Finiteness of pi1 and geometri inequalities in almost positive Ri i urvature

Erwann Aubry

Description

Finiteness of pi1 and geometri inequalities in almost positive Ri i urvature Erwann AUBRY ? Abstra t We show that omplete n-manifolds whose part of Ri i urva- ture less than a positive number is small in Lp norm (for p > n/2) have bounded diameter and nite fundamental group. On the on- trary, omplete metri s with small Ln/2-norm of the same part of the Ri i urvature are dense in the set of metri s of any ompa t dierentiable manifold. Keywords: Ri i urvature, omparison theorems, fundamental group 1 Introdu tion A lassi al problem in Riemannian geometry is to nd topolog- i al, geometri al or analyti al ne essary onditions for the exis- ten e on a manifold of a Riemannian metri satisfying a given set of urvature bounds. For instan e, S. Myers showed that a omplete n-manifold with Ric≥k(n?1) (where k>0) is ompa t (the diameter is bounded by π√ k ) and has nite π1, whereas, on the ontrary, J. Lohkamp showed in [11? that on every n-manifold with n≥3 there exists a metri with negative Ri i urvature. This paper is devoted to the study of the Riemannian manifolds satis- fying only an Lp-pin hing on the negative lower part of their Ri i urvature tensors.

• eitherm has small


• stronger ur


• π1


• nite


• volume


• urvature bounded


• r0 ? π


• riemannian metri

Sujets

• Volume

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

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∗
n
pL p>n/2
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n Ric≥k(n−1) k>0
π√ π1k
n
n≥3
pL
Ric(x) = inf Ric (X,X)/g(X,X)x
X∈T Mx
x ∈ M
f (x)=max(−f(x),0) f−
∗ ◦
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nC(p,n) C(p,n,k) VolS
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Theorem
2ρp
VolM
1ρ Sp
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p = 1 n = 2 π1
p =n/2 n≥ 3
n(M ,g) n
n≥ 3
(g ) M gm
ρ (g )n/2 m
→ 0
Volgm
n−1
∞L
∞L
π n1
n−1
π1
the

olume
v
v
olume
the
nite
[8
and
b
,
h
ology
on
top
natural
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has
theorem
on
is
of
still
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v
presen
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the
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with
w
-
o
niteness
an
ob
use
viously
the
follo
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ws
Sob
from
of
the
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Gauss-Bonnet
sition
theorem),
wn
but
Gallot
in
requiring

℄
manifold
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v
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er
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Theorem
an
1.3
b
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pro
ula

of
By
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to
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b
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any
and

hniques,
omp
a
act
on
R

iemannian
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elop
-manifold
generaliza-
(
My
olume
the
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form
nite
℄
).
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Ther
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only
se
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olev
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ere
of
an

trol
omplet