Download Convex Sets and Convex Functions on Complete Manifolds

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].
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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
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