ANNALES DE
L’INSTITUT FOURIER
LesAnnales de l’institut Fouriersont membres du Centre Mersenne pour l’édition scientifique ouverte
Gilles Carron
Geometric inequalities for manifolds with Ricci curvature in the Kato class
Tome 69, no7 (2019), p. 3095-3167.
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GEOMETRIC INEQUALITIES FOR MANIFOLDS WITH RICCI CURVATURE IN THE KATO CLASS
by Gilles CARRON
Abstract. —We obtain Euclidean volume growth results for complete Rie- mannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, and Betti number estimates for closed manifolds whose Ricci curvature is controlled in the Kato class.
Résumé. —On démontre qu’une variété riemannienne complète vérifiant une inégalité de Sobolev euclidienne et dont la courbure de Ricci est petite dans une classe de Kato et à croissance euclidienne du volume. On obtient aussi des estima- tions spectrales, du noyau de la chaleur et du premier nombre de Betti des variétés riemanniennes compactes dont la courbure de Ricci est controlée dans une classe de Kato.
1. Introduction 1.1. Volume growth
1.1.1. Motivation
One of our motivations was a quest for a higher dimensional analogue of the following beautiful result of P. Castillon ([11]):
Theorem 1.1. — Let(M2, g) be a complete noncompact Riemannian surface with nonnegative Laplacian ∆, Gaussian curvature Kg and Rie- mannian measure dAg. Assume that there is some λ > 14 such that the Schrödinger operator
∆ +λKg
Keywords:Sobolev inequalities, volume growth, Green kernel, Doob transform.
2020Mathematics Subject Classification:53C21, 58J35, 58C40, 58J50.
is nonnegative, i.e.,
∀ψ∈ C0∞(M) : Z
M
|dψ|2g+λKgψ2
dAg>0.
Then, for allR>0and allx∈M,
Areag(B(x, R))6c(λ)R2, wherec(λ)is a constant which only depends onλ.
In fact, P. Castillon has shown that such a surface is either conformally equivalent to the plane or to the cylinder:
(M2, g) '
conf.C or (M2, g) '
conf.C∗.
More generally, when the operator ∆+λKgis only assumed to have finitely many negative eigenvalues for some λ > 1/4, then the same conclusion holds but in this case supRAreag(B(x,R))R2 depends onM andλ.
The purpose of this paper is to investigate a possible higher dimensional analogue of Theorem 1.1. Indeed, it is sometimes crucial to get a Euclidean volume growth estimate, recall that we say that a complete Riemannian manifold (Mn, g) has Euclidean volume growth, if
(EVG) ∀R >0 : volB(x, R)6CRn
where the constantCmay depend on the pointxor not ; for instance this kind of estimate was one of the difficult results obtained by G. Tian and J. Viaclovsky ([53]), and this was a key point toward the description of the moduli spaces of critical Riemannian metrics in dimension four ([54]).
1.1.2. Definitions
According to the Bishop–Gromov comparison theorem, a complete Rie- mannian manifold (Mn, g) with nonnegative Ricci curvature has at most Euclidean volume growth,
∀x∈M, R >0 : volB(x, R)6ωnRn,
whereωn is the Euclidean volume of the unit Euclideann−ball.
If (M, g) is a Riemannian manifold, we introduce the function Ric− de- fined by
Ric−(x) := max{−κ(x),0}
where
κ(x) := inf
~v∈TxM,gx(~v,~v)=1
Riccix(~v, ~v);
so that we have Riccix>−Ric−(x)gx.
We are looking for a spectral condition on the Schrödinger operator Lλ= ∆−λRic−
that would imply a Euclidean volume growth. We do not think that it is possible to prove a result similar to Theorem 1.1 in higher dimension, under the sole assumption thatLλ is nonnegative inL2.
Recall that Lλ is nonnegative on L2 if and only if there is a positive functionhonM such thatLλh= 0. We introduce a stronger condition:
Definition 1.2. — The Schrödinger operatorLλis said to begaugeable if there is a functionh: M →Rand a constantγsuch that16h6γ and Lλh= 0. The constantγis called the gaugeability constant ofLλ.
The behavior of the heat semigroup associated with the Schrödinger operatorLλ can be quite different onL2and on Lp. For instance the fact that the heat semigroup is uniformly bounded onL∞:
sup
t>0
e−tLλ
L∞→L∞ <+∞
implies the nonnegativity of the Schrödinger operatorLλonL2. It may hap- pen that a Schrödinger operator is nonnegative onL2 but the associated semigroup is not uniformly bounded on L∞ ([23]). The uniform bound- edness of the semigroup e−tLλ
t>0 onL∞ is strongly related to the fact that the Schrödinger operatorLλis gaugeable (see [57] and Theorem 2.14).
Hence the gaugeability condition could be interpreted as an L∞ spectral condition.
It is well known that a Sobolev inequality is useful in order to control the behavior of the heat semigroupe−t∆.
Definition 1.3. — We say that a complete Riemannian manifold (Mn, g) satisfies the Euclidean Sobolev inequality with Sobolev constant µif
(Sob) ∀ψ∈ C0∞(M) : µ Z
M
ψn−22n dvg
1−2n
6 Z
M
|dψ|2gdvg
According to a celebrated result of J. Nash and N. Varopoulos ([45, 55]), the Euclidean Sobolev inequality is equivalent to a Euclidean type upper bound on the heat kernel:
(EUB) ∀t >0, x, y∈M: H(t, x, y)6C t−n2e−d(x,y)2Ct .
That is to say, given the Sobolev constantµand the dimension n, there is a constantC=C(n, µ) such that the Euclidean type upper bound on the heat kernel (EUB) holds. Conversely, if the Euclidean type upper bound
on the heat kernel (EUB) holds for some constantC, then (Mn, g) satisfies the Sobolev inequality (Sob) with some constantµ=µ(n, C).
1.1.3. Main result
Theorem 1.4. — Let (Mn, g)be a complete Riemannian manifold of dimensionn >2. Assume that
• (Mn, g) satisfies the Euclidean Sobolev inequality (Sob) with Sobolev constantµ.
• There is aδ >0 such that the Schrödinger operator∆−(n−2)× (1 +δ) Ric− is gaugeable with gaugeability constant γ.
Then, there is a constantθdepending only onn, δ, γ andµ, such that for allx∈M andR>0:
1
θRn 6volB(x, R)6θRn.
In fact, we already know that the Sobolev inequality (Sob) implies a lower bound on the volume of geodesic balls ([2, 7]): there is a constantcn
such that for allx∈M andR >0,
cnµn2Rn6volB(x, R).
The log-Sobolev inequality ([46, Proposition 5.1]) also yields the same con- clusion. Hence, the crucial point in the proof of Theorem 1.4 is to get the upper bound.
Remark 1.5. — If (Mn, g) satisfies the Euclidean inequality (Sob) with constantµthen any of the following conditions implies thatLλis gaugeable for someλ >(n−2):
(1) There is some ∈(0,1) such that Ric− ∈Ln2(1−)∩Ln2(1+) and the Schrödinger operator ∆−(n−2)(1 +) Ric− is nonnegative in L2.
(2) There is some∈(0,1) such that Ric−∈Ln2(1−)∩Ln2(1+)and Z
M
Ric−n2 dvg6 µ
n−2 n2
(1−).
(3) sup
x∈M
Z ∞ 0
1 rn−1
Z
B(x,r)
Ric−(y) dvg(y)
!
dr <nµn2, wheren is a computable constant depending only onn.
The first two conditions are due to B. Devyver ([25]). The last one is an easy consequence of Green kernel estimates (see for instance [20, Theorem 3.1]).
1.1.4. Overview of the proof (see Section 4 for more details) Following T. Colding ([14]), when (Mn, g) is a nonparabolic manifold, we introducebo(x) =G(o, x)−n−21 whereG(o,·) is the Green kernel with pole ato ∈ M, normalized so thatb(x)'x→o d(o, x). When (Mn, g) has nonnegative Ricci curvature, T. Colding has shown that
|dbo|61.
The crucial point in the proof of Theorem 1.4 is to prove a uniform bound on the gradient ofbo:
|dbo|6Γ.
HenceB(o, R)⊂ {x∈M, bo(x)6ΓR}, and using [7, Proposition 1.14], we know that
volg{x∈M, bo(x)6ΓR}6C(µ, n)ΓnRn.
1.1.5. Other definitions
Definition 1.6. — A complete Riemannian manifold (Mn, g) is said to be doubling if there is a constantθ such that
(D) ∀x∈M, R >0 : volB(x,2R)6θvolB(x, R).
Definition 1.7. — A complete Riemannian manifold(Mn, g)satisfies the Poincaré inequality if there is a constantγsuch that for any geodesic ballB of radiusr, we have
(PI) ∀ψ∈ C1(B) : Z
B
(ψ−ψB)2dvg6γr2 Z
B
|dψ|2gdvg. Here and thereafter, for an arbitrary O ⊂M with 0<volgO<+∞,
ψO = 1 volO
Z
O
ψdvg.
Recall that the heat kernel of (M, g) is the Schwartz kernel of the heat operatore−t∆. It is the positive functionH: (0,+∞)×M×M →Rsuch that for anyf ∈L2(M,dvg):
e−t∆f (x) =
Z
M
H(t, x, y)f(y) dvg(y).
Definition 1.8. — We say that the heat kernel of(Mn, g)satisfies the Li–Yau estimates if there are positive constantsc, C such that
(LY) c
volB(x,√
t)e−d(x,y)2ct 6H(t, x, y)6 C volB(x,√
t)e−d(x,y)2Ct .
Definition 1.9. — We say that the heat kernel of (Mn, g) admits a Gaussian upper bound if there is a positive constantCsuch that
(GUB) H(t, x, y)6 C
volB(x,√
t)e−d(x,y)2Ct .
Remark that if (M, g) has nonnegative Ricci curvature, then the Bishop–
Gromov comparison theorem implies that it is doubling. According to P. Buser ([6]), (M, g) satisfies the Poincaré inequality (PI) and, according to a famous result of P. Li and S-T. Yau [42], its heat kernel satisfies the Li–Yau estimates. Furthermore, the estimates (LY) are equivalent to the conditions (D and PI) ([29, 51]). Moreover, according to a nice observation by T. Coulhon ([16]), we note that the lower bound
∀t >0, x, y∈M: H(t, x, y)>c t−n2e−d(x,y)2ct yields a Euclidean upper bound on the volume of geodesic balls.
1.1.6. Consequences of Theorem 1.4
Theorem 1.10. — If(M, g)is a complete Riemannian manifold which satisfies the conditions of Theorem 1.4, then:
• (M, g)is doubling and satisfies the Poincaré inequality (PI).
• The heat kernel of(M, g)satisfies the Li–Yau estimates (LY).
• For anyp∈(1,+∞), the Riesz transformd∆−12:Lp(M)→Lp(T∗M) is a bounded operator.
Remark 1.11. — According to D. Bakry ([4]), on a complete Riemann- ian manifold with nonnegative Ricci curvature, the Riesz transform is a bounded operator onLp for anyp∈(1,+∞).
1.1.7. Gaugeability and the Kato constant
The gaugeability of the Schrödinger operator ∆−λRic− is strongly re- lated to Kato constants. These constants measure the size of the potential Ric− relative to ∆. For a nice introduction to Kato constants, we recom- mend [34, Chapter 6].
Definition 1.12. — LetG(·,·)be the positive minimal Green kernel of(Mn, g). The Kato constant ofRic− is defined by
K(Ric−) := sup
x∈M
Z
M
G(x, y) Ric−(y) dvg(y).
Definition 1.13. — Let{H(t, x, y)}(t,x,y)∈R+×M×M be the heat kernel of(Mn, g). The parabolic Kato constant ofRic− at time T is defined by
kT(Ric−) = sup
x∈M
Z T 0
Z
M
H(t, x, y) Ric−(y) dvg(y)dt.
As we haveG(x, y) =R∞
0 H(t, x, y)dt, we easily deduce that
T→+∞lim kT(Ric−) = K(Ric−).
The observation is as follows.
Lemma 1.14. — Assume that K(Ric−) < n−21 , and that λ > n−2 is such that λK(Ric−)< 1. Then, the Schrödinger operator ∆−λRic− is gaugeable with gaugeability constant
γ= λK(Ric−) 1−λK(Ric−).
The conditions for gaugeabily given in Remark 1.5 imply an estimate of the Kato constant of Ric−.
1.1.8. Localization at infinity
It is possible to get a slightly weaker result, if we only get a control of the Ricci curvature outside a compact set.
Theorem 1.15. — Let (M, g) be a complete Riemannian manifold which satisfies the Euclidean Sobolev inequality (Sob). Assume that there is a compact subsetK⊂M such that
sup
x∈M\K
Z
M\K
G(x, y) Ric−(y) dvg(y) < 1 16n. Then,
• there is a constantθsuch that, for allx∈M, R>0, 1
θRn 6volB(x, R)6θRn.
• (Mn, g)is doubling,
• its heat kernel satisfies(GUB),
• for n > 4 and p ∈ (1, n), the Riesz transform d∆−12: Lp(M) → Lp(T∗M)is a bounded operator.
Remarks 1.16.
• The value 16n1 is not optimal but quite explicit.
• The constantθdepends on (M, g). It cannot be estimated from the dimension, the Sobolev constant. Indeed the geometry of (M, g) on a neighborhood of the compact set K has some influence on this constantθ.
According to [25], the assumptions of Theorem 1.15 are satisfied by com- plete Riemannian manifolds satisfying a Euclidean Sobolev inequality, and such that for some∈(0,1), Ric−∈Ln2(1−)∩Ln2(1+).
1.2. The case of closed manifolds
Recents papers have emphasized how a control on the Kato constant of the Ricci curvature can be useful in order to control some geometrical quantities for closed or complete Riemannian manifolds ([5, 17, 20, 25, 47, 48, 49, 58, 59]). For a closed Riemannian manifold (M, g), we will explain how the works of Qi S. Zhang and M. Zhu [59], together with some classical ideas, can be used in order to obtain geometric and topological estimates based on the Kato constant of the Ricci curvature. Recently C. Rose has also obtained similar results based on this idea ([48]).
Definition 1.17. — Let (Mn, g) be a closed Riemannian manifold of diameterD. The scale invariant geometric quantityξ(M, g)is the smallest positive real numberξ such that, for allx∈M,
Z D
2 ξ2
0
Z
M
H(t, x, y) Ric−(y) dvg(y)dt6 1 16n.
If T(M, g)>0is the largest time, such thatkT(Ric−)6 16n1 then, we have ξ(M, g)p
T(M, g) =D.
For instance if the Ricci curvature is bounded from below, Ricci >
−(n−1)κ2g, then ξ(M, g) 6 4κD. In this case, it is well known that the geometry of (Mn, g) is well controlled by the geometrical quantityκD.
We obtain almost the same results in terms of the new quantityξ(M, g).
Theorem 1.18. — Let (M, g)be a closed Riemannian manifold of di- mension n and diameter D. Then there is a constant γn, which depends only onn, such that
(1) the first nonzero eigenvalueλ1 of the Laplacian satisfies λ1> γ−1−ξ(M,g)n
D2 ; (2) the first Betti number ofM satisfies
b1(M)6n+1
4+ξ(M, g)γ1+ξ(M,g)n .
In particular, there existsn>0 such that ifξ(M, g)<n then b1(M)6n.
(3) (M, g)is doubling: for anyx∈M and06R6D/2, volB(x,2R)6γn1+ξ(M,g)volB(x, R);
(4) for allt >0 andx∈M,
H(t, x, x)6 γn1+ξ(M,g)
volB(x,√ t).
We can get a slight improvement of the previous theorem with a stronger control on the Ricci curvature.
Proposition 1.19. — Let (M, g)be a closed Riemannian manifold of dimensionnand diameterD. Letp >1, and assume that for someT,Λ>0,
D2p−2 sup
x∈M
Z T 0
H(s, x, y) Ricp−(y) dvgds6Λp. Then (withq the exponent dual top:pq=p+q),
ξ(M, g)6α(n, D,Λ, T, p) := max D
√T,(16nΛ )q/2
.
Moreover, there is a constantθ, depending only onα(n, D,Λ, T, p)and n, such that for anyx∈M and06r6R6D,
volB(x, R)
Rn 6θvolB(x, r) rn 6θ2.
A quick comparison between Theorem 1.4 and Theorem 1.18 naturally leads to the question whether the Euclidean Sobolev inequality is necessary in Theorem 1.4. According to Qi S. Zhang and M. Zhu in [59], the results obtained in Theorem 1.18 could be generalized to complete Riemannian manifold provided one has good approximations of the distance function:
there exists c > 0 such that, for all o ∈ M, there exists χo: M → R+
satisfying
d(o, x)/c6χo(x)6cd(o, x),
|dχo|2+|∆χo|6c.
This is a very strong hypothesis. Our proof of Theorem 1.4 and Theo- rem 1.15 yields a comparison between the level sets of the Green kernel and geodesic spheres. As a consequence, we prove the existence of such an approximation of the distance function.
Our estimates on the first Betti number is a generalization of the one ob- tained by M. Gromov under a lower bound on the Ricci curvature. Accord- ing to T. Colding [12, 13], one knows that there exists an(n)>0 such that if (Mn, g) is a closedn-dimensional manifold with Ric−diam2(M)<(n) andb1(M) =n, thenM is diffeomorphic to a torusTn. A natural question is then to ask what can be said on a closed Riemannian manifold satisfying ξ(M, g)<<1 andb1(M) =n. We believe that such a manifold should be close to a torusTnin the Gromov–Hausdorff topology. In order to say more, it would need to understand spaces which are Gromov–Hausdorff limits of Riemannian manifolds (Mn, g), with ξ(M, g)6Ξ and diamM 6D. Note that our results yield a precompactness result in the Gromov–Hausdorff topology for these class of spaces. We hope that the results of this paper will turn out useful to give some answers to a question of G. Tian and J. Vi- aclosky about critical metrics in higher dimension (see [54, Section 8.2]).
A lower bound on the Ricci curvature also yields some isoperimetric inequalities. In [10], we have shown that a bound on the Kato constant of the Ricci curvature yields some isoperimetric inequality.
In the pioneering paper ([27]), S. Gallot proved isoperimetric inequali- ties, eigenvalue and heat kernel estimates for closed Riemannian manifold (Mn, g), under a control of Ric− inLp (forp > n/2). It would be interest- ing to know whether one can directly get a control of the Ricci curvature in some Kato class from a control of Ric− in Lp (for p > n/2), without using Gallot’s isoperimetric inequality.
1.3. Localization in a geodesic ball
Our ideas can be adapted to understand the geometry of a geodesic ball under some stronger control on the Ricci curvature.
Theorem 1.20. — Let(Mn, g)be a Riemannian manifold. Assume that B(o,3R) ⊂ M is a relatively compact geodesic ball. Let p > 1 and let q:=p/(p−1). The Green kernel for the Laplacian∆ onB(o,3R)for the Dirichlet boundary condition is denotedG3R. Define the constant Λby
Λp:=R2p−2 sup
x∈B(o,3R)
Z
B(o,3R)
G3R(x, y) Ric−(y)pdvg(y).
Assume that
(1) for someδ > (q(n−2)−2)8q(n−2) 2, the operator∆−(1 +δ)(n−2) Ric− is nonnegative onB(o,3R),
(2) the ball B(o,3R) satisfies the Sobolev inequality (Sob) with con- stantµ.
Then, there exist constants θ and γ, which only depend onn, p, δ,Λ, µ and on the volume density volB(o,3R)Rn , such that for any x∈B(o, R) and anyr∈(0, R),
• θ1rn 6volB(x, r)6θrn,
• ∀ψ∈ C1(B(x, r)) :R
B(x,r)(ψ−ψB(x,r))2dvg6γr2R
B(x,r)|dψ|2gdvg. 1.4. Organization of the paper
In the next section, we review and collect some analytical tools which will be used in the paper. For instance, we describe Agmon’s type volume estimate which are mainly due to P. Li and J. Wang ([40, 41]). These estimates will be crucial in the proof of Theorem 1.4. We also prove a new elliptic estimate based on a variation of the De Giorgi–Nash–Moser iteration scheme which will be useful in the proof of Theorem 1.20. The third section is devoted to the proof of Theorem 1.18. Theorem 1.4 and the first part of Theorem 1.15 are proved in the fourth section. The last sections are devoted to the end of the proofs of Theorem 1.15 and Theorem 1.20.
Acknowledgments
I wish to thank F. Bernicot, P. Castillon, B. Devyver and M. Herzlich for valuable conversations and useful suggestions. I would also like to thank P. Bérard for his wise advice leading to the final version of the document.
I am also grateful to the referee for their helpful comments on the manu- script. I thank the Centre Henri Lebesgue ANR-11-LABX-0020-01 for cre- ating an attractive mathematical environment. I was partially supported by the ANR grant: ANR-12-BS01-0004“Geometry and Topology of Open Manifolds”.
2. Preliminaries
In this section we review some classical results which will be used throughout the paper. We consider (M, g) a Riemannian manifold, and
a measure dm = Φ dvg where dvg is the Riemannian measure and Φ a positive Lipschitz function; theLp norm associated with this measure will be notedk · kp or k · km,p.
2.1. Laplacians
2.1.1.
The Laplacian ∆mor ∆Φis the differential operator defined by the Green formula:
∀ψ∈ C0∞(M) : Z
M
|dψ|2gdm = Z
M
(∆mψ)ψdm. It is associated with the quadratic form:
(QF) ψ∈ C0∞(M)7−→q(ψ) :=
Z
M
|dψ|2gdm.
The geometric Laplacian will be noted ∆ = ∆1 and we have the formula
∆mψ= ∆ψ− hd log Φ,dψig.
The Friedrichs realization of the operator ∆m is associated with the minimal extension of the above quadratic form. We introduceD(q) to be the completion ofC∞0 (M) with respect to the norm,
ψ7→
q
q(ψ) +kψk22. The domain of the operator ∆mis given by
D(∆m) ={v∈ D(q), ∃C such that∀ϕ∈ C0∞(M) : |hv,∆mϕi|6Ckϕk2}.
Remarks 2.1.
• If (M, g) is geodesically complete, then
∆m:C0∞(M)−→L2(M,dm) has a unique selfadjoint extension.
• If M is the interior of a compact manifold with boundary M = X\∂X and if g and Φ have Lipschitz extensions to X then the Friedrichs realization of the operator ∆mis the Laplacian associated with the Dirichlet boundary condition.
2.1.2. Chain rule
Whenv∈ C∞(M) andf ∈ C∞(R,R), by a direct computation, we have
∆mf(v) =f0(v) ∆mv−f00(v)|dv|2g.
In particular iff is nondecreasing, convex and if ∆mv 6V where V is a nonnegative function then ∆mf(v)6f0(v)V. By approximation, this can be generalized to weak solutions and nonsmooth convex functions. Recall that ifv∈L1loc we say that
∆mv6V weakly if for any nonnegativeϕ∈C0∞(M):
Z
M
v∆mϕdm6 Z
M
V ϕdm. Whenv∈Wloc1,2 then
∆mv6V weakly
if and only if for any nonnegativeϕ∈C0∞(M) (orϕ∈ D(q)):
Z
M
hdv,dϕigdm6 Z
M
V ϕdm.
Then it is classical to get the following (where forx∈R:x+= max{x,0}) Lemma 2.2. — Letv∈Wloc1,2∩ C0and letV ∈L1loc be such thatV >0 and∆mv6V. Then for any α>1, we get:
∆mv+α6αV v+α−1.
∆m(v−1)α+6αV(v−1)α−1+ .
2.1.3. Integration by parts formula The formula
(2.1) |d(χv)|2g =|dχ|2gv2+hdv,d(χ2v)ig, implies the following integration by parts inequality.
Lemma 2.3. — Letv∈Wloc1,2, and let V ∈L1loc be a nonnegative func- tion such that:
∆mv6V weakly.
Then, for every Lipschitz functionχ with compact support, Z
M
|d(χv)|2gdm6 Z
M
|dχ|2gv2dm + Z
M
χ2vV dm.
We would like to make sure that this inequality is also valid for Lips- chitz functions which are constant at infinity. The notion of parabolicity is precisely what is needed.
Definition 2.4. — A Borel measure dµ on a Riemannian manifold is called parabolic if there is a sequence(χk)of smooth functions with com- pact support such that,
• 06χk 61,
• χk→1 uniformly on compact sets,
• limk→∞R
M|dχk|2gdµ= 0.
Then we have the following refinement of Lemma 2.3.
Lemma 2.5. — Letv∈Wloc1,2, and let V ∈L1loc be a nonnegative func- tion such that
∆mv6V weakly.
If the measurev2dmis parabolic then, for every Lipschitz functionχwhich is constant outside a compact set and such thatR
Mχ2vV dm<∞, Z
M
|d(χv)|2gdm6 Z
M
|dχ|2gv2dm + Z
M
χ2vV dm.
Remark 2.6. — If (M, g) is geodesically complete and if M(r) :=
R
B(o,r)v2dm satisfies
M(r) =O r2 or
Z ∞ 1
rdr
M(r) = +∞
then the measurev2dm is parabolic (see [31]).
2.2. Sobolev inequalities
Definition 2.7. — We say that a weighted complete Riemannian man- ifold (Mn, g,m) satisfies the Euclidean Sobolev inequality with Sobolev constantµif
(Sobm) ∀ψ∈ C0∞(M) : µ Z
M
ψn−22n dm 1−n2
6 Z
M
|dψ|2gdm We recall here some classical results which hold in the presence of the Sobolev inequality.
Theorem 2.8. — Let(M, g,m)be a weighted Riemanian manifold. As- sume it satisfies the Euclidean Sobolev inequality (Sobm) with Sobolev constant µ. Then there is a positive constant cn, such that the following properties hold.
(1) The heat kernel associated with the Laplacian∆m satisfies:
∀x∈M,∀t >0 : Hm(t, x, x)6 cn
(µt)n2. (2) The associated positive minimal Green kernel satisfies:
(a) ∀x, y∈M: Gm(x, y)6 µcnn2 dn−21(x,y).
(b) ∀x∈M,∀t >0 : m ({y∈M;Gm(x, y)> t})6(µt)−n−2n . (c) Ifα∈(0, n/(n−2))and ifΩ⊂M has finitem-measure, then Z
Ω
Gαm(x, y) dm(y) 1α
6
n (n−2)α−n
α1 1
µ(m(Ω))α1−1+2n. (3) IfB(x, r)⊂M is a relatively compact geodesic ball in M, and if
v∈Wloc1,2(B(x, r))satisfies
∆mv60
then, forp>2, there is a positive constantcn,p such that:
|v(x)|p6 cn,p
õ rn Z
B(x,r)
|v|p(y) dm(y).
(4) IfB(x, r)⊂M is a relatively compact geodesic ball inM, then 1
c(n)µn2rn6m (B(x, r)). Remarks 2.9.
• The upper bound on the heat kernel comes essentially from an adap- tation in this setting of old ideas of J. Nash ([45]). In fact both properties are equivalent ([55]).
• The estimate on the heat kernel implies a Gaussian upper bound for the heat kernel ([21]):
∀x, y∈M, ∀t >0 : Hm(t, x, y)6 cn
(µt)n2e−d2 (x,y)5t , and the formula
Gm(x, y) = Z +∞
0
Hm(t, x, y)dt yields the estimate (2a) on the Green kernel.
• The property (2b) is equivalent to the Sobolev inequality ([7]).
• The elliptic estimate (3) is proved by a classical De Giorgi–Nash–
Moser iteration method. The lower bound (4) on the volume is a consequence of this elliptic estimate applied to the constant function 1 (see [2, 7]).
2.3. Schrödinger operators and the Doob transform
2.3.1. Schrödinger operators
Assume thatV ∈L∞loc is a nonnegative function such that the quadratic form
ψ∈ C0∞(M)7−→qV(ψ) :=
Z
M
|dψ|2g−V ψ2 dm, is bounded from below; i.e., there is a constant Λ such that
∀ψ∈ C0∞(M) :qV(ψ)>−Λ Z
M
ψ2dm.
With the Friedrichs extension procedure, we get a self-adjoint operator which will be denoted:
L:= ∆m−V.
An easy consequence of the maximum principle, or of its weak formulation, is that
∀x, y∈M,∀t >0 : Hm(t, x, y)6HL(t, x, y), whereHL denotes the heat kernel of the operatorL.
Definition 2.10. — The operator Lis said to be subcritical, if Lhas a positive minimal Green kernelGL.
Remark 2.11 ([31]). — The weighted Laplacian is subcritical if and only if the measure dm is not parabolic in the sense of Definition 2.4. In that case, we say that the weighted Riemannian manifold (M, g,m) is nonparabolic.
WhenLis subcritical, we have
∀x, y∈M: Gm(x, y)6GL(x, y).
2.3.2. The Doob Transform
Assume that (M, g) is complete noncompact, and that the operatorLis nonnegative,
∀ψ∈ C0∞(M) : Z
M
|dψ|2g−V ψ2
dm>0.
Then, the Agmon–Allegretto–Piepenbrink theorem ([1, 26, 44]) implies that there is a positive functionh∈Wloc1,2such that
Lh= 0.
Remark that becauseV is assumed to be locally bounded, we also have h∈Wloc2,pfor anyp <∞.
Using the formula (2.1) and integrating by parts, we get that for any ψ∈ C0∞(M) :
(2.2)
Z
M
|d(hψ)|2g−V h2ψ2 dm =
Z
M
|dψ|2gh2dm.
Hence the Schrödinger operatorLand the Laplacian ∆h2mare conjugate : L(hψ) =h∆h2mψ
and we have:
HL(t, x, y) =h(x)h(y)Hh2m(t, x, y).
This conjugacy is called the Doob transform (or Doobh-transform) and it is the key point in order to get estimates on the Green and heat kernels of the Schrödinger operatorL.
2.3.3. The Kato condition and uniform boundedness inL∞ The Laplacian ∆mis sub-Markovian, that is to say,
(2.3) ∀t >0,∀x∈M: Z
M
Hm(t, x, y) dm(y)61.
An equivalent formulation is that
ke−t∆mkL∞→L∞ 61.
We are interested in similar properties for Schrödinger operators. The nonnegativity of L implies that the semigroup
e−tL t is uniformly bounded onL2; but it is not necessarily uniformly bounded onL∞. How- ever the above Doob transform guarantees that if the Schrödinger operator Lhas a zero eigenfunctionhsatisfying
16h6γ, then the semigroup
e−tL tis uniformly bounded onL∞.
Definition 2.12. — We say that the Schrödinger operatorL= ∆m−V, with nonnegative potentialV, is uniformly stable if any of the following equivalent conditions is satisfied.
(1) supt>0 e−tL
L∞→L∞<∞.
(2) supt>0 e−tL
L1
→L1<∞.
(3) There is a constantγ such that, for allt >0 and allx∈M, e−tL1
(x) = Z
M
HL(t, x, y) dm(y)6γ.
The equivalences follow from the study of Qi S. Zhang and Z. Zhao ([57], see also [60]).
Definition 2.13. — Assume thatV >0is not identically zero. We say that the Schrödinger operatorL= ∆m−V is gaugeable with gaugeability constantγ>1 if any of the following equivalent conditions is satisfied.
(a) There is anh∈Wloc1,2 such that
Lh= 0 and 16h6γ.
(b) Lis subcritical, i.e., it has a positive minimal Green kernelGL, and there is a constantγsuch that
∀x∈M: Z
M
GL(x, y)V(y) dm(y)6γ−1.
Proof of the equivalences in Definition 2.13. — If we assume that prop- erty (b) holds, then
h(x) = 1 + Z
M
GL(x, y)V(y) dm(y)
is a bounded solution of the equationLh= 0 such that h>1, hence the property (a) holds.
If we assume that property (a) holds, the Doob transform implies thatL is nonnegative. We have assumed thatV is not identically zero, hence the nonnegativity ofLimplies that ∆mis subcritical, now the Doob transform and the fact thath is bounded insure that the operator L is subcritical.
For a relatively compact domain Ω⊂M, we consider the solution of the Dirichlet boundary problem:
(∆mhΩ=V hΩ on Ω, hΩ= 1 on∂Ω.
LetGL(·,·; Ω) denote the Green function of the operatorLon Ω, with the Dirichlet boundary condition. Then we havehΩ= 1 +vΩwhere
vΩ= Z
M
GL(x, y; Ω)V(y) dm(y).
The maximum principle implies that h
γ 6hΩ6h,
and that Ω7→hΩis increasing, hence we can define eh(x) = lim
Ω→MhΩ(x)
and we have
eh(x)6γ, and
eh(x) = 1 + Z
M
GL(x, y)V(y) dm(y).
Hence the property (b) holds.
Theorem 2.14([57], see also [60]). — LetLbe a Schrödinger operator with nonnegative potentialV. The following relations hold.
(a) The gaugeability condition implies the uniform stability condition.
(b) If(M, g,m)is stochastically complete, i.e.,∀t >0 : e−t∆m1
= 1, then the gaugeability condition is equivalent to the uniform stability condition.
(c) If the operator∆m is subcritical and if the Kato constant of V is smaller than1,
K(V) := sup
x∈M
Z
M
Gm(x, y)V(y) dm(y)<1, thenL= ∆m−V is gaugeable
Remark 2.15. — According to [31], (M, g,m) is stochastically complete provided that there is someo∈M and some positive constantc such that for anyR >0:
m (B(o, R))6cecR2.
Proof of Theorem 2.14. — Let’s explain why under the stochastically completeness assumption, the uniform stability implies the gaugeability.
The stochastic completeness condition implies that for all x∈ M, the functiont7→R
MHL(t, x, y) dm(y) is nondecreasing. Indeed, the semigroup property implies that ift, τ >0 then
Z
M
HL(t+τ, x, y) dm(y) = Z
M×M
HL(t, x, z)HL(τ, z, y) dm(z) dm(y).
UsingHL(τ, z, y)>Hm(τ, z, y) andR
MHm(τ, z, y) dm(y) = 1, one gets:
Z
M
HL(t+τ, x, y) dm(y)>
Z
M
HL(t, x, y) dm(y).
Hence if the condition (c) is satisfied then we can define h(x) = sup
t>0
Z
M
HL(t, x, y) dm(y) = lim
t→+∞
Z
M
HL(t, x, y) dm(y).
We have 16h6γand for allτ >0:
Z
M
HL(τ, x, y)h(y) dm(y) =h(x).
HenceLh= 0.
Remark 2.16. — The set
{λ>0,∆m−λV is gaugeable (resp. uniformly stable)}
is an interval of the type [0, ω) or [0, ω].
2.3.4. Subcriticality, Green kernel and parabolicity
The subcriticality of a Schrödinger operatorL is a strengthening of the nonnegativity property.
Proposition 2.17 ([60]). — For a Schrödinger operator L with non- negative potentialV, we have the following equivalent properties:
(1) Lis subcritical.
(2) There is a non-empty open set Ω⊂M and a positive constant κ such that
∀ψ∈ C∞0 (M) :κ Z
Ω
ψ2dm6 Z
M
|dψ|2g−V ψ2 dm.
(3) For all relatively compact open subsetΩ⊂M, there is a positive constantκsuch that
∀ψ∈ C∞0 (M) :κ Z
Ω
ψ2dm6 Z
M
|dψ|2g−V ψ2 dm. (4) There is a positive Green kernel forL.
(5) If h ∈ Wloc1,2 is a positive solution of the equation Lh = 0, then (M, g, h2m)is nonparabolic (see Remark 2.11).
(6) Ifh∈Wloc1,2 is a positive solution of the equationLh= 0, then the operator∆h2mhas a positive Green kernel.
2.3.5. Elliptic estimates for Schrödinger operators
The Euclidean Sobolev inequality and the gaugeability property imply good estimates on the Green kernel of the operatorL.
Theorem 2.18. — Let(M, g,m) be a weighted Riemannian manifold and assume that it satisfies the Euclidean Sobolev inequality(Sobm)with constantµ. LetL= ∆m−V be a Schrödinger operator with nonnegative potentialV and assume thatLis gaugeable with gaugeability constant γ.
Then there is a positive constantcnsuch that the following properties hold:
(1) The heat kernel associated with the Laplacian∆L satisfies
∀x∈M,∀t >0 : HL(t, x, x)6 cnγn (µt)n2. (2) The associated positive minimal Green kernel satisfies:
∀x, y∈M: GL(x, y)6 cn µn2
γn dn−2(x, y).
(3) LetB(x, r)⊂M be a relatively compact geodesic balls inM, and Assume thatv∈Wloc1,2(B(x, r))satisfies
Lv60.
Then, forp>2there is a positive constant C(n, p)such that
|v(x)|p6 C(n, p)
√µ rnγn−2+p Z
B(x,r)
|v|p(y) dm(y).
All these results follow from the Doob transform and the fact that the new measuredm =e h2dm satisfies the Sobolev inequality:
∀ψ∈ C0∞(M) : µγ−n2(n−2) Z
M
|ψ|n−22n dme 1−2n
6 Z
M
|dψ|2dm.e
2.3.6. Estimate on the gaugeability constant
In [25], B. Devyver has studied conditions under which a nonnegative Schrödinger operator is gaugeable. One of his results is the following.
Theorem 2.19. — Let(Mn, g,m)be a complete weighted Riemannian manifold andV∈L∞loca nonnegative function. Assume that the Schrödinger operatorL = ∆m−V is strongly positive: there is some δ >0 such that the operator∆m−(1 +δ)V is nonnegative:
∀ψ∈ C0∞(M) : (1 +δ) Z
M
V ψ2dm6 Z
M
|dψ|2gdm.
Assume moreover that the Kato constant ofV is small at infinity: there is a compact subsetK⊂M and someε∈(0,1)such that
∀x6∈K: Z
M\K
Gm(x, y)V(y) dm(y)61−ε, thenLis gaugeable.
Moreover [24], B. Devyver has shown:
Theorem 2.20. — Let(M, g,m) be a weighted Riemannian manifold and assume that it satisfies the Euclidean Sobolev inequality (Sobm). Let V ∈L∞loc be a nonnegative function such that
• for someε∈(0,1),V ∈L(1±ε)n2,
• ker
Ln−22n L=
v∈Ln−22n (M,dm) :Lv= 0 ={0}. Then,Lis gaugeable.
For geometric applications, it is sometimes useful to obtain explicit bounds on the function h used in the Doob transform. The second hy- pothesis in Theorem 2.20 is satisfied whenever
Z
M
Vn2 dm6(1−ε)µ,
and in this case we can follow the argument given in [24] in order to get an estimate of khk∞ which only depends on n, µ, ε, R
MV(1−)n2 dm and R
MV(1+)n2 dm. The next proposition gives such a local estimate.
Proposition 2.21. — Let(Mn, g,m)be a weighted Riemannian mani- fold andB(x,2R)⊂M be a relatively compact geodesic ball. LetV ∈L∞loc be a nonnegative function. Assume that for some constantµ, δ >0,p >1 andΛ>0, the following conditions hold.
• The ballB(x,2R)satisfies the Euclidean Sobolev inequality(Sobm) with Sobolev constantµ.
• The Schrödinger operatorLis strongly positive:
∀ψ∈ C0∞(B(x,2R)) : (1 +δ) Z
B(x,2R)
V ψ2dm6 Z
B(x,2R)
|dψ|2gdm.
• If Gm(z, y) is the Dirichlet Green kernel of the Laplacien ∆m on B(x, R), then :
R2(p−1) sup
z∈B(x,R)
Z
B(x,R)
Gm(z, y)Vp(y) dm(y)6Λp. Then there is a constantγdepending only onn, p,Λ, δ,m(B(x,2R))
µn2Rn such that the solution of the Dirichlet boundary problem:
(∆mh−V h= 0 onB(x, R)
h= 1 on∂B(x, R)
satisfies
16h6γ.
Proof. — By scaling, we can suppose thatR = 1 and let B :=B(x,1) and 2B:=B(x,2).
We first get an integral estimate on v:=h−1. IfW01,2(B) is the closure ofC0∞(B) for the normψ7→ kdψk2+kψk2, we havev∈W01,2(B) and
∆mv−V v=V, hence
Z
B
|dv|2g−V v2 dm =
Z
B
V vdm6kvk∞ Z
B
Vdm. We let
L:=kvk∞. Using the strong positivity and the function
ξ(y) = min{2−d(x, y),1}, we get
Z
B
V dm6 Z
2B
V ξ2dm6 1 1 +δ
Z
2B
|dξ|2gdm6m(2B).
Using again the strong positivity and the Sobolev inequality we get:
µδ 1 +δ
Z
B
vn−22n dm 1−n2
6 δ 1 +δ
Z
B
|dv|2gdm6 Z
B
|dv|2g−V v2 dm. So that we get
(2.4)
Z
B
vn−22n dm 1−2n
6Lm(2B) µδ . The functionv is a solution of the integral equation:
(2.5) v(z) = Z
B
Gm(z, y)V(y) dm(y) + Z
B
Gm(z, y)V(y)v(y) dm(y).
Letq =p/(p−1), using Hölder inequality and the integral estimate (2c) in Theorem 2.8, we estimate the first term in the RHS of (2.5)
Z
B
Gm(z, y)V(y) dm(y)6Λ Z
B
G(z, y) dm(y) 1q
6Λ
m(B) µn2
nq2 .
Introducing
I= Λ
m(2B) µn2
nq2 , we get
(2.6)
Z
B
Gm(z, y)V(y) dm(y)6I.
For the second term in the RHS of (2.5), we introduce:
ψ(z) :=
Z
B
Gm(z, y)v(y) dm(y).
Then using again the Hölder inequality, we get:
(2.7) Z
B
Gm(z, y)V(y)v(y) dm(y) 6
Z
B
Gm(z, y)Vp(y)v(y) dm(y) 1p
ψ1q(z) 6ΛL1pψ1q(z).
Ifβ is such that
β > n
2 andβ> 2n n−2
then with α = β/(β−1) and the integral estimate (2c) in Theorem 2.8 we get:
ψ(z)6
n (n−2)α−n
1α 1
µ(m(B))α1−1+2nkvkβ 6
n (n−2)α−n
1α 1
µ(m(2B))α1−1+n2 kvkβ. The estimate (2.4) implies that:
kvkβ6L1−n−2n β1
m(2B) δµ
n−2n β1 .
After a bit of arithmetic, we get that:
(2.8) Z
B
Gm(z, y)V(y)v(y) dm(y) 6IL1−n−2n qβ1
m(2B) (δµ)n2
n−22 1
qβ n
(n−2)α−n qα1
.
With (2.4) and (2.8), we get
L6I+CκL1−κ whereκ=n−2n qβ1 and
Cκ=I
m(2B) (δµ)n2
n−22 1
qβ n
(n−2)α−n qα1
.
