Eigenvalues of perturbed Laplace operators on compact manifolds

0.24 MB·English

by  Asma Hassannezhad

EIGENVALUES OF PERTURBED LAPLACE OPERATORS
ON COMPACT MANIFOLDS
3
ASMA HASSANNEZHAD
1
0
2 Abstract. WeobtainupperboundsfortheeigenvaluesoftheSchr¨odinger

n operator L = ∆g +q depending on integral quantities of the potential
a q and a conformal invariant called the min-conformal volume. More-
J over,whentheSchr¨odingeroperatorLispositive,integral quantitiesof

3 q which appear in upperbounds, can be replaced by the mean value of
the potential q. The upper bounds we obtain are compatible with the

] asymptoticbehavioroftheeigenvalues. Wealsoobtainupperboundsfor
G
the eigenvalues of the weighted Laplacian or the Bakry–E´mery Lapla-
D cian ∆φ =∆g+∇gφ·∇g using two approaches: First, we use the fact
h. that∆φ isunitarilyequivalenttoaSchr¨odingeroperatorandwegetan
t upper bound in terms of the L2-norm of ∇gφ and the min-conformal
a volume. Second, we use its variational characterization and we obtain
m upper bounds in terms of the L∞-norm of ∇gφ and a new conformal
[ invariant. ThesecondapproachleadstoaBusertypeupperboundand
also gives upper bounds which do not depend on φ when the Bakry-
2 E´mery Ricci curvatureis non-negative.
v
3
1
7
7 Contents
.
0
1 1. Introduction and statement of the results 1
2 Acknowledgements 7
1
2. Preliminaries and technical tools 8
:
v 3. Eigenvalues of Schro¨dinger operators 13
Xi 4. Eigenvalues of Bakry–E´mery Laplace operators 17
r Appendix A. Buser type upper bound on Bakry-E´mery manifolds 18
a
References 20
1. Introduction and statement of the results
In this paper, we study upper bound estimates for the eigenvalues of
Schro¨dingeroperatorsandweightedLaplaceoperatorsorBakry–E´meryLaplace
operators.
2010 Mathematics Subject Classification. 58J50, 35P15, 47A75 .
Key words and phrases. Schr¨odinger operator, Bakry–E´mery Laplace operator, eigen-
value,upperbound,conformal invariant.
The author has benefited from the support of the boursier du gouvernement Franc¸ais
duringherstay in Tours.
1
2 ASMAHASSANNEZHAD
Schr¨odinger Operator. Let (M,g) be a compact Riemannian manifold of
dimension m and q C0(M). The eigenvalues of the Schro¨dinger operator
∈
L := ∆ +q acting on functions constitute a non-decreasing, semi-bounded
g
sequence of real numbers going to infinity.
λ (∆ +q) λ (∆ +q) λ (∆ +q) .
1 g 2 g k g
≤ ≤ ··· ≤ ≤ ··· ր ∞
The well-known Weyl law which describes the the asymptotic behavior of
theeigenvalues oftheLaplacian [2]canbeeasily extendedtotheeigenvalues
of Schro¨dinger operators on compact Riemannian manifolds:
2
µg(M) m
lim λ (∆ +q) = α , (1)
k g m
k→∞ k
(cid:18) (cid:19)
where α = 4π2ω−m2 and ω is the volume of the unit ball in Rm.
m m m
2
It describes that normalized eigenvalues, λ (∆ +q) µg(M) m, asymp-
k g k
totically tend to a constant depending only on the dim(cid:16)ension(cid:17). However,
upper bounds of normalized eigenvalues in general cannot be independent
ofgeometricinvariantsandthepotentialq (see[5]ortheintroductionof[9]).
Weshallobtainupperboundsfornormalizedeigenvalues dependingonsome
geometric invariants and integral quantities of the potential q. Moreover,
these upper bounds are compatible with the asymptotic behavior in (1) i.e.
they tend asymptotically to a constant depending only on the dimension as
k goes to infinity.
Numerous articles are devoted to study how the eigenvalues of L can be
controlled in terms of geometric invariants of the manifold and quantities
depending on the potential. From the variational characterization of eigen-
values, it is easy to see that
1
λ (∆ +q) qdµ .
1 g g
≤ µ (M)
g ZM
For the second eigenvalue λ (∆ +q), an upperbound in terms of the mean
2 g
value of the potential q and a conformal invariant was obtained by El Soufi
and Ilias [7, Theorem 2.2]:
2
λ (∆ +q) m Vc([g]) m + M qdµg, (2)
2 g
≤ µ (M) µ (M)
(cid:18) g (cid:19) R g
whereV ([g]) is the conformal volume defined by Li and Yau [13] which only
c
depends on the conformal class of g, denoted [g].
For a compact orientable Riemannian surface (Σ ,g) of genus γ, they
γ
obtained the following inequality as a consequence of Inequality (2):
λ (∆ +q) 8π γ+3 + Σγ qdµg, (3)
2 g
≤ µg(Σγ)(cid:20) 2 (cid:21) Rµg(Σγ)
where [γ+3] is the integer part of γ+3.
2 2
EIGENVALUES OF PERTURBED LAPLACE OPERATORS 3
ForhighereigenvaluesofSchro¨dingeroperators,Grigor’yan,Netrusovand
Yau [8] proved a general and abstract result that can be stated in the case
of Schro¨dinger operators as follows: Given positive constants N and C ,
0
assumethat a compact Riemannian manifold (M,g) has the (2,N)-covering
property (i.e. each ball of radius r can be covered by N balls of radius r/2)
and µ (B(x,r)) C r2 for every x M and every r > 0. Then for every
g 0
≤ ∈
q C0(M) we have [8, Theorem 1.2 (1.14)]:
∈
Ck+δ−1 q+dµ δ q−dµ
λ (∆ +q) M g − M g, (4)
k g
≤ µ (M)
R g R
where δ (0,1) is a constant which depends only on N, C > 0 is a constant
∈
which depends on N and C , and q± = max q ,0 .
0
{|± | }
Moreover, if L is a positive operator [8, Theorem 5.15], then
Ck+ qdµ
λ (∆ +q) M g, (5)
k g
≤ ǫµ (M)
gR
where ǫ (0,1) depends only on N and C depends on N and C .
0
∈
The above inequalities in dimension 2 have special feature as follows. Let
Σ be a compact orientable Riemannian surface of genus γ. Then for every
γ
Riemannian metric g on Σ and every q C0(Σ ) we have [8, Theorem 5.4]:
γ γ
∈
Q(γ +1)k+δ−1 q+dµ δ q−dµ
λ (∆ +q) Σγ g − Σγ g,
k g
≤ Rµg(Σγ) R
where δ (0,1) and Q > 0 are absolute constants.
∈
Inequalities (4) and (5) are not compatible with the asymptotic behavior
regarding to the power of k, except in dimension 2. Yet, for surfaces, the
limit of the above upper bound for normalized eigenvalues depends on the
genus γ as k goes to infinity. Therefore, it is not compatible with (1).
We obtain upperboundswhichgeneralize andimprovetheabove inequal-
itieswithoutimposinganyconditiononthemetricandwhicharecompatible
with the asymptotic behavior. Before stating our theorem, we need to re-
call the definition of the min-conformal volume. For a compact Riemannian
manifold (M,g), its min-conformal volume is defined as follows [9].
V([g]) = inf µ (M) : g [g], Ricci (m 1) .
{ g0 0 ∈ g0 ≥ − − }
Theorem 1.1. There exist positive constants α (0,1), B and C de-
m m m
∈
pending only on m such that for every compact m-dimensional Riemannian
manifold (M,g), every potential q C0(M) and every k N∗, we have
∈ ∈
α−1 q+dµ α q−dµ
λ (∆ +q) m M g − m M g
k g
≤ µ (M)
R g R
2 2
V([g]) m k m
+B +C , (6)
m m
µ (M) µ (M)
(cid:18) g (cid:19) (cid:18) g (cid:19)
4 ASMAHASSANNEZHAD
In particular, when the potential q is nonnegative one has
2 2
λ (∆ +q) A M qdµg +B V([g]) m +C k m , (7)
k g m m m
≤ µ (M) µ (M) µ (M)
R g (cid:18) g (cid:19) (cid:18) g (cid:19)
where A = α−1.
m m
We also obtain upper bounds for eigenvalues of positive Schro¨dinger op-
erators. Note that the positivity of the Schro¨dinger operator L = ∆ +q
g
implies that q 0 and q here may not be nonnegative. The following
M ≥
upper bound generalizes Inequalities (5) and (7).
R
Theorem 1.2. There exist constants A > 1, B and C depending only
m m m
onm suchthat ifL = ∆ +q, q C0(M) isapositive operator thenforevery
g
∈
compact m-dimensional Riemannian manifold (Mm,g) and every k N∗ we
∈
have
2 2
λ (∆ +q) A M qdµg +B V([g]) m +C k m .
k g m m m
≤ µ (M) µ (M) µ (M)
R g (cid:18) g (cid:19) (cid:18) g (cid:19)
GiventheSchro¨dingeroperatorL = ∆ +q,foreveryε > 0,theSchro¨dinger
g
operator L˜ = ∆ +q λ (L)+ε is positive and λ (L˜) = λ (L) λ (L)+ε.
g 1 k k 1
− −
When ε goes to zero, Theorem 1.1 leads to the following:
Corollary 1.1. Under the assumptions of Theorem 1.1 we get
qdµ
λ (∆ +q) A M g +(1 A )λ (∆ +q)
k g m m 1 g
≤ µ (M) −
R g
2 2
V([g]) m k m
+B +C .
m m
µ (M) µ (M)
(cid:18) g (cid:19) (cid:18) g (cid:19)
In the 2-dimensional case, for a compact orientable Riemannian surface
(Σ ,g) of genus γ, thanks to the uniformization and Gauss-Bonnet theo-
γ
rems, one has V([g]) 4πγ. Therefore, in compact orientable Riemannian
≤
surfaces, one can replace the min-conformal volume by the topological in-
variant 4πγ in the above inequalities.
Corollary 1.2. There exist absolute constants a (0,1), A and B such
∈
that, for every compact orientable Riemannian surface (Σ ,g) of genus γ,
γ
every potential q C0(M) and every k N∗, we have
∈ ∈
λ (∆ +q)µ (Σ ) (aq+ a−1q−)dµ +Aγ +Bk. (8)
k g g γ g
≤ −
ZΣγ
And if L is a positive operator then
λ (∆ +q)µ (Σ ) a qdµ +Aγ +Bk.
k g g γ g
≤
ZΣγ
An interesting application of Theorem 1.1 is the case of weighted Laplace
operators or Bakry–E´mery Laplace operators.
EIGENVALUES OF PERTURBED LAPLACE OPERATORS 5
Bakry–E´mery Laplacian. Let (M,g) be a Riemannian manifold and φ
∈
C2(M). The corresponding weighted Laplace operator is defined as
φ
△
follows.
∆ = ∆ + φ .
φ g g g
∇ ·∇
This operator is associated with the quadratic functional f 2e−φdµ
M |∇g | g
i.e.
R
∆ fhe−φdµ = f, h e−φdµ .
φ g g g g
h∇ ∇ i
ZM ZM
This operator is an elliptic operator on C∞(M) L2(e−φdµ ) and can
c ⊆ g
be extended to a self-adjoint operator with the weighted measure e−φdµ .
g
In this sense, it arises as a generalization of the Laplacian. The weighted
Laplace operator ∆ is also known as the diffusion operator or the Bakry–
φ
E´mery Laplace operator which is used to study the diffusion process (see
for instance, the pioneering work of Bakry and E´mery [1], the paper of Lott
[14], and Lott and Villani [15] on this topic). The triple (M,g,φ) is called
a Bakry–E´mery manifold where φ C2(M) and (M,g) is a Riemannian
∈
manifold with the weighted measure e−φdµ (see [16], [18]). The interplay
g
between geometry of M and the behavior of φ is mostly taken into account
by means of new notion of curvature called the Bakry–E´mery Ricci tensor1
that is defined as follows
Ricci = Ricci +Hessφ.
φ g
Ouraimistofindupperboundsfortheeigenvaluesof∆ denotedbyλ (∆ )
φ k φ
in terms of the geometry of M and of properties of φ.
Upper bounds for the first eigenvalue λ (∆ ) of complete non-compact
1 φ
Riemannian manifolds have been recently considered in several works (see
[17],[19],[20], [22] and [23]). These upper bounds depend on the L∞-norm
of φ and a lower bound of the Bakry–E´mery Ricci tensor:
g
∇
Let (M,g,φ) be a complete non-compact Bakry–E´mery manifold of dimen-
sion m with Ricci κ2(m 1) and φ σ for some constants κ 0
φ g
≥ − − |∇ | ≤ ≥
and σ > 0. Then we have [20, Proposition 2.1] (see also [17], [22] and [23]):
1
λ (∆ ) ((m 1)κ+σ)2. (9)
1 φ
≤ 4 −
In particular, if Ricci 0, then we have
φ
≥
1
λ (∆ ) σ2. (10)
1 φ
≤ 4
We consider compact Bakry–E´mery manifolds and we present two ap-
proaches to obtain upper bounds for the eigenvalues of the Bakry–E´mery
Laplace operator in terms of the geometry of M and of the properties of φ.
1 The Bakry–E´mery Ricci tensor Ricci is also referred to as the ∞-Bakry–E´mery
φ
Riccitensor. Wewill denote Ricciφ and Hessφ by Ricciφ(M,g) and Hessgφ wherever any
confusion might occur.
6 ASMAHASSANNEZHAD
First approach. One can see that ∆ is unitarily equivalent to the
φ
Schro¨dinger operator L = ∆ + 1∆ φ+ 1 φ2 (see for example [19, page
g 2 g 4|∇g |
28]). Therefore, as a consequence of Theorem 1.2 we obtain an upperbound
for λ (∆ ) in terms of the min-conformal volume and the L2-norm of φ.
k φ g
∇
Theorem 1.3. There exist constants A , B and C depending on m
m m m
N∗, suchthatforeverym-dimensionalcompact Bakry–E´merymanifold(M,∈g,φ),
we have
2 2
1 V([g]) m k m
λ (∆ ) A φ 2 +B +C .
k φ ≤ mµ (M)k∇g kL2(M) m µ (M) m µ (M)
g (cid:18) g (cid:19) (cid:18) g (cid:19)
It is worth noticing that in full generality, it is not possible to obtain
upper bounds which do not depend on φ (see for instance [20, Section 2]).
However, we will see that for compact manifolds with nonnegative Bakry–
E´mery Ricci curvature we can find upper bounds which do not depend on
φ (see Corollary 1.4 below).
In the 2-dimensional case, as a result of Corollary 1.2 we obtain
Corollary 1.3. There exist absolute constants a (0,1), A and B such
∈
that, for every compact orientable Riemannian surface (Σ ,g) of genus γ
γ
and every k N∗, we have
∈
λ (∆ )µ (Σ ) a φ 2 +Aγ+Bk.
k φ g γ ≤ k∇g kL2(Σγ)
Second approch. It is based on using the technique introduced in [9]
which was successfully applied for the Laplace operator ∆ on Riemannian
g
manifolds in [9, Theorem 1.1]. We obtain upper bounds for eigenvalues of
∆ in terms of a conformal invariant. We also obtain a Buser type upper
φ
bound for λ (∆ ) (see below Corollary 1.5).
k φ
Definition 1.1. Let (M,g,φ) be a compact Bakry–E´mery manifold. We
define the φ min conformal volume as
−
V ([g]) = inf µ (M,g ): g [g],Ricci (M,g ) (m 1) , (11)
φ φ 0 0 φ 0
{ ∈ ≥ − − }
where µ (M,g ) is the weighted measure2 of M with respect to the metric
φ 0
g .
0
Note that up to dilations3 there is always a Riemannian metric g [g]
0
∈
suchthatRicci (M,g ) (m 1). Wearenowreadytostateourtheorem.
φ 0
≥ − −
2ForaBakry–E´merymanifold(M,g,φ),whenµ istheweightedmeasurewithrespect
φ
to the metric g, we simply denote the weighted measure of a measurable subset A of M
byµ (A) instead of µ (A,g).
φ φ
3NoticeHessφandRiccigdonotchangeunderdilations. IfRicciφ(M,g)≥−κ2(m−1)g,
then ∀α>0, Ricciφ(M,g0):=Ricciφ(M,αg)=Ricciφ(M,g)≥−κ2(m−1)g=−κα2(m−
1)g0.
EIGENVALUES OF PERTURBED LAPLACE OPERATORS 7
Theorem 1.4. There exist positive constants A(m) and B(m) depending
onlyonm N∗ suchthatforeverycompact Bakry–E´merymanifold(M,g,φ)
∈
with φ σ for some σ 0 and for every k N∗, we have
g
|∇ | ≤ ≥ ∈
2 2
λ (∆ ) A(m)max σ2,1 Vφ([g]) m +B(m) k m . (12)
k φ
≤ { } µ (M) µ (M)
(cid:18) φ (cid:19) (cid:18) φ (cid:19)
Ifametricg is conformally equivalent toametricg withRicci (M,g )
0 φ 0
≥
0, then V ([g]) = 0. Therefore, an immediate consequence of Theorem 1.4
φ
is the following.
Corollary 1.4. There exists a positive constant A(m) which depends only
on m N∗ such that for every compact Bakry–E´mery manifold (M,g,φ)
∈
with V ([g]) = 0, and for every k N∗
φ
∈
2
k m
λ (∆ ) A(m) . (13)
k φ
≤ µ (M)
(cid:18) φ (cid:19)
The above upper bound is similar to the upper bound for the eigenval-
ues of the Laplacian in Riemannian manifolds (M,g) when V([g]) = 0 (see
[11]).
If Ricci (M) > κ2(m 1) for some κ 0, then for g = κ2g one has
φ 0
− − ≥
Ricci (M,g ) > (m 1) and V ([g]) µ (M,g ) = κmµ (M,g). Replac-
φ 0 φ φ 0 φ
− − ≤
ing in Inequality (12), we get a Buser type upper bound for the eigenvalues
of the Bakry–E´mery Laplacian.
Corollary 1.5 (Buser type upper bound). There are positive constants
A(m) and B(m) depending only on m N∗ such that for every compact
∈
Bakry–E´mery manifold (M,g,φ) with Ricci (M) > κ2(m 1) and φ
φ g
− − |∇ | ≤
σ for some κ 0 and σ 0, and for every k N∗, we have
≥ ≥ ∈
2
k m
λ (∆ ) A(m)max σ2,1 κ2 +B(m) .
k φ
≤ { } µ (M)
(cid:18) φ (cid:19)
A weaker version of Corollary 1.5 can be proved directly by the classic
idea used by Buser [3], Li and Yau [12]. We refer the reader to Appendix A
where we give a simple direct proof.
Remark 1.1. Notice that all of the results have been mentioned above for
compact manifolds are also valid when bounded sudomains of complete man-
ifolds with the Neumann boundary condition are considered.
Acknowledgements
This paper is part of the author’s PhD thesis under the direction of Pro-
fessors Bruno Colbois (Neuchaˆtel University), Ahmad El Soufi (Franc¸ois
Rabelais University), and Alireza Ranjbar-Motlagh (Sharif University of
Technology) and she acknowledge their support and encouragement. The
author wishes to express her thanks to Bruno Colbois and Ahmad El Soufi
8 ASMAHASSANNEZHAD
for suggesting the problem and for many helpful discussions. She is also
very grateful to the referee for helpful comments on the first version of the
paper.
2. Preliminaries and technical tools
We begin by recalling some definitions.
Basic definitions. A capacitor is a couple of Borel sets (F,G) in a topo-
logical space X such that F G.
We say that a metric space (X,d) satisfies the (κ,N;ρ)-covering property if
each ball of radius 0 < r ρ can be covered by N balls of radius r. We
≤ κ
sometimes call it local covering property when ρ < .
∞
For any x X and 0 r R, we define the annulus A(x,r,R) as
∈ ≤ ≤
A(x,r,R) := B(x,R) B(x,r)= y X : r d(x,y) < R .
\ { ∈ ≤ }
Note that A(x,0,R) = B(x,R). For any annulus A(x,r,R) and λ 1, set
≥
λA:= A(x,λ−1r,λR). For F X and r > 0, we denote the r-neighborhood
⊆
of F by Fr, that is
Fr = x X :d(x,F) r .
{ ∈ ≤ }
Here, we state the key method that we use in order to obtain our results.
This method was introduced in [9] and was inspired by two elaborate con-
structions given in [6] and [8]. It leads to construct a “nice” family of
capacitors which is crucial to estimate the eigenvalues of Schro¨dinger oper-
ators and Bakry–E´mery operators via capacities.
Capacity on Riemannian manifolds. For each capacitor (F,G) in a
Riemannian manifold (M,g) of dimension m, we define the capacity and
the m-capacity by:
cap (F,G) = inf ϕ2dµ , and cap(m)(F,G) = inf ϕmdµ ,
g ϕ∈T ZM |∇g | g [g] ϕ∈T ZM |∇g | g
(14)
respectively, where = (F,G) is the set of all functions ϕ C∞(M) such
T T ∈ 0
that supp ϕ G, 0 φ 1 and ϕ 1 in a neighborhood of F. If (F,G)
⊂ ≤ ≤ ≡ T
(m)
is empty, then cap (F,G) = cap (F,G) = + .
g [g] ∞
Proposition 2.1. ([10,Theorem1.2.1], seealso[9])Let(X,d,µ) beametric
measure space with a non-atomic Borel measure µ satisfying the (2,N;ρ)-
covering property. Then for every n N∗, there exists a family of capacitors
∈
= (F ,G ) n with the following properties:
A { i i }i=1
µ(X)
(i) µ(F ) ν := , where c is a constant depending only on N ;
i ≥ 8c2n
(ii) the G ’s are mutually disjoint ;
i
(iii) the family is such that either
A
(a) all the F ’s are annuli and G = 2F , with outer radii smaller than
i i i
ρ, or
EIGENVALUES OF PERTURBED LAPLACE OPERATORS 9
(b) all the F ’s are domains in X and G = Fr0, with r = ρ .
i i i 0 1600
The following lemma is a consequence of the above proposition.
Lemma 2.1. Let (Mm,g,µ) be acompact Riemannian manifold with anon-
atomic Borel measure µ. Then there exist positive constants c(m) (0,1)
∈
and α(m) depending only on the dimension such that for every k N∗
∈
there exists a family (F ,G ) k of mutually disjoint capacitors with the
{ i i }i=1
following properties:
µ(M)
(I) µ(F ) >c(m) ;
i k
2 2
(II) cap (F ,G ) µg(M) 1 V([g]) m +α(m) k m ,
g i i ≤ k r02 µg(M) µg(M)
(cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
where r = 1 .
0 1600
Proof of Lemma 2.1. Take themetric measurespace(M,d ,µ), whereg
g0 0 ∈
[g] with Ricci (m 1) and d is the distance associated to the Rie-
g0 ≥ − − g0
mannian metric g . It is easy to verify that (M,d ,µ) has the (2,N;1)-
0 g0
covering property where N is a constant depending only on the dimension
[9]. Therefore, Proposition 2.1 implies that for every k N∗ there is a fam-
∈
ily of 3k mutually disjoint capacitors (F ,G ) 3k , satisfying the following
{ i i }i=1
properties.
µ(M)
(a) µ(F ) > c(m) , where c(m) (0,1) is a positive constant de-
i k ∈
pending only on the dimension ;
(b) all the F ’s are annuli, G = 2F with outer radii smaller than 1 and
i i i
(m)
cap (F ,2F ) Q , where Q is a constant depending only on
[g] i i ≤ m m
the dimension, or
(c) all the F ’s are domains in M and G = Fr0 is the r -neighborhood
i i i 0
of F , where r = 1 .
i 0 1600
We refer the reader to [9, Proposition 3.1] for more details on the proof of
the part (b). Hence, the family of (F ,G ) 3k has the property (I).
{ i i }i=1
We now show that at least k of them satisfy the property (II). We first find
(m)
an upper bound for the m-capacity cap (F ,G ). If all F ’s are annuli, we
[g] i i i
already have an estimate by the part (b). In the case (c), one can define a
family of functions ϕ (F ,G ), 1 i 3k so that ϕ 1 . Then
i ∈ T i i ≤ ≤ |∇g0 i|≤ r0
1
cap(m)(F ,G ) ϕ mdµ µ (G ).
[g] i i ≤ |∇g0 i| g0 ≤ rm g0 i
ZM 0
Since G ,...,G are mutually disjoint, there exist at least 2k of them so
1 3k
that µ (G ) µ (M)/k. Similarly, there exist at least 2k sets (not nec-
g0 i ≤ g0
essarily the same ones) such that µ (G ) µ (M)/k. Therefore, up to
g i g
≤
re-ordering, we assume that the first k of them (i.e. G ,...,G ) satisfy
1 k
both of the two following inequalities
µ (G ) µ (M)/k, µ (G ) µ (M)/k.
g i ≤ g g0 i ≤ g0
10 ASMAHASSANNEZHAD
Hence, in general, there exist k capacitors (F ,G ),1 i k with
i i
≤ ≤
1 µ (M)
cap(m)(F ,G ) Q + g0 .
[g] i i ≤ m rm k
0
The left hand-side of the above inequality is a conformal invariant. Now,
taking infimum over g [g] with Ricci (m 1) we get
0 ∈ g0 ≥ − −
1 V([g])
(m)
cap (F ,G ) Q + .
[g] i i ≤ m rm k
0
Now, for every ε > 0, we consider plateau functions f k , f (F ,G )
{ i}i=1 i ∈ T i i
with
f mdµ cap(m)(F ,G )+ε.
|∇g i| g ≤ [g] i i
ZM
Therefore,
2 1−2
m m
cap (F ,G ) f 2dµ f mdµ 1 dµ
g i i ≤ |∇g i| g ≤ |∇g i| g suppfi g
ZM (cid:18)ZM (cid:19) (cid:18)ZM (cid:19)
2
≤ cap[(gm])(Fi,Gi)+ε m µg(Gi)1−m2
(cid:16) (cid:17) 2
≤ Qm+ r1mV(k[g]) +ε m µg(Gi)1−m2
(cid:18) 0 (cid:19)
2 1−2
2 1 V([g]) m 2 µg(M) m
≤ Qmm + r2 k +εm k . (15)
" 0 (cid:18) (cid:19) #(cid:18) (cid:19)
where Inequality (15) is due to the well-know fact that
(a+b)s as+bs
≤
when a,b are nonnegative real numbers and 0 < s 1. Letting ε tend to
zero, we obtain the property (II). It completes the p≤roof. (cid:3)
Capacity on Bakry–E´mery manifolds. In an analogous way, we define
thecapacityinaBakry–E´merymanifold(M,g,φ). Foreachcapacitor(F,G)
in a Bakry–E´mery manifold (M,g,φ) of dimension m, the capacity and the
m-capacity is defined as:
cap (F,G) = inf ϕ2dµ , and cap(m)(F,G) = inf ϕmdµ ,
φ ϕ∈T ZM |∇g | φ φ ϕ∈T ZM |∇g | φ
(16)
respectively, where = (F,G) is the set of all functions ϕ C∞(M) such
T T ∈ 0
that supp ϕ G, 0 φ 1 and ϕ 1 in a neighborhood of F. If (F,G)
⊂ ≤ ≤ ≡ T
(m)
is empty, then cap (F,G) = cap (F,G) = + .
φ φ ∞
We shall prove a similar lemma as Lemma 2.1. We start by showing that
every compact Bakry–E´mery manifold satisfies the assumptions of Proposi-
tion 2.1. Thanks to volume comparison theorem proved by Wei and Wylie
[21]forBakry–E´mery manifolds,onecanshowthatBakry–E´mery manifolds
have local covering property (see Lemma 2.2 below).