Table Of Content5 The Boundary between Compact and
0
0
2 Noncompact Complete Riemann Manifolds
n
a
J
D. Holcman C. Pugh
∗ †
4
2
]
G
Abstract
D
.
h In 1941 SumnerMyers proved that if the Ricci curvature of a com-
at plete Riemann manifold has a positive infimum then the manifold is
m compact and its diameter is bounded in terms of the infimum. Sub-
[ sequently the curvature hypothesis has been weakened, and in this
paper we weaken it further in an attempt to find the ultimate, sharp
1
v result.
4
1
4
1 Introduction
1
0
5
Myers’ Theorem [13] states that a complete Riemann manifold (M,g) of
0
/ dimension n 2 is compact if its Ricci curvature is uniformly positive, and
h
≥
t furthermore it has diameter π/√C if its Ricci curvature satisfies
a ≤
m
Ric (n 1)C (1.1)
: p
v ≥ −
i
X everywhere on M, C being a positive constant. (Here and below we adopt
r the shorthand that Ric c means that for all X T M,
a p ≥ ∈ p
Ric (g,X,X) c X,X ,
p p
≥ h i
where , is the g-inner product on T M). Later, asymptotic condi-
p p
h i
tions on the curvature were found that still imply compactness, [5], [6], [3],
∗Weizmann Institute of Science, Department of Mathematics, Rehovot 76100, Is-
rael.email:[email protected]
†MathematicsDepartment,UniversityofCalifornia,BerkeleyCalifornia,94720,U.S.A.
email:[email protected]
1
although definitive conditions remain unknown. The idea is to fix an origin
O M and study the curvature along geodesics emanating from O. One
∈
always assumes the curvature is positive, but permits it to decay to zero far
from O.
To be more specific, we set
Ric(r) = inf Ric : p = exp (v) and v = r ,
{ p O | | }
andassume throughoutthatRic(r) > 0. Hypotheses thatimplycompactness
and give a diameter estimate are:
(a) (Cheeger-Gromov-Taylor [5]) For some ν > 0, some r > 0, and all
0
r r ,
0
≥
n 1 1+4ν2
Ric(r) − .
≥ 4 (cid:16) r2 (cid:17)
(b) (Cheeger-Gromov-Taylor [5]) For some ν > 0, some r > 1, and all
0
r r ,
0
≥ n 1 1 1+4ν2
Ric(r) − + .
≥ 4 (cid:16)r2 (rlnr)2(cid:17)
(c) (Boju-Funar [3]) For some ν > 0, some integer k, some r > e , and all
0 k
r r ,
0
≥
n 1 1 1 1+4ν2
Ric(r) − + + + ,
≥ 4 (cid:16)r2 (rlnr)2 ··· (rln(r)lnln(r) lnk(r))2(cid:17)
···
where lnk is the kth iterated logarithm, lnk(r) = ln ln ln(r), and
◦ ◦···◦
lnk(e ) = 0.
k
It is natural to set ln0(r) = r. Then (a) and (b) are (c) with k = 0 and
k = 1. The diameter estimates on M involve ν, r , and k. When k = 0, one
0
has diam(M) eπ/νr . See [5] and [3].
0
≤
Remark. In [6], Dekster and Kupka prove that the estimate in (a) is sharp.
Remark. In terms of decay rates, these results are nearly optimal. For
example, there exist noncompact complete manifolds whose Ricci curvature
satisfies a Boju-Funar equality with ν = 0,
n 1 1 1
Ric(r) = − + + , (1.2)
4 (cid:16)r2 ··· (rln(r)lnln(r) lnk(r))2(cid:17)
···
See [3].
2
Remark. Dekster and Kupka consider also the sectional curvature K, and
show that all noncompact complete Riemann manifolds of positive curvature
satisfy
1
liminfk(r)r2 ,
r→∞ ≤ 4
where k(r) = inf K : p = exp (v) and v = r . The constant 1/4 is sharp,
{ p O | | }
[6].
Our first results, the Kick Theorems, state that asymptotic estimates are
not the only way to guarantee compactness. Instead of curvature that decays
to zero at a positive rate (ν > 0) as the radius tends to infinity, it is enough
that in addition to Ricci curvature obeying (1.2), there is a certain amount
of extra curvature on a finite shell p = exp (v) : a v b . We refer to
{ O ≤ | | ≤ }
the extra curvature as a “kick.” See Sections 2 and 3 for details.
Remark. None of these conditions is truly optimal; in Section 7 we show
that asurface approximating the cappedcylinder has theproperty that every
morecurved surface is compact, but thisis implied by none oftheasymptotic
or kick conditions.
Nevertheless, it is tempting to postulate some kind of a boundary in the
space of positive Ricci curvature functions with all compact manifolds on one
side and all non-compact complete manifolds on the other.1 A manifestation
of such a boundary would be a topology on the space of Ricci curvature
R
functions which are defined on a fixed tangent space T M, and a closed
O
subset such that through each R there is a curve R of Ricci
0 0 0 t
R ⊂ R ∈ R
curvature functions, and
(a) If t > 0 and M has Ricci curvature R then it is compact.
t
(b) If t 0 and M has Ricci curvature R then it is non-compact.
t
≤
(c) This transverse, single-point crossing from non-compact to compact
persists for all nearby curves R .
t
1 Readers familiar with Walter Rudin’setext, Principles of Mathematical Analysis, will
recognize this phrase, in which Rudin asserts that there is no such boundary dividing
convergent and divergent series. A difference between series and curvature functions is
that local perturbations have no global effect on series, while for curvature functions this
isnotso. Perturbationofafinitenumberoftermsinaseriesdoesnotchangeconvergence,
but a compactly supported perturbation of curvature can affect the manifold’s topology
at a long distance from the perturbation’s support, and hence such a curvature boundary
is not unreasonable.
3
In Section 6 we establish this kind of result for planar curves, finding a
boundary between embedded curves and nonembedded immersed curves; in
Section 7 we pass to surfaces embedded in 3-space.
Our second result partially identifies the boundary postulated above.
0
R
We call a function b : [0, ) (0, ) an SL-bifurcator if the solution to
∞ → ∞
the Sturm-Liouville equation
w′′ +b(r)w = 0, w(0) = 0, w′(0) = 1
is monotone and bounded. An example is
2r
b(r) = .
(1+r2)2arctanr
See Sections 5 and 7 for more on SL-bifurcators.
Definition. A function f(x) exceeds a function g(x), if for all x, f(x)
≥
g(x), and for some x, f(x) > g(x).
Boundary Theorem. Let M be a complete Riemann manifold with positive
Ricci curvature, and let b(r) be an SL-bifurcator.
(a) M is noncompact if for all r 0,
≥
sup Ric : p = exp(v) and v = r b(r).
p
{ | | } ≤
(b) M is compact if Ric(r) exceeds b(r). (As above, Ric(r) denotes the
infimum of Ric such that p = exp (v) and v = r.)
p O | |
Corollary. SL-bifurcators distinguish compact and noncompact complete
Riemann manifolds with positive Ricci curvature.
See Section 5 for the simple proofs.
2 Linear Kick
In this section we deal with the kick condition when k = 0. Thus we assume
that for all r r > 0,
0
≥
n 1 1
Ric(r) − , (2.3)
≥ 4 (cid:16)r2(cid:17)
4
and we find a sufficient amount of extra curvature on a shell p = exp (v) :
{ O
a v b (with r a) that implies M is compact. Define λ to be the
0
≤ | | ≤ } ≤
smallest positive root of the equation
cot(λln(b/a)) = λln(a/r ). (2.4)
0
Note that λ = λ(a,b,r ) exists, lies in (0,π/2ln(b/a)], and is unique. For,
0
as µln(b/a) varies from 0 to π/2, its cotangent decreases monotonically from
to 0, while µln(a/r ) is non-decreasing.
0
∞
Linear Kick Theorem. In addition to (2.3), assume that for all r [a,b],
∈
we have
n 1 1+4λ2
Ric(r) > − ,
4 (cid:16) r2 (cid:17)
where λ = λ(a,b,r ). Then M is compact.
0
Remark. As an example of the kick we can take a = e and b = e2. Then
λ is approximately .46. Also, if the interval [a,b] is small, in the sense that
b a = ǫ, it is not hard to check that a kick sufficient for compactness
−
increases like ǫ−1/2 as ǫ 0.
→
The proof is based on analyzing a kicked Sturm-Liouville equation
1 1+4µ2χ (r)
y′′ + [a,b] y = 0. (2.5)
4(cid:16) r2 (cid:17)
Lemma 2.1. If λ = λ(a,b,r ) is determined as in (2.4) and if µ > λ then the
0
solution y(r) of (2.5) with initial conditions y(r ) = 0, y′(r ) > 0, necessarily
0 0
vanishes at some r > r .
0
Proof. For any constants c,k > 0, the function w(r) = cy(r/k) satisfies (2.5)
and has initial conditions
cy′(kr )
w(kr ) = 0 w′(kr ) = 0 > 0.
0 0
k
Taking k = 1/r and c = k/y′(r ), we can assume without loss of generality
0 0
that r = 1 and y′(1) = 1. We do so.
0
For r [a,b], the solution of (2.5) is of the form
∈
y(r) = Ar1/2cos(µlnr)+Br1/2sin(µlnr) (2.6)
5
where A,B are constants. For µ = 0, the solution degenerates as
y(r) = Ar1/2 +Br1/2lnr, (2.7)
which can be seen by replacing B with B/µ in (2.6) and letting µ tend to
zero. Matching initial conditions at r = 1, r = a, and r = b gives
r1/2ln(r) if 1 r a
≤ ≤
1
y(r) = r1/2 ln(a)cos(µln(r/a))+ sin(µln(r/a) if a r b
(cid:16) µ (cid:17) ≤ ≤
(r/b)1/2(α+βln(r/b)) if b r <
≤ ∞
where α,β are constants
1
α = b1/2 ln(a)cos(µln(b/a))+ sin(µln(b/a)) (2.8)
(cid:16) µ (cid:17)
β = b1/2 cos(µln(b/a)) µln(a)sin(µln(b/a)) (2.9)
(cid:16) − (cid:17)
By the Sturm Comparison Theorem, a second zero of y(r), if it exists,
is a monotone decreasing function of µ. Thus it is no loss of generality to
assume that µ λ is small. Since λ is the smallest positive root of
−
cot(λln(b/a)) = λlna 0,
≥
and since λln(b/a) π/2, we can assume µ λ so small that µln(b/a) < π
≤ −
and
1
cot(µln(b/a)) > − .
µlna
Since the cotangent is monotone decreasing on (0,π), this implies that
1
cot(µln(r/a)) > −
µlna
for a r b, and hence that y(r) > 0 on [a,b]. For the same reasons,
≤ ≤
cot(µln(b/a)) < cot(λln(b/a)) = λlna µlna,
≤
which implies that β < 0. But y(b) > 0 and β < 0 implies that y(r) = 0 for
some r > b.
6
Proof of the Linear Kick Theorem. We must show that M is compact.
By the Hopf-Rinow Theorem, it suffices to show that every geodesic through
O contains a pair of conjugate points. For if M is not compact then it
contains an everywhere distance minimizing geodesic γ from O to infinity
(M is complete), and this is contrary to conjugate pairs on γ. See [12].
By the assumption on the Ricci curvature and compactness of [a,b], there
exists µ such that
Ric(γ′(r),γ′(r)) > µ > λ = λ(a,b,r )
0
for r [a,b]. Fix such a µ, and let y(r) be a solution of the kicked Sturm-
∈
Liouville equation (2.5) with initial conditions y(r ) = 0, y′(r ) > 0. By
0 0
Lemma 2.1, y(r ) = 0 for some r > r .
1 1 0
Following Myers’ use of the Index Theorem, this gives a pair of conjugate
points on γ. We recapitulate his proof.
Chooseanorthonormalbasis e ,...,e ofT M withe = γ′(0), andlet
1 n O n
{ }
E (r),...,E (r) be the corresponding parallel vector fields along γ. Define
1 n−1
vector fields along γ,
X (r) = y(r)E (r).
j j
We will check that
n−1
I(X ,X ) < 0 (2.10)
j j
X
j=1
where
r1
I(X,X) = X 2 R(X,γ′),X dr
Z (cid:16)|∇t | −h i(cid:17)
r0
is the index of a vector field X along γ. (R is the curvature tensor.) To
verify (2.10) we evaluate the Ricci curvature hypothesis on (γ′(r),γ′(r)) as
R(E (r),γ′(r))γ′(r),E (r)
j j
h i
X
n 1 1+4µ2χ (r)
= Ric(γ′(r),γ′(r)) > − [a,b] .
4 (cid:16) r2 (cid:17)
7
Then we write
I = X′,X′ R(X ,γ′)γ′,X dr
j Z h j ji−h j ji
X X
= (y′)2 R(E ,γ′)γ′,E y2dr
Z −h j ji
X
= (n 1) (y′)2dr R(E ,γ′)γ′,E y2dr
− Z −Z h j ji
X
1 1+4µ2χ
< (n 1) (y′)2dr (n 1) [a,b] y2dr
− Z − − Z 4(cid:16) r2 (cid:17)
= (n 1) (y′)2 +y′′ydr
− Z
r1
= (n 1)(y′y) = 0,
− (cid:12)r0
(cid:12)
(cid:12)
where I = I(X ,X ), all integrands are evaluated at r, all sums range from
j j j
j = 1 to j = n 1, and all integrals are taken from r to r . This verifies
0 1
−
(2.10).
Negativity of a sum implies negativity of at least one term, so (2.10)
implies that for some j , I(X ,X ) < 0, and so by Jacobi’s Theorem there
0 j0 j0
exists a point γ(r) with r < r < r which is conjugate to γ(r ).
0 1 0
Remark. It is straightforward to estimate the diameter of M as follows.
All points of M lie inside the geodesic ball at O of radius r , such that the
1
Sturm-Liouville solution y(r) above has its second zero at r . Here is how
1
this reads in two cases.
Case 1.r (a,b],atrivialsituation. Therootr occurswhentan(µln(r/a)) =
1 1
∈
µlna. If a = 1 then this gives
−
D 2eπ/µ.
≤
Case 2. y(r) > 0 for a r b. The second zero of y(r) occurs at the
≤ ≤
first root of
α+βln(r/b) = 0
beyond b. (As in the theorem, we have β < 0.) This is equivalent to r = beF
where
α lnacosθ+(sinθ)/µ
F = = θ = µln(b/a),
β µlnasinθ cosθ
− −
8
and thus,
D 2beF.
≤
When a = 1 we get
D 2be−tan(µlnb)/µ.
≤
If the curvature hypotheses are valid at all origins O then the factors 2 in
these diameter estimates are superfluous. Note that as µ decreases to λ, the
second diameter estimate tends to + .
∞
3 Logarithmic Kick
Lete < r a < bbegiven, wheree isthekth superpower ofe, lnk(e ) = 0.
k 0 k k
≤
Define λ = λ (r ,a,b) as the smallest positive solution of the equation
k 0
cot(λ(lnkb lnka)) = λ(lnka lnkr ).
0
− −
Define
1 1 1 1+4µ2
F (r,µ) = + + + .
k 4(cid:16)r2 (rlnr)2 ··· (rln(r) lnk(r))2(cid:17)
···
Logarithmic Kick Theorem. Assume that for all r r we have Ric(r)
0
≥ ≥
(n 1)F (r,0) and that for all r [a,b] we have
k
− ∈
Ric(r) > (n 1)F (r,λ),
k
−
where λ = λ (r ,a,b) as above. Then M is compact.
k 0
When µ > 0, the general solution to the Boju-Funar equation
y′′ +F (r,µ)y = 0 (3.11)
k
is of the form
Φ (r) Acos(µlnkr)+Bsin(µlnkr) ,
k
(cid:0) (cid:1)
where A, B are constants and
Φ (r) = rln(r) lnk−1(r) 1/2.
k
···
(cid:0) (cid:1)
When µ = 0 the solution degenerates to
Φ (r) A+Blnk(r) .
k
(cid:0) (cid:1)
See [3].
9
Lemma 3.1. If y(r) solves the Boju-Funar equation (3.11) with initial con-
ditions y(r ) = 0 and y′(r ) > 0 and if µ > λ (r ,a,b) then y(r) = 0 for
0 0 k 0
some r > r .
0
Proof. Linear rescaling of the r-variable is invalid in the logarithmic context,
but still the proof is similar to that of Lemma 2.1. The solution is
lnk(r) lnk(r ) if r r a
0 0
− ≤ ≤
y(r) = Φ (r)Acos(µlnkr)+Bsin(µlnkr) if a r b
k ≤ ≤
α+βlnkr if b r < .
≤ ∞
Matching values at a, gives
lnk(a) lnk(r ) = Acos(µlnka)+Bsin(µlnka) (3.12)
0
−
after canceling the common factor Φ (r). Matching derivative values gives
k
Φ′(a)[lnk(a) lnk(r )]+Φ (a)(lnk)′(a)
k − 0 k
= Φ′(a) Acos(µlnka)+Bsin(µlnka) (3.13)
k
(cid:0) (cid:1)
+Φ (a) Asin(µlnka)+Bcos(µlnka) µ(lnk)′(a).
k
−
(cid:0) (cid:1)
Plugging in (3.12), discarding the equal terms, and then canceling the com-
mon factor Φ (a)(lnk)′(a) gives
k
1
= Asin(µlnka)+Bcos(µlnka).
µ −
From this and (3.12) it follows that
1
A = cos(µlnka)[lnka lnkr ] sin(µlnka)
0
− − µ
(3.14)
1
B = sin(µlnka)[lnka lnkr ]+ cos(µlnka).
0
− µ
Similarly at r = b we have
Acos(µlnkb)+Bsin(µlnkb) = α+βlnkb,
and through more canceling we get
β
Asin(µlnkb)+Bcos(µlnkb) = .
− µ
10
