ESTIMATES FOR WEIGHTED VOLUMES
AND APPLICATIONS
By ZHONGMIN QIAN1
[Received 3 March 1995; in revised form 29 April 1996]
§1. Introduction and results
WE give a simple approach to weighted volume measures on a Reimannian manifold. Weighted volume measures arise naturally from the study
of confonnal deformation of a Riemannain metric. Let (M, g) be an
n -dimensional and complete Riemannian manifold, and let Ag and ftg
denote the Laplace-Beltrami operator and the Riemannian volume
measure respectively. In a local chart, write g = (gtj). Then
where (g^) = (g//)"1 and g = det (gq). Suppose the metric g is confonnally
deformed by a positive smooth function a on M, that is, let g be a new
metric on M defined by
g-(X, Y) = (Tg(X, Y),
VZ, Y e TM.
Then the volume measure fig is the weighted volume measure cr^2 d/ig,
and the transform formula of Ricci curvature is given by
ic, = Ric,
n—2
n—2
— Hess hi O- + - — V i n a ® Vln o-
where the Hessian and gradient are computed using the metric g.
We want to establish geometric results relating to the metric g~ by using
the data associated to the original metric g.
To this end we need a concept of curvature associated to a weighted
1
The research was supported by EPSRC grant GR/J55946.
Quart. J. Math. Oxford (2), 48 (1997), 235-242
© 1997 Oxford Univenity Press
236
Z. QIAN
Laplacian. Such a concept has been introduced by Bakry and Emery [1].
Let L be a diffusion operator. Then the metric T and the curvature
operator T2 are defined by
and
r2(/. g) = W<J, g) - r(Lf, g) - Y(j, Lg)},
respectively.
For simplicity, use A* to denote the weighted Laplacian A + Vh, so that
Aft/ = A/ + <V/i, V/). Then if L = A\ we have (cf. [1] Proposition 3,
p. 187])r(/ ) g ) = <V/,Vg), and
W , g) = (Hess/, Hessg) + (Ric - Hess/i)(V/, Vg),
V/, g e C2(M).
We call Ric,, = Ric - Hess/i the "Ricci curvature" of the weighted
Laplacian A*. It is natural to generalize known results for A to Ah using
Ric - Hess/i. For example we have the following several well known
results using Ricci curvature.
THEOREM 1. (Yau [7]) Every non-compact manifold with non-negative
Ricci curvature possesses infinite volume.
THEOREM 2. (Myers [6]) Let M be an n-dimensional and complete
RUmannian manifold, and Ric > k2 for some positive constant k. Then M
is compact and the diameter d(M) =s VnAT1.
THEOREM 3. (Milnor [5]) Let M be an n-dimensional and compact
manifold with nonnegative Ricci curvature. Then the fundamental group
7Ci(M) has polynomial growth whose degree is at most n.
The following simple example shows that such results are no longer
true for a weighted Laplacian if we replace Ricci curvature by R i c Hess/i.
Let M = R2 be the Euclidean space with the standard metric,
and let h(x,y) = -(x2 + y2). Then Ric - HessA = 2 but VolH(M) < »,
EXAMPLE.
although M is non-compact, where Vol^(Af) = I eh dx, the volume
associated to A . Moreover, Theorem 2 does not hold if we replace Ricci
curvature by Ric - HessA.
DEFINITION 1. Let a > 0 . The a-Ricci curvature of the weighted
Lapacian A* is defined to be
Ric? » Ric - Hess/i - - Vh ® Vh.
a
ESTIMATES FOR WEIGHTED VOLUMES AND APPLICATIONS
237
Using a-Ricci curvature, we have the following.
THEOREM 4. Let (M, g) be a complete and non-compact Riemannian
manifold, and let h e C2(M). If Ric£ 5» 0 for some positive constant a,
then the weighted volume Vol*(A/) = °°, where of course VolA = fig with
(2//)
THEOREM 5. Let (M, g) be a complete and connected Riemannian
manifold. Assume that there is a h e C2(M), a >0 and C > 0 , such that
C2. Then M is compact and the diameter d(M) =s V« + a - \nC~\
THEOREM 6. Let M be a compact Riemannian manifold and let Ric£ s* 0
for some h e C2(M), a>0. Then the fundamental group n^M) has
polynomial growth whose degree is at most n + a.
Theorem 4 has the following corollary.
COROLLARY 1. Let (M, g) be a complete and non-compact Riemannian
manifold. Assume that there is a upper bounded function h E C2(M) such
that h>0 and Ric - — Hess/i > 0 for some positive constant a. Then
h
Vol(M) = ».
There are several works by various authors on extensions of Theorem 2
and Theorem 3, cf. [2,4] and references therein.
In Section 2, we will give proofs of Theorem 4, Theorem 5 and
Theorem 6, and establish several estimates for a weighted volume
measure as well.
§2. Estimates of weighted volumes
We recall several basic facts in Riemannian geometry. Let p e M and
let cut(p) denote the cut locus of the Riemannian manifold M with
respect to the point p. For £ e TpM, |f | = 1, let y((t) = expp (f£) the
geodesic starting at p with direction £ and let Ap(t, £) be the solution of
the Jacobi equation:
^- + KfA=0,
A(0) = 0,
A'(0) = I,
where Kf(t): ^ ^ t , K((t)r, = r7'R(y((t), T,7,)-yf(0 for any TJ e f± =
{X e TPM: (X, f) = 0}, where t, denotes the parallel translation along the
geodesic y( and R denotes the curvature tensor.
238
Z. QIAN
Let ^gp(t, f) = det Ap(t, £). Then Vgp(r, f) is the volume density of the
Riemannain manifold M, and
(0) = ^ In ^ ( ' - £),
Vr > 0,
7f
(0 * art (p),
where p{x) = dist (p, x).
The purpose of this section is to give a similar Bishop's comparison
theorem for the weighted volume measure fig, see Theorem 7.
By Gauss lemma we have
(0) = | h V & ' , f)
V/ > 0,
yt(0 « cirf(p),
where V^(f, f) = Vgp(/, f)exp[/i(y f (/))] is the density of the weighted
volume ehfjLg.
It is well known that
r
i
J
2
y f (0)« J (« - l)^'(*) dj - J 9(*)2Ric(yf(5), yf(s)) ds
(1)
o
o
for any t > 0, yf (t) « cnt(p), and for any continuous and piece-wisely
smooth function <p satisfying the conditions that <p(0) = 0 and that
<p(t) = 1. Using the facts that
and that
^h(yt(t))
= Hessh(y((t),
yf(t)),
we deduce from equation 1 that
j )2
t
-j <p(s)2Rich(y((s), y((s)) ds
/
0
0
J ,
VpXy((s))<p'(s)<p(s) ds,
(2)
o
for t > 0 such that y((t) $ cut(p), and for any continuous and piece-wisely
smooth function <p on [0, t] such that <p(0) = 0 and <p(t) = 1. Using Cauchy
inequality we derive that
i
f
2
«(0)« (« + a - 1) J ?'(*) d5 - J .p(s)2RicJ(yf (5), yf (*)) ds.
o
o
(3)
ESTIMATES FOR WEIGHTED VOLUMES AND APPLICATIONS
239
In particular, if RicjJ > 0, then equation 3 yields that
M -4- /V — 1
A"pa£
P
, on M-cut(p).
(4)
We are now in a position to prove Theorem 4 and Theorem 5.
Proof of Theorem 4. The main point is that indeed equation 4 holds in
distribution. In fact, there is a sequence of domains Dn having compact
closures and smooth boundaries, such that
dp
\JDn = M-cut(p),
and -f->0 on dDn,
aV
n
n = 1, 2 , . . . , where v denotes the outer normal vector. Using integration
by parts formula on Dn, and then taking limit, we get
lit
M
for any nonnegative <p e C%(M). The conclusion now follows the same
argument as in [7], Theorem 7, p. 667. The proof is completed.
Return to equation 2. Suppose RicJ > C2 for some C, C «* 0 or C = ic0
with c0 > 0.
Let M(p, h, r)» max \Vh\, so that M(p, h, r)-» \Vh\ (p) as r-*0. Let
l*B,(r)
<Pc,p,i(.s) = 8c,p(.s)/8c,p(t)> arj d l e t tfc^C*) be the solution of the differential
equation:
(5)
Letting tp = <pc,n,i in equation 2, we get that:
Yd')) « (n " 1) ^ ^ + M ( P . h> r >''
(6>
for any 0< t « r <——— n, y((t) * cut(p), where we use the convention
that ——— = » if C = ic0.
On the other hand, using equation 3 we can conclude that
A*p(yf(0) * ( « + « - ! ) ^ 7 ^ ,
for any t > 0 with y{(r) ? cut(p).
(7)
240
z. CHAN
In fact, we only have to prove that if C > 0 , and if t0>0 is the first zero
Vn + a - 1
point of gCin+a: to =
n and if y((tx) $ cut(p), then f, </ 0 Indeed if t} s» r0, then equation 3 implies that
Hence we have
for any small r, f > r > 0. Letting f-> t0 to get that "VgJ(r0, £) = 0, so that
"Vgp('o» £) = 0. It is a contradition to the hypothesis that rj>r 0 and
y^tt) & cut(p). The above argument in particular implies that the
diameter of the manifold M is less than t0 if the manifold M is connected.
Thus we have proved Theorem 5.
Set
iff>r,
Sc.n+a
Sc.n(t)
Equation 6 and equation 7 together imply that
M'P^fc.n,a.H.Ap)>
on M-cut(p).
(9)
7. For any f e 7^,M with |f | = 1, we Ziove
1- "^(f. f)/(gc^.+a(0)'1+a'"1 « fl decreasing function in t for
yf(t) <t cut(p),
THEOREM
2- y/£(t, O^-^Z^eh^(gc^a(0)n+a-1
for any t>0,
t>0,
y((t)*cut
(P)Proof. By equation 7 we have
d
VWf £1
dV.u-Vr--'* 0 '
Vl>0
' ^" ea " ( ")'
(10)
which yields Conclusion 1. Integrating the two sides of equation 10 over
[0, t] and using equation 8 and equation 9, we get that
Jc,n,aj>J,A
(ID
ESTIMATES FOR WEIGHTED VOLUMES AND APPLICATIONS
241
y/n - 1
for any 0 < r < ——— n, where we have used the fact that
— In vgp(t, £)
, as t -* 0.
Letting r—>0 in equation 11 we obtain Conclusion 2.
COROLLARY
2 Ler Ric£ > C2, and let R > r > 0. 77ien
0
and
(13)
In particular, if R i c ^ O , then equation 13 implies that the measure
VoL. possesses the doubling property:
VoU(*,(2*»
Proof of Theorem 6. If G is a group of finite type and (gu ... , gk) is a
set of generators of G, we denote by
= ]•>/: g = tf'• • • *?*},
n(A) = # { g e G : |g|*A}.
Let G be any finitely generated subgroup of isometrics in (M, g~) which
acts properly discontinuously on M, where g~ = erg with o- = exp(2h/n).
Then we have the following basic fact:
Hence we have
for some constants r > 0, ^ > 0, for detail cf. [3]. Now Theorem 6 follows
from equation 13 immediately.
Equation 12 is not a good estimate for a small geodesic ball, however
we can prove the following estimates for small geodesic ball, whose
proofs will be omitted.
242
THEOREM
Z. QIAN
8. We have
- 3V/z ® V/i]
as t—*0, where s(p) denotes the sealer curvature at p.
A cknowledgement
The author would like to thank Professor K. D. Elworthy for helpful
discussions.
REFERENCES
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Lect Notes in Math. 1123 (1985), 117-206.
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abstract Markov generator1, Duke Math. J. 85 (1) (1996), 253-270.
3. I. Chavel, Riemannian Geometry: a Modern Introduction, Cambridge University Press
(1993).
4. K. D. Elworthy, X. M. Ii and S. Rosenberg, 'Curvature and topology: spectral
positivity', Warwick preprint 47/1993 (1993).
5. J. Milnor, 'A note on curvature and fundamental group', J. Diff. Geom. 2 (1968), 1-7.
6. S. B. Myers, 'Riemann manifolds with positive mean curvature'. Duke Math, J. 8 (1941),
401-404.
7. S. T. Yau, 'Some function-theoretic properties of complete Riemannian manifolds and
their applications to geometry', Indiana Math. J. 25 (1976), 659-670.
Department of Mathematics
Imperial College of Science, Technology and Medicine
180 Queen's Gate
London SW7 2BZ, UK