheorems, M admits large convex sets which I want to discuss in § 1. As is seen there large convex sets enable us to generalize the sphere and rigidity theorems, which are obtained by K. Grove and the author [11], [16]. 444 Katsuliiro Shiohama In the next place I shall deal with convex functions on complete and noncompact Riemannian manifolds and generalize the theorems obtained by Gromoll-Meyer [10] and Greene-Wu [8]. Let y: [0, <*>)-> M be a ray emanating from a fixed point p. A Busemann function F : M->R with respect to y is defined by Fy(x)= lim[/-d(jc,y(f))], x£M. /-»-co If the sectional curvature K of M is nonnegative everywhere, then it follows from Toponogov's triangle comparison theorem that Fy is convex. Obviously it is not constant on any open set of M. Moreover the function F: M^R defined to be F(x) = sup[Fy(x); y(0)=p] is convex and exhaustion, where the sup is taken over all rays emanating from p. Gromoll-Meyer proved that a noncompact M is diffeomorphic to Rn if Ä>0. Then Cheeger and Gromoll proved that there exists on a noncompact M with K^O a compact totally geodesic submanifold S without boundary which is totally convex. Furthermore M is diffeomorphic to the total space of the normal bundle v(S) over S. Recently Greene and Wu showed in [7] that the above F can be replaced by a convex exhaustion function whose second difference quotient along every geodesic is bounded away from 0 on every compact set provided K>0. And in [8] they approximated a convex function with positive second difference quotient along every geodesic by a smooth convex function with positive second derivative along every geodesic. Therefore if M admits a convex exhaustion function with positive second difference quotient along every geodesic, then M is diffeomorphic to Rn. These results are generalized in § 2, which I worked with Robert E. Greene. (See [6].) 1. Convex sets. Throughout this section let M be a compact and connected Riemannian manifold of dimension n^2. Assume that the sectional curvature K and the diameter d(M) of M satisfy K^ S >0 and d(M)^n/2fô. Let p, p£M be such that d(p9 p)=d(M)9 where d is the distance function. A large convex set is defined to be Ap={x£M; d(p9 x)^nj2io?\. Ap is clearly a nonempty closed convex set, and has a nonempty boundary if the diameter assumption is inequality. More generally if a closed convex set in a complete manifold of positive sectional curvature has a nonempty boundary, then the soul of it is a single point and hence it is homeomorphic to a closed disc (see [5]). We next define B=f} {Aq\ q€Ap}. Then B is a nonempty closed convex set. It turns out that if M is not simply connected then both Ap and Ap have no boundary (see [15]). In any case we can choose for a closed convex set C a neighborhood of C which is either an embedded open «-disc with smooth boundary (if C has a nonempty boundary) or else a normal disc bundle over C (if C has no boundary). In order to see what happens in between Ap and B9 we define a function / : M—R by fM = id(p9x)-d(p9x) \d(Ap9x)-d(B9x) if d(M)>Tzl2fö if d(M) = n/2fô. Convex Sels and Convex Functions on Complete Manifolds 445 / is continuous on M and smooth outside a closed set Q of measure zero. Choose neighborhoods U and F (of p and p if d(M)^7i/2]fô and of B and Ap if d(M) = n/2yS) which have the properties stated above. By a smoothing convolution process we can approximate / by a family {fQ:M-+R; Q£(09Q0)} of smooth functions so that if QX is taken to be sufficiently small then V/ e ^0 in M—Uu V and Vfe is transversal to dl/udV for any e€(0, gt). Thus we can prove the following THEOREM 1. Let M be a connected and compact Riemannian manifold and let K^S>0. (a) (see [11]). If d(M)>7i/2fô9 then M is a topological sphere. (b) (see [17]). Assume that d(M) = n/2]/ô. Then we have b-1. IfAp has nonempty boundary or d i m ^ p = 0 and if B has nonempty boundary or dim B=09 then M is a topological sphere. b-2. If one of the Ap and B has no boundary and its dimension is greater than 0 then M has the same cohomology structure as that of a symmetric space of compact type of rank 1. b-3. If both Ap and B have no boundary and if dim Ap and dim!? are positive, then M is exhibited as a union of two normal disk bundles over Ap and B joined along their common boundary. REMARKS (1). Note that the condition in (a) is the best possible one for a compact manifold of positive sectional curvature to be a topological sphere. (2). In case b-2, M admits a complete metric and a point on the convex set with nonempty boundary with respect to which the tangent cut locus at that point is a sphere. Then the result follows from [14] or [2]. (3). If M is not simply connected, then b-3 occurs. In the case where M is not simply connected, we have the 2. Assume that M is not simply connected and K^o>0 and d(M) = 7i/2]/(5. If dim Ap=2 and if there exists a pair of points x£B and y£Ap such that there are at most finitely many minimizing geodesies joining x to y9 then M is of constant curvature S and its fundamental group has a fully reducible representation. THEOREM 2. Convex functions. Throughout this section let M be a complete, noncompact Riemannian manifold without boundary. We want to investigate the topology of M which admits a convex function. A function cp: M-+R is said to be convex if for every normal geodesic y:R-+M and every tl9 t2Ç:R9 and /l£[0, 1], p°y((i-A)'i+A'«) < 0-A)p°y('i)+Apoy(f«)If the above inequality is strict for X 6(0, 1), then <p is called to be strictly convex. It does not necessarily follow that a strictly convex function has positive second difference quotient along a geodesic. In order to investigate topology of M which 446 Katsuhiro Shiohama admits a convex function, it is natural to restrict ourselves to consider the case where it is not locally constant. Let cp: M-+R be a convex function which is not locally constant. A convex function is continuous and Lipschitz continuous on every compact set. Let M^((p) = (p~1(a) and Ma(<p)={xÇ:M; cp(x)^a}. Then Ma((p) is totally convex. Since a monotone increasing bounded convex function defined on [0, ~ ) is constant, we have the LEMMA 1. If M^(q>) is compact for some a£<p(M)9 then so is Mjj(<p) for all b^a. A perpendicular geodesic from a point to a closed convex set is uniquely determined if the point is close to the set. This fact together with the Lipschitz continuity of a convex function on a compact set will imply the following THEOREM 3. If M"(cp) is not connected for some a£(p(M)9 then we have the following statements. (1) (p attains its minimum, say, m0. (2) Mm (cp) is a complete totally geodesic hypersurface without boundary and has a trivial normal bundle. (3) M%((p) has exactly two components for all b>m0. As a direct consequence of the above theorem, we have COROLLARY TO THEOREM 3. Let \I/:M-+R be strictly convex. Then every level set is connected. The existence of a strictly convex function gives a strong restriction to the Riemannian metric. For instance there is no compact totally geodesic submanifold without boundary if M ? Jmits a strictly convex function. Moreover we can prove the following THEOREM 4. Let \j/: M-+R be strictly convex. exp p : Mp-*M at every point p is proper. Then the exponential map As a consequence of Theorem 4, we have 4. If there is a compact level set for a strictly convex function, then every level set is compact. COROLLARY TO THEOREM In the case where the sectional curvature of M is nonnegative mor egenerally if the Ricci curvature of M is nonnegative, then M has at most two ends. We can also discuss the ends of M which admits a convex function. THEOREM 5. If M admits a strictly convex function, then M has at most two ends. Finally we shall state a structure theorem as follows. Convex Sets and Convex Functions on Complete Manifolds 447 THEOREM 6. Let \j/: M-+R be a strictly convex function. If there is a compact level set of xj/, then M is diffeomorphic to either Rn (if \j/ attains its minimum) or else a cylinder NXR, where N is a compact hypersurface homeomorphic to a level set of xj/. In the case where M admits a convex exhaustion function, we have 7. Let cp\M-+R be a convex exhaustion function. Assume that there is a level set which is not connected. Then M is diffeomorphic to a cylinder Mm (q>)XR. THEOREM References 1. M. Berger, Les variétés riemanniennes (ì/4)-pincées, Ann. Scuola Nonn. Sup. Pisa HI 14 (1960), 161—170. 2. A. Besse, Manifolds all of whose geodesies are closed, Springer-Verlag, Berlin, 1978. 3. J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61—74. 4. J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Amsterdam, 1975. 5. J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvatine, Ann. of Math. 96 (1972), 413—443. 6. R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds (preprint). 7. R. E. Greene and H. Wu, Integrals of subharmonic functions on manifolds of nonnegative curvatine, Invent. Math. 24 (1974), 265—298. 8. C°° convex functions and mani j olds of positive curvature, Acta Math. 137 (1976), 209—245. 9. D. Gromoll and W. Klingenberg and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Math. vol. 55, Springer-Verlag, Berling and New York, 1968. 10. D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. 90 (1969), 75—90. 11. K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of. Math. 106 (1977), 201 — 211. 12. W. Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver Kìummung, Comment. Math. Helv. 35 (1961), 47—54. 13. G. Margulis, Proceedings of the International Congress of Mathematicians, (Vancouver. Canada, 1974), part 2, Canad. Math. Soc, 1976 pp. 21—34. 14. H. Nakagawa and K. Shiohama, Geodesic and curvature structures characterizing projective spaces, Diff. Geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972. 15. T. Sakai and K. Shiohama, On the structure oj positively curved mani j olds with certain diameter, Math. Z. 127(1972), 75—82. 16. K. Shiohama, The diameter of ö-pinched manifolds, J. Differential Geometry, 5 (1971), 61—74. 17. Topology of positively curved manifolds with certain diameter, Proc. of the Japan— U.S.A. Seminar on Minimal Submanifolds and Geodesies Kaigai Pubi., Tokyo, 1978, pp. 217—228. 18. A. Weinstein, On the homotopy type oj positively pinched manifolds, Arch. Math. 18 (1967), 523—524. UNIVERSITY OF TSUKUDA 300—31, IBARAKI, JAPAN