On the bottom of spectra under coverings

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by  Werner Ballmann

ON THE BOTTOM OF SPECTRA UNDER COVERINGS
7
WERNER BALLMANN,HENRIKMATTHIESEN,
1
0 ANDPANAGIOTISPOLYMERAKIS
2

b Abstract. For a Riemannian covering M1 → M0 of connected Rie-
e mannian manifolds with respective fundamental groups Γ1 ⊆ Γ0, we
F showthatthebottomsofthespectraofM0 andM1 coincideiftheright

3 action of Γ0 on Γ1\Γ0 is amenable.
1

]
G
1. Introduction
D
. In this article, we study the behaviour under coverings of the bottom of
h
the spectrum of Schro¨dinger operators on Riemannian manifolds.
t
a
Let M be a connected Riemannian manifold, not necessarily complete,
m
and V : M → R be a smooth potential with associated Schro¨dinger operator
[
∆ + V. We consider ∆ + V as an unbounded symmetric operator in the
2 space L2(M) of square integrable functions on M with domain C∞(M), the
c
v
space of smooth functions on M with compact support.
0
For a non-vanishing Lipschitz continuous function on M with compact
3
1 support in M, we call
2
(|∇f|2+Vf2)
0 (1.1) R(f)= M
1. R f2
M
0 R
the Rayleigh quotient of f. We let
7
1 (1.2) λ (M,V) = infR(f),
0
:
v
where f runs through all non-vanishing Lipschitz continuous functions on
i
X M with compact support in M. If λ (M,V) > −∞, then ∆+V is bounded
0
∞
r frombelowonC (M)andλ (M,V)is equaltothebottom ofthespectrum
a c 0
of theFriedrichs extension of ∆+V. If λ (M,V) = −∞, then the spectrum
0
of any self-adjoint extension of ∆+V is not bounded from below.
∞
Recall that ∆+V is essentially self-adjoint on C (M) if M is complete
c
and infV > −∞. Then the unique self-adjoint extension of ∆ +V is its
closure. In the case where M is the interior of a complete Riemannian
manifold N with smooth boundary and where V extends smoothly to the
boundary of N, λ (M,V) is equal to the bottom of the Dirichlet spectrum
0
of ∆+V on N.
In the case of the Laplacian, that is, V = 0, we also write λ (M) and call
0
itthebottom of the spectrum of M. Itiswellknownthatλ (M)isthesupre-
0
mum over all λ ∈ R such that there is a positive smooth λ-eigenfunction
Date: February 14, 2017.
2010 Mathematics Subject Classification. 58J50, 35P15, 53C99.
Key words and phrases. Bottom of spectrum, amenable covering.
We would like to thank the Max Planck Institute for Mathematics and the Hausdorff
Center for Mathematics in Bonn for their support.
1
2 WERNERBALLMANN,HENRIKMATTHIESEN,ANDPANAGIOTISPOLYMERAKIS
f: M → R(see, e.g., [4,Theorem7], [5,Theorem1], or[6, Theorem2.1].) It
is crucial that these eigenfunctions are not required to be square-integrable.
In fact, λ (M) is exactly the border between the positive and the L2 spec-
0
trum of ∆ (see, e.g., [6, Theorem 2.2]).
Suppose now that M is simply connected and let π : M → M and
0 0
π : M → M be Riemannian subcovers of M. Let Γ and Γ be the groups
1 1 0 1
of covering transformations of π and π , respectively, and assume that
0 1
Γ ⊆ Γ . Then the resulting Riemannian covering π: M → M satisfies
1 0 1 0
π◦π = π . Let V : M → R be a smooth potential and set V = V ◦π.
1 0 0 0 1 0
Since the lift of a positive λ-eigenfunction of ∆ on M to M is a positive
0 1
λ-eigenfunction of ∆, we always have λ (M ) ≤ λ (M ) by the above char-
0 0 0 1
acterization of the bottom of the spectrum of ∆ by positive eigenfunctions.
InSection4,wepresentashortandelementary proofoftheinequalitywhich
does not rely on the characterization of λ by positive eigenfunctions:
0
Theorem 1.3. For any Riemannian covering π: M → M as above,
1 0
λ (M ,V )≤ λ (M ,V ).
0 0 0 0 1 1
Brooksshowedin[3,Theorem1]thatλ (M ) = λ (M )inthecasewhere
0 0 0 1
M is complete, has finite topological type, and π is normal with amenable
0
group Γ \Γ of covering transformations. B´erard and Castillon extended
1 0
this in [1, Theorem 1.1] to λ (M ,V ) = λ (M ,V ) in the case where M is
0 0 0 0 1 1 0
complete, π (M ) is finitely generated (this assumption occurs in point (1)
1 0
of their Section 3.1), and the right action of Γ on Γ \Γ is amenable. We
0 1 0
generalize these results as follows:
Theorem 1.4. If the right action of Γ on Γ \Γ is amenable, then
0 1 0
λ (M ,V )= λ (M ,V ).
0 0 0 0 1 1
Here a right action of a countable group Γ on a countable set X is said
to be amenable if there exists a Γ-invariant mean on L∞(X). This holds if
and only if the action satisfies the Følner condition: For any finite subset
G ⊆ Γ and ε > 0, there exists a non-empty, finite subset F ⊆ X, a Følner
set, such that
(1.5) |F \Fg| ≤ ε|F|
for all g ∈ G. By definition, Γ is amenable if the right action of Γ on itself
is amenable, and then any action of Γ is amenable.
Incomparison withtheresultsof Brooks, B´erard, andCastillon, themain
point of Theorem 1.4 is that we do not need any assumptions on metric and
topology of M . A main new point of our arguments is that we adopt our
0
constructions more carefully to the different competitors for λ separately.
0
2. Fundamental domains and partitions of unity
ChooseacompleteRiemannianmetrichonM . Inwhatfollows,geodesics,
0
distances, and metric balls in M , M , and M are taken with respect to h
0 1
and its lifts to M and M, respectively.
1
Fix a point x in M . For any y ∈ π−1(x), let
0
(2.1) D = {z ∈M | d(z,y) ≤ d(z,y′) for all y′ ∈π−1(x)}
y 1
BOTTOM OF SPECTRUM 3
bethefundamental domain of π centered at y. ThenD is closed in M , the
y 1
boundary ∂D of D has measure zero in M , and π: D \∂D → M \C is
y y 1 y y 0
an isometry, where C is a subset of the cut locus Cut(x) of x in M . Recall
0
that Cut(x) is of measure zero. Moreover, M1 = ∪y∈π−1(x)Dy, y ∈ π−1(x).
Lemma 2.2. For any ρ > 0, there is an integer N(ρ) such that any z in
M is contained in at most N(ρ) metric balls B(y,ρ), y ∈ π−1(x).
1
Proof. Letz ∈B(z ,ρ)∩B(y ,ρ) withy 6= y in π−1(x)and γ ,γ : [0,1] →
1 2 1 2 1 2
M be minimal geodesics from y to z and y to z, respectively. Then σ =
1 1 2 1
π◦γ and σ = π◦γ are geodesic segments form x to π(z). Since y 6= y ,
1 2 2 1 2
σ and σ are not homotopic relative to {0,1}. Hence, if z lies in in the
1 2
intersection of n pairwise different balls B(y ,ρ) with y ,...,y ∈ π−1(x),
i 1 n
then the concatenations σ−1 ∗ σ represent n pairwise different homotopy
1 i
classes of loops at x of length at most 2ρ. Hence n is at most equal to
the number N(ρ) of homotopy classes of loops at x with representatives of
length at most 2ρ. (cid:3)
Lemma 2.3. If K ⊆ M is compact, then π−1(K)∩D is compact. More
0 y
precisely, if K ⊆ B(x,r), then π−1(K)∩D ⊆ B(y,r).
y
Proof. Chooser > 0 such that K ⊆ B(x,r). Let z ∈ π−1(K)∩D andγ be
y 0
aminimalgeodesicfromπ(z) ∈ K tox. Letγ betheliftofγ toM starting
0 1
in z. Then γ is a minimal geodesic from z to some point y′ ∈π−1(x). Since
z ∈ D , this implies
y
′
d(z,y) ≤ d(z,y ) ≤ L(γ) = L(γ ) < r.
0
Hence π−1(K)∩D ⊆ B(y,r). (cid:3)
y
Let K ⊆ M be a compact subset and choose r > 0 such that K ⊆
0
B(x,r). Let ψ: R → R be the function which is equal to 1 on (−∞,r],
to t + 1 − r for r ≤ t ≤ r + 1, and to 0 on [r + 1,∞]. For y ∈ π−1(x),
let ψ = ψ (z) = ψ(d(z,y)). Note that ψ = 1 on π−1(K)∩D and that
y y y y
suppψ = B¯(y,r+1).
y
Lemma 2.4. Any z in M is contained in the support of at most N(r+1)
1
of the functions ψ , y ∈ π−1(x).
y
Proof. This is clear from Lemma 2.2 since suppψ is contained in the ball
y
B(y,r+1). (cid:3)
In particular, each point of M lies in the support of only finitely many
1
of the functions ψ . Therefore the function ψ = max{1− ψ ,0} is well
y 1 y
defined. By Lemma 2.3, we have suppψ ∩π−1(K) = ∅. ToPgether with ψ ,
1 1
the functions ψ lead to a partition of unity on M with functions ϕ and
y 1 1
ϕ , y ∈ π−1(x), given by
y
ψ ψ
1 y
(2.5) ϕ = and ϕ = .
1 y
ψ + ψ ψ + ψ
1 z∈π−1(x) z 1 z∈π−1(x) z
P P
Note that suppϕ = suppψ and suppϕ = suppψ for all y ∈ π−1(x).
1 1 y y
Lemma 2.6. The functions ϕ , y ∈ π−1(x), are Lipschitz continuous with
y
Lipschitz constant 3N(r+1).
4 WERNERBALLMANN,HENRIKMATTHIESEN,ANDPANAGIOTISPOLYMERAKIS
Proof. Thefunctionsψ ,y ∈ π−1(x),areLipschitzcontinuouswithLipschitz
y
constant 1 and take values in [0,1]. Hence ψ is Lipschitz continuous with
1
Lipschitz constant N = N(r+1), by Lemma 2.4, and takes values in [0,1].
Therefore the denominator χ = ψ + ψ in the fraction defining
1 z∈π−1(x) z
the ϕ is Lipschitz continuous and takPes values in [1,N]. Hence
y
|(χ(z )−χ(z ))ψ (z )+χ(z )(ψ (z )−ψ (z ))|
2 1 y 1 1 y 1 y 2
|ϕ (z )−ϕ (z )| ≤
y 1 y 2
χ(z )χ(z )
1 2
(2N +N)d(z ,z )
≤ 1 2 ≤ 3Nd(z ,z ). (cid:3)
1 2
χ(z )χ(z )
1 2
As a consequence of Lemma 2.6, we get that ϕ = 1 − ϕ is also
1 y
Lipschitz continuous with Lipschitz constant 6N(r+1)2. P
3. Pulling up
Let f be a non-vanishing Lipschitz continuous function on M with com-
0
pact support and let f = f ◦π. We will construct a cutoff function χ on
1
M such that R(χf ) is close to R(f).
1 1
Let g be the given Riemannian metric on M and h be a complete back-
ground Riemannian metric on M as in Section 2. Then there is a constant
A≥ 1 such that
(3.1) A−1g ≤ h ≤ Ag
on the support of f. We continue to take distances and metric balls in M ,
0
M , and M with respect to h and its respective lifts to M and M.
1 1
Fix a point x in M . With K = suppf and r > 0 such that K ⊆ B(x,r),
0
we get a partition of unity with functions ϕ and ϕ , y ∈ π−1(x), as above.
1 y
Fix preimages u ∈ M and y = π (u) ∈ M of x under π and π, respec-
1 1 0
tively. Write π−1(x) = Γ u as the union of Γ -orbits Γ gu, where g runs
0 0 1 1
through a set R of representatives of the right cosets of Γ in Γ , that is, of
1 0
the elements of Γ \Γ . Then π−1(x) = {π (gu) | g ∈ R}. Let
1 0 1
S = {s ∈R | d(y,π (su)) ≤ 2r+2}
1
= {s ∈R | d(u,tsu)) ≤ 2r+2 for some t ∈ Γ },
1
T = {t ∈ Γ | d(u,tgu) ≤ 2r+2 for some s∈ S},
1
G= TS ⊆ Γ .
0
Since the fibres of π and π are discrete, S and T are finite subsets of Γ ,
0 0
hence also G.
Let ε > 0 and F ⊆ Γ \Γ be a Følner set for G and ε satisfying (1.5).
1 0
Let
P = {g ∈ R |Γ g ∈ F} ⊆ R
1
and set
χ = ϕ .
π1(gu)
gX∈P
Since |P| = |F| < ∞, suppχ is compact. Hence, by Lemma 2.6, χf is
1
compactly supported and Lipschitz continuous on M . Let
1
Q = {y ∈ π−1(x) |(χf )(z) 6= 0 for some z ∈ D }.
1 y
BOTTOM OF SPECTRUM 5
To estimate the Rayleigh quotient of χf , it suffices to consider χf on the
1 1
union of the D , y ∈ Q. We first observe that
y
P = {π (gu) |g ∈ P} ⊆ Q.
1 1
To show this, let y = π (gu) and observe that f does not vanish identically
1 1
on π−1(K)∩D and that ϕ is positive on π−1(K)∩D . Since R is a set
y y y
of representatives of the right cosets of Γ in Γ , there exists a one-to-one
1 0
correspondence between P and P , and hence
1
|P| = |P | ≤ |Q|.
1
The problematic subset of Q is
Q− = {y ∈ Q |0 < χ(z) < 1 for some z ∈ π−1(K)∩Dy}.
Let now y ∈ Q− and z ∈ π−1(K)∩Dy with 0 < χ(z) < 1. Since π1(gu),
g ∈ R, runs through all points of π−1(x), we have ϕ (z) = 1.
g∈R π1(gu)
Hence there are g ,...,g ∈R\P such that ϕ (zP) 6= 0 and
1 k π1(giu)
χ(z)+ ϕ (z) = 1.
π1(giu)
X
Furthermore,therehastobeag ∈P withϕ (z) 6= 0. Thenthesupports
π1(gu)
ofthefunctionsϕ andϕ intersectandwegetd(π (gu),π (g u)) ≤
π1(gu) π1(giu) 1 1 i
2r+2. Thatis, we have d(gu,h g u)≤ 2r+2 for someh ∈ Γ . We conclude
i i i 1
that
d(u,g−1h g u)) = d(gu,h g u)≤ 2r+2.
i i i i
Since π is distance non-increasing, we get that there are s ∈ S and t ∈ T
1 i i
such that g−1h g = t s , and then h g = gt s . Since g ∈/ P, we conclude
i i i i i i i i i
that Γ gt s ∈/ F, i.e., Γ g ∈ F \F(t s )−1. Since (t s )−1 ∈ G, there are at
1 i i 1 i i i i
mostε|F||G|suchelementsg ∈ P. Sinced(y,z) ≤randd(z,π (gu)) ≤ r+1,
1
weconcludewithLemma2.2thatforfixedg ∈ P thereareatmostN(2r+1)
such y ∈ Q. We conclude that
|Q−| ≤ ε|F||G|N(2r+1)
(3.2)
= ε|P||G|N(2r+1) ≤ ε|Q||G|N(2r+1).
We now estimate the Rayleigh quotient of χf1. For any y ∈ Q+ = Q\Q−,
we have χ = 1 on π−1(K)∩D and therefore
y
{|∇(χf )|2+V (χf )2}= {|∇f |2+V f2}
Z 1 1 1 Z 1 1 1
Dy Dy
= {|∇f|2+Vf2}
Z
M0
and
χ2f2 = f2 = f2,
Z 1 Z 1 Z
Dy Dy M0
where, here and below, integrals, gradients, and norms are taken with re-
spect to the original Riemannian metric g on M.
For any y ∈ Q−, we have
χ2f2 ≤ f2 and |V |χ2f2 ≤ C f2,
Z 1 Z Z 1 1 0Z
Dy M0 Dy M0
6 WERNERBALLMANN,HENRIKMATTHIESEN,ANDPANAGIOTISPOLYMERAKIS
whereC isthemaximumof|V |onsuppf = K. ByLemma2.4,Lemma2.6,
0 0
and (3.1), we have |∇χ|2 ≤ 9N(r+1)4A. Therefore
|∇(χf )|2 ≤ 2 {|∇χ|2f2+χ2|∇f ◦π|2|}
Z 1 Z
Dy Dy
≤ 18N(r+1)4A f2+2 |∇f|2.
Z Z
M0 M0
In conclusion,
{|∇(χf )|2 +|V |χ2f2}≤ C
Z 1 1 1
Dy
for any y ∈ Q−, where C >0 is an appropriate constant, which depends on
f, but not on y or the choice of ε and F. With D = |G|N(2r+1), we obtain
from (3.2) that
εD
|Q−| ≤ |Q+|,
1−εD
and conclude that
{|∇(χf )|2+V χ2f2}
R(χf ) = 1 1 1
1 R (χf )2
1
R {|∇f |2+V f2}
= Py∈QRDy 1 1 1
f2
y∈Q Dy 1
P R
{|∇f |2+V f2}+εCD|Q |/(1−εD)
≤ Py∈Q+RDy 1 1 1 +
f2
y∈Q+ Dy 1
P R
{|∇f|2+Vf2}+εCD/(1−εD)
= M0
R
f2
M0
R
εCD
= R(f)+ .
(1−εD) f2
M0
R
For ε → 0, the right hand side converges to R(f).
Proof of Theorem 1.4. By Theorem 1.3, we have λ (M ,V ) ≤ λ (M ,V ).
0 0 0 0 1 1
By(1.1),thebottomofthespectrumofSchro¨dingeroperatorsisgivenbythe
infimum of corresponding Rayleigh quotients R(f) of Lipschitz continuous
functions with compact support. The arguments above show that, for any
suchfunctionf onM andanyδ > 0,thereisaLipschitzcontinuousfunction
0
χf on M with compact support and Rayleigh quotient at most R(f)+δ.
1 1
Therefore we also have λ (M ,V ) ≥ λ (M ,V ). (cid:3)
0 0 0 0 1 1
4. Pushing down
Let f be a Lipschitz continuous function on M with compact support.
1
Define the push down f : M → R of f by
0 0
f (x)= f(y)2 1/2.
0
(cid:0)y∈πX−1(x) (cid:1)
Since suppf is compact, the sum on the right hand side is finite for all
x ∈ M , and hence f is well defined. We have suppf = π(suppf), and
0 0 0
hence suppf is compact. Furthermore, f is differentiable at each point x,
0 0
BOTTOM OF SPECTRUM 7
wheref isdifferentiable atall y ∈ π−1(x)andf(y)6= 0for somey ∈π−1(x),
and then
1
∇f0(x) = f(y)π∗(∇f(y)).
f (x)
0 y∈πX−1(x)
For the norm of the differential of f at x, we get
0
2
1
|∇f0(x)|2 ≤ f (x)2(cid:12) f(y)π∗(∇f(y))(cid:12)
0 (cid:12)(cid:12)y∈πX−1(x) (cid:12)(cid:12)
1 (cid:12) (cid:12)
≤ f(y)2 |∇f(y)|2
f (x)2
0 y∈πX−1(x) y∈πX−1(x)
= |∇f(y)|2.
y∈πX−1(x)
Furthermore, f is differentiable with vanishing differential at almost any
0
point of {f = 0}. Therefore f is Lipschitz continuous and
0 0
f2 = f2, V f2 = V f2, |∇f |2 ≤ |∇f|2.
Z 0 Z Z 0 0 Z 1 Z 0 Z
M0 M1 M0 M1 M0 M1
In particular, we have R(f ) ≤ R(f).
0
Proof of Theorem 1.3. For any non-vanishing Lipschitz continuous function
f on M with compact support, the push down f as above is a Lipschitz
1 0
continuous function on M with compact support and Rayleigh quotient
0
R(f ) ≤ R(f). Theassertedinequalityfollowsnowfromthecharacterization
0
of the bottom of the spectrum by Rayleigh quotients as in (1.1). (cid:3)
5. Final remarks
It is well-known that any countable group is the fundamental group of a
smooth 4-manifold. (A variant of the usual argument for finitely presented
groups, taking connected sums of S1×S3 and performing surgeries, can be
used to produce 5-manifolds with fundamental group any countable group.)
In particular, for a non-finitely generated, amenable group G, e.g., G =
Z or G = Q, there is a smooth manifold M with π (M) ∼= G. In
n∈N 1
Lcontrasttotheresultsin[1,3],ourmainresultalsoappliestosuchexamples.
Moreover, we do not assume λ (M ,V ) > −∞. Given any non-compact
0 0 0
manifold M , it is indeed easy to construct a smooth potential V such that
0 0
λ (M ,V ) = −∞. In fact, it suffices that V (x) tends to −∞ sufficiently
0 0 0 0
fast as x → ∞.
8 WERNERBALLMANN,HENRIKMATTHIESEN,ANDPANAGIOTISPOLYMERAKIS
References
[1] P.B´erard and P.Castillon, Spectral positivity and Riemannian coverings. Bull.
Lond. Math. Soc. 45 (2013), no. 5, 1041–1048.
[2] R.Brooks,ThefundamentalgroupandthespectrumoftheLaplacian.Comment.
Math. Helv. 56 (1981), no. 4, 581–598.
[3] R.Brooks, The bottom of the spectrum of a Riemannian covering. J. Reine
Angew. Math. 357 (1985), 101–114.
[4] S.Y.Cheng and S.T.Yau, Differential equations on Riemannian manifolds and
theirgeometricapplications.Comm.PureAppl.Math.28(1975),no.3,333–354.
[5] D.Fischer-Colbrie and R.Schoen, Thestructureof complete stable minimal sur-
faces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math.
33 (1980), no. 2, 199–211.
[6] D.Sullivan,RelatedaspectsofpositivityinRiemanniangeometry.J.Differential
Geom. 25 (1987), no. 3, 327–351,
WB:MaxPlanckInstituteforMathematics,Vivatsgasse7, 53111Bonnand
Hausdorff Center for Mathematics, Endenicher Allee 60, 53115 Bonn.
E-mail address: [email protected]
HM:MaxPlanckInstitutefor Mathematics,Vivatsgasse7, 53111Bonnand
Hausdorff Center for Mathematics, Endenicher Allee 60, 53115 Bonn.
E-mail address: [email protected]
PP: Institut fu¨r Mathematik, Humboldt-Universita¨t zu Berlin, Unter den
Linden 6, 10099 Berlin.
E-mail address: [email protected]