A QUANTITATIVE ANALYSIS OF METRICS WITH ALMOST CONSTANT
SCALAR CURVATURE ON THE SPHERE
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
Abstract. We prove a quantitative version of Struwe’s theorem on the structure of metrics
on the sphere which are conformal to the standard metric and have almost constant scalar
curvature. As an application we prove the convergence of the Yamabe flow on the sphere by
means of a entropy-entropy production inequality and of the improved Sobolev inequality by
Bianchi-Egnell [BE91].
1. Introduction
On a n-dimensional Riemannian manifold (M, g0 ), n ≥ 3, the problem of finding a conformal
metric g to g0 whose scalar curvature Rg is equal to a prescribed function R boils down to showing
the existence of a positive solution u to the nonlinear PDE
n−2
n−2
∆g0 u +
Rg u =
R up
(1.1)
4(n − 1) 0
4(n − 1)
where ∆g0 = −div (∇g0 u) denotes the Laplace-Beltrami operator on M and
n+2
2n
= 2⋆ − 1 ,
2⋆ =
.
n−2
n−2
Indeed, if u solves (1.1), then the metric g = up−1 g0 satisfies Rg = R. By means of a stereographic projection onto Rn , (1.1) on the sphere S n takes the form
p=
−∆u = K up
on Rn ,
(1.2)
where K is equal to [(n − 2)/4(n − 1)] R composed with the inverse stereographic projection.
We look for solutions of (1.2) satisfying
∫
n
u > 0 on R and
|∇u|2 < ∞ .
(1.3)
Rn
In general, for a given function K (and even when K is just a small perturbation of a constant)
there may exist no solution to (1.2)–(1.3) (see [KW74]), and indeed there is a vast literature
dedicated to finding necessary and sufficient conditions on K in order to guarantee the solvability
of (1.2)–(1.3), see for example [Mos73, Ni82, DN85, ES86, Che89, CY91, BC91, BE93, Li95, Li96,
Bah96, CL97, AGAP99, AM01, LL09, LZ15]. At the same time, when K is constantly equal to
some κ > 0, the problem is completely rigid. Indeed, by [Oba72, GNN79], if u solves (1.2)–(1.3)
with K ≡ κ > 0, then there exist λ > 0 and z ∈ Rn such that
u(x) = λ(n−2)/2 vκ (λ(x − z))
where
( n(n − 2) )(n−2)/4
∀x ∈ Rn ,
1
∀x ∈ Rn .
(1.4)
κ
(1 + |x|2 )(n−2)/2
In particular, up−1 dx (dx being the Euclidean metric) is isometric to the standard metric on Sn
after stereographic projection on Rn .
The goal of this paper is obtaining a quantitative description of solutions u to the prescribed
scalar curvature equation (1.2)–(1.3) in the regime when K is close (in a suitable sense) to a
vκ (x) =
1
2
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
constant. A qualitative description of such solutions is provided by a celebrated result of Struwe
[Str84], stating that solutions u corresponding to almost-constant K must be close to sums of
(translated and rescaled) “bubbles” of the form (1.4). More precisely, for a given constant κ > 0
let us consider the functional
∫
⋆
|u|2
|∇u|2
Iκ (u) =
−κ ⋆
(1.5)
2
2
Rn
defined on the homogeneous Sobolev space
∫
}
{
1
n
1
n
H̊ (R ) = u ∈ Lloc (R ) :
|∇u|2 < ∞ .
Rn
The Euler-Lagrange equation of Iκ at a positive function u is −∆u = κ up on Rn , as the first
variation of Iκ at u is given by
∫
(δIκ )u (φ) =
∇u · ∇φ − κ up φ .
Rn
By Struwe’s theorem (see, e.g. [Heb14, Theorem 3.3]) if {uj }j∈N is a sequence of positive
functions such that
sup Iκ (u(j) ) < ∞
lim
(j)
sup
(δIκ )u(j) (φ) = 0 ,
j→∞ ∥∇φ∥
(1.6)
L2 (Rn ) ≤1
j∈N
(j)
then there exist m ∈ N, zi ∈ Rn and λi > 0 (where j ∈ N and 1 ≤ i ≤ m) such that
m
∑
(j) (j) lim ∇u(j) −
∇vκ [zi , λi ] 2 n = 0 ,
j→∞
(j)
lim
j→∞
λi
(j)
λk
+
i=1
(j)
λk
+
(j)
λi
L (R )
(1.7)
(j) (j) (j)
λi λk |zi
−
(j)
zk |2
= +∞ ,
∀i ̸= k ,
where vκ is as in (1.4) and where for any v : Rn → R, λ > 0 and z ∈ Rn we have set
v[z, λ](x) = λ(n−2)/2 v(λ(x − z)) ,
x ∈ Rn .
The problem addressed in this paper is quantifying the rate of convergence in (1.7) in terms
of that in (1.6). In order to quantify the latter, given u ∈ C ∞ (Rn ) ∩ H̊ 1 (Rn ) with u > 0, we
define the constant scalar curvature deficit δ(u) of u by setting
∫
2n/(n+2) ⋆ )(n+2)/2n
( ∫ ∆u
|∇u|2
2
∫n
δ(u) =
u
K0 (u) = R
,
(1.8)
p + K0 (u)
2⋆
Rn u
Rn u
where
2n
2⋆
= (2⋆ )′ = ⋆
.
n+2
2 −1
The definition of K0 (u) is motivated by the fact that, as an integration by parts shows, if
−∆u/up is constant on Rn then, necessarily, −∆u/up = K0 (u). Notice that
K0 (u[z, λ]) = K0 (u)
δ(u[z, λ]) = δ(u)
for every z ∈ Rn and λ > 0 ,
and that δ(u) = 0 if and only if u = vK0 (u) [z, λ] for some z ∈ Rn and λ > 0. Moreover, by
Hölder inequality, and for κ = K0 (u), we find
sup
(δIκ )u (φ) ≤
∥∇φ∥L2 (Rn ) ≤1
δ(u)
S
where S denotes the Sobolev constant on Rn ,
{ ∥∇v∥ 2 n
}
L (R )
S = inf
: v ̸= 0 , |{|v| > t}| < ∞ ∀t > 0 .
∥v∥L2⋆ (Rn )
(1.9)
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
3
We recall that by [Aub76, Tal76, CENV04] {vκ [z, λ]}κ ,λ ,z is the family of minimizers in (1.9),
a fact that can be used together with −∆vκ = κ vκp to check that
∫
∫
(∫
)2/2⋆
Sn
Sn
⋆
2
2
2⋆
|∇vκ | = S
vκ
= (n−2)/2
vκ2 = n/2 .
(1.10)
κ
κ
Rn
Rn
Rn
With these remarks at hand, we are ready to state our main result, where we set

if n ̸= 6 or m = 1
s
(1)
(1.11)
Φn (m, s) =
 s log2/3
if n = 6 and m > 1 .
s
Theorem 1.1. Given L ∈ N and n ≥ 3, there exists C0 = C0 (L, n) > 0 with the following
property. If u > 0, u ∈ C ∞ (Rn ) ∩ H̊ 1 (Rn ), is such that
∫
(
1) n
K0 (u) = 1
and
|∇u|2 ≤ L +
S ,
2
Rn
n
m
then there exist m ∈ {1, ..., L}, {zi }m
i=1 ⊂ R , and {λi }i=1 ⊂ (0, ∞) such that
u=
m
∑
v1 [zi , λi ] + ρ ,
i=1
where
∥∇ρ∥L2 (Rn ) ≤ C0 Φn (m, δ(u))
and
(1.12)
)min{2,(n−2)/2}
λi /λj
≤ C0 δ(u)
if λi ≤ λj .
1 + λ2i |zi − zj |2
Moreover, one has the energy estimate
∫
(∫
)2/2⋆
2
2/n 2
2⋆
−C0 δ(u) ≤
|∇u| − m S
u
≤ C0 Φn (m, δ(u))2 .
(
Rn
(1.13)
(1.14)
Rn
Geometrically speaking, the conformal factor up−1 , and thus the metric properties of
(M, g) = (Rn , up−1 dx), are quantitatively described in terms of the integral oscillation of Rg
represented by δ(u). Inequality (1.12) says that, from the metric point of view, (Rn , up−1 dx) is
obtained by gluing together (through patches of small metric size) the (finitely many) standard
spheres (Rn , vK0 (u) [zi , λi ]p−1 dx). The interaction between these metrics becomes weaker as δ(u)
becomes smaller as either the scaling parameters λi and λj or the centers zi and zj become
increasingly far from each other, see (1.13). From the qualitative point of view, this description is already contained in Struwe’s theorem. The new aspect here is that everything can be
quantified in terms of δ(u). We now make some remarks.
Remark 1.2. Theorem 1.1 is optimal, at least when n ̸= 6 or m = 1. Indeed, let U1 = v1 [0, 1]
and U2 = v1 [z, 1] with |z| ≫ 1, set σ = U1 + U2 , and consider φ solving
∆φ = σ p−1 ψ
on Rn
for some fixed ψ ∈ Cc∞ (Rn ) with ψ ̸≡ 0. If we set
u = (1 + ε φ) σ
u = U1 + U2 + ρ
then
−∆u = K up ,
Notice that, if ε ≪ 1,
where
K=−
ρ = εφσ ,
1
U1p + U2p
εψ
−
.
(1 + εφ)p−1
σp
(1 + εφ)p
(
Up + Up )
|K − 1| ≤ C ε + C 1 − 1 p 2 .
σ
4
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
Noticing that u ≤ 2 σ for ε ≪ 1, we see that
∫ ∫ U p + U p 2n/(n+2) 2∗
U p + U p 2n/(n+2) 2∗
∗
u
≤ 22
σ
1 − 1 p 2 1 − 1 p 2 σ
σ
Rn
Rn
∫ 2n/(n+2)
p
p
p
2∗
= 2
→0
σ − U1 + U2 Rn
as |z| → ∞ .
Hence, choosing first ε > 0 small enough, and then |z| extremely large with respect to 1/ε so
that the last term is negligible with respect to ε, it follows that
(∫ )(n+2)/2n
K − 12n/(n+2) u2∗
≤ C ε.
(1.15)
Rn
Since K =
−∆u/up
we see that
∫
∫
∗
2
K u2
n |∇u|
R
∫
=
= K0 (u) .
2∗
2⋆
Rn u
Rn u
R
∫n
Therefore, by (1.15) and Hölder inequality,
∫
(∫
)
2n/(n+2) u2∗ (n+2)/2n
n (K − 1) u2∗ |K
−
1|
n
∫
≤ R
≤ C ε,
|K0 (u) − 1| = R ∫
2∗
2∗
Rn u
Rn u
from which we deduce that δ(u) ≤ C ε. Since ∥∇ρ∥ = ε∥∇(φσ)∥L2 (Rn ) ≃ ε, this shows that apart
from the logarithmic factor appearing in (1.11) when m > 1 and n = 6, our result is optimal in
terms of the dependence in δ(u).
We note that the logarithmic factor in (1.11) naturally arise from the interaction of two
bubbles in dimension 6 (see Proposition A.1). Therefore, it seems very likely to us that Theorem
1.1 is sharp also in that case. However, the construction of an example showing this optimality
would require a more delicate argument than the one presented above, and this goes beyond the
scope of this paper. In addition, for most applications, the presence of a logarithmic correction
is usually irrelevant.
Note that (1.14) allows one∫to deduce∫the exact number of bubbles when δ(u) ≪ 1. More
⋆
precisely, using (1.8) to rewrite Rn u2 as Rn |∇u|2 , the following is an immediate consequence
of (1.14):
Remark 1.3. Let {uj }j∈N be a sequence of smooth positive functions satisfying
∫
(
1) n
K0 (uj ) = 1
and
|∇uj |2 ≤ L +
S
∀j ∈ N.
2
Rn
Also, assume that δ(uj ) → 0 as j → ∞. Then there exists an integer number m ∈ {1, . . . , L}
such that
∫
1
|∇uj |2 → m
as j → ∞.
S n Rn
To show an application of our result, in section 3 we exploit Theorem 1.1 to give a new
short and self-contained proof of the exponential convergence of the Yamabe flow on the sphere.
We point out that the convergence of the Yamabe flow to a conformal metric of constant scalar
curvature is of course a very well-known problem, which has already been addressed in full
generality in the series of papers [Cho92, Ye94, SS03, Bre05, Bre07a]. In particular, for the case
of interest to us, this was shown first by Ye [Ye94] in the more general case of locally conformally
flat initial metrics, and then obtained again by Brendle [Bre07b] on the sphere with a simpler
proof.
Our goal here is to show how our quantitative stability estimates can be applied to give
a very elementary and short proof of the same result, using only entropy-entropy production
inequalities. Let us recall the setting of the problem. We are given a metric g0 on Sn , which by
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
5
stereographic projection can be written as a conformal metric u0p−1 dx to the Euclidean metric
dx of Rn , where u0 is smooth and positive on Rn . As shown by Hamilton [Ham89] there exists
a curve of metrics g(t) = u(t)p−1 dx, t ≥ 0, evolving accordingly to the Yamabe flow
d
g = −(Rg − rg ) g
t ≥ 0,
(1.16)
dt
and such that g(0) = g0 . Here rg is the average of Rg on (Rn , g). In terms of the conformal
factor u, Rg and rg take the form
4(n − 1) ∆u
Rg = −
p
∫ n−2 u
∫
4(n − 1)
4(n − 1) Rn |∇u|2
Rn Rg dvol g
∫
rg =
=
=
K0 (u) .
2⋆
vol g (Rn )
n−2
n−2
u
Rn
A simple, direct computation using (1.16) shows that
d
vol g (Rn ) = 0
dt
and
d
n−2
rg = −
dt
2
∫
Rn
(Rg − rg )2 dvol g ,
(1.17)
⋆
where vol g = u2 dx. In particular, by the first identity in (1.17), we consider without loss of
generality the case that vol g0 (Rn ) = 1, so that
∫
∫
⋆
2⋆
u(t) =
u20 = 1
∀t ≥ 0 .
(1.18)
Rn
Rn
The relation between (1.16) and Theorem 1.1 is clarified by the second identity in (1.17), which
gives, combined with Hölder inequality and (1.18),
∫
∫ (
)2 ⋆
d
∆u
2
+
K
(u)
u2 ≤ −2(n − 1) δ(u)2 .
|∇u| = −2(n − 1)
(1.19)
0
dt Rn
up
Rn
In other words, the Dirichlet energy of the conformal factor u(t) is dissipated along the Yamabe
flow at a rate proportional to the square of the constant scalar curvature deficit δ(u). By
applying Theorem 1.1 to a suitable normalization of u(t) and exploiting (1.19), we see that the
number of “bubbles” m(t) needed to decompose u(t) is decreasing in time. At the same time,
combining (1.19) with some elementary differential inequalities, we find that
∫ ∞
δ(u(t)) dt < ∞ ,
0
which, by the same simple remark used in [SS03], immediately rules out concentration of volume
and shows that u(t) is close to a single bubble, that is m(t) ≡ 1, for every t ≥ t0 . In light of
this last fact, (1.19) and Theorem 1.1 imply that starting from the critical time t0 , the Sobolev
deficit of u(t) is exponentially decaying in time. Here, given a function U ∈ H̊ 1 (Rn ) we denote
by
∫
(∫
)
⋆ (n−2)/n
δSobolev (U ) =
|∇U |2 − S 2
U2
,
(1.20)
Rn
Rn
(so that δSobolev (U ) = 0 if and only if U = vκ [z, λ] for some z ∈ Rn and κ , λ > 0 thanks to
[Aub76, Tal76].) By the quantitative Sobolev inequality of Bianchi and Egnell [BE91], for every
U ∈ H̊ 1 (Rn ) one has
∫
CBE (n) δSobolev (U ) ≥
inf
|∇U − ∇vκ [z, λ]|2 .
(1.21)
n
z∈R ,κ λ>0 Rn
Combining this last inequality with (1.19), the fact that m(t) ≡ 1, and (1.14), we come to obtain
the following theorem.
6
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
Theorem 1.4. Let g0 be a smooth metric on Sn which is conformally equivalent to the standard
metric on Sn , and let u(0)p−1 dx be the representation of g0 under stereographic projection.
Assume without loss of generality that
∫
⋆
u20 = 1 .
Rn
Then there exist positive constants λ0 and C∗ , and a vector z0 ∈ Rn , all depending on u0 , and
a dimensional constant c(n) > 0, such that
∫ 2
∀t ≥ 0 .
(1.22)
∇u(t) − ∇vS 2 [z0 , λ0 ] ≤ C∗ e−c(n) t
Rm
Moreover, if we assume in addition that
∫
|∇u0 |2 ≤ (2 − τ )2/n S 2
Rn
for some τ ∈ (0, 1), then
∫ 2
(∫
∇u(t) − ∇vS 2 [z0 , λ0 ] ≤ C(n)
Rn
Rn
)
|∇u0 |2 − S 2 e−c(n) t
∀t ≥ 0 .
(1.23)
Acknowledgment: This work has been done while GC was visiting the University of Texas at
Austin under the support of NSF-DMS FRG Grant 1361122, the Oden Fellowship at ICES, the
GNAMPA of the Istituto Nazionale di Alta Matematica (INdAM), and the FIRB project 2013
“Geometrical and Qualitative aspects of PDE”. AF is supported by NSF Grant DMS-1262411
and FM is supported by NSF-DMS Grant 1265910.
2. Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. Recall the notation
2n
n+2
4
n ≥ 3,
2⋆ =
,
p = 2⋆ − 1 =
= 1 + s,
s=p−1=
.
n−2
n−2
n−2
Through this section L ∈ N is fixed.
We begin by observing that it is enough to prove the theorem for u : Rn → R satisfying
u ∈ C ∞ (Rn ) ∩ H̊ 1 (Rn ) ,
u > 0 on Rn ,
∫
and such that
(
1) n
|∇u|2 ≤ L +
S ,
δ(u) ≤ δ0 .
(2.1)
2
Rn
for a suitably small constant δ0 = δ0 (L, n).
Indeed we note that if δ(u) > δ0 , the theorem is trivially true with any number m ∈
{1, . . . , L} of bubbles simply by choosing λi = 1 and zi = 0 for all i = 1, . . . , m, setting
K0 (u) = 1 ,
ρ = u − m v1 ,
and then simply choosing C0 = C0 (L, n) large enough.
Step one: Let us set I = I1 for the functional Iκ with κ = 1 defined in (1.5). We notice that
|(δI)u (φ)| ≤ ∥φ∥L2⋆ (Rn ) δ(u) ≤
while K0 (u) = 1 gives
∫
⋆
I(u) =
Rn
|∇u|2 u2
1
− ⋆ =
2
2
n
∫
δ(u)
,
S
(
1 ) Sn
|∇u|2 ≤ L +
.
2 n
Rn
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
7
If we set for the sake of simplicity U = v1 , so that, see (1.4),
−∆U = U p
on Rn ,
then, thanks to Struwe’s result, for a constant ε0 = ε0 (L, n) > 0 to be fixed later on we can
choose δ0 depending on ε0 in such a way that, for some m ≤ L,
m
∑
∇U [zi , λi ] 2 n ≤ ε0 ,
∇u −
L (R )
i=1
(2.2)
λi
λk
1
+
+ λi λk |zi − zk |2 ≥
,
∀i ̸= k .
λk
λi
ε0
Here zi ∈ Rn and λi ∈ (0, ∞) are chosen in such a way that
m
m
∑
∑
min
∇U [zi , λi ] 2 n =
∇u
−
∇U
[w
,
µ
]
∇u −
i
i
n
L (R )
i=1
wi ∈R , µi >0
L2 (Rn )
i=1
.
(2.3)
In particular, if we set
σ=
m
∑
ρ=u−σ =u−
Ui ,
i=1
∂U [w, µ] Vi =
,
∂µ w=zi ,µ=λi
Ui = U [zi , λi ] ,
then by (2.3) we find that
∫
∫
∫
∇Ui · ∇ρ =
∇Vi · ∇ρ =
Rn
Rn
Rn
∇Wiα · ∇ρ = 0 ,
m
∑
Ui
i=1
Wiα
∂U [w, µ] =
,
∂wα w=zi ,µ=λi
∀1 ≤ i ≤ m , 1 ≤ α ≤ n ,
(2.4)
while the first bound in (2.2) gives
∥∇ρ∥L2 (Rn ) ≤ ε0 .
By a spectral analysis argument (see, e.g., the appendix to [BE91]), (2.4) implies that
∫
∫
2
|∇ρ| ≥ Λ
Uip−1 ρ2
∀1 ≤ i ≤ m ,
Rn
(2.5)
(2.6)
Rn
where Λ = Λ(n) is such that
Λ > p.
(2.7)
Our first goal, achieved in step two and step three of the proof, will be to improve the lower
bound (2.6), see in particular (2.21). Although this last estimate can be found, for example, in
[Bah89, Proposition 3.1], we have thought of including a justification for the sake of clarity.
Step two: In this step we construct a useful localization of the various bubbles Ui in the form of
a family of cut-off Lipschitz functions φi : Rn → [0, 1], i = 1, ..., m, satisfying
∫
|∇φi |n ≤ C(L, n) η ,
Rn
{sptφi }m
i=1 is a disjoint family of sets ,
∫
⋆
Ui2 + |∇Ui |2 + |∇Vi |2 + |∇Wiα |2 ≤ C(L, n) η 2 .
(2.8)
{φi <1}
for a positive parameter η = η(L, n) to be chosen in later steps of the proof. In the construction
of the φi we shall introduce various subsidiary parameters R, r, M and R∗ , all depending on η.
The parameter ε0 will be small enough with respect to η and its subsidiary parameters R, r, M
8
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
and R∗ , and of course δ0 will be picked small with respect to ε0 in order to apply step one. The
choice of R > r is such that
∫
∫
∫
{∫
2⋆
2
2
α 2}
BR (z)c |∇U |
BR (z)c U
BR (z)c |∇V |
BR (z)c |∇W |
∫
, ∫
max max
, ∫
, max ∫
≤ η2 ,
2⋆
2
2
α |2
1≤α≤n
z∈Rn
U
|∇U
|
|∇V
|
|∇W
n
n
n
n
R
R
(2.9)
∫
∫R
∫R
{∫
2⋆
2
2
α 2}
U
|∇U
|
|∇V
|
Br (z)
Br (z)
Br (z)
Br (z) |∇W |
∫
max max
, ∫
, ∫
, max ∫
≤ η2 ,
2⋆
2
2 1≤α≤n
α |2
z∈Rn
U
|∇U
|
|∇V
|
|∇W
n
n
n
n
R
R
R
R
where we have set (recall that U = U [w, λ]|w=0,λ=1 )
∂U [w, µ] ∂U [w, µ] α
V =
, W =
∂µ w=0 ,µ=1
∂wα w=0 ,µ=1
1 ≤ α ≤ n.
If we set
Ri =
R
,
λi
ri =
r
,
λi
then we see that (2.9) holds by scaling with Ui , Vi , Wiα , Ri and ri in place of U , V , W α , R
⋆
and r for every i = 1, ..., m. In other words, the densities Ui2 , |∇Ui |2 , |∇Vi |2 and |∇Wiα |2 are
almost entirely concentrated on BRi (zi ) and they do not significantly charge any ball of radius
less than ri .
Without loss of generality we may assume that
λ1 ≤ λ2 ≤ · · · ≤ λm .
(2.10)
For a constant M ≥ 1 to be fixed later depending on η we partition {1, ..., m} by setting
I0 = {i : λi ≤ M λ1 }
= {1, ..., N (1) − 1} ,
where N (0) = 1 , N (1) ≥ 2 ,
I1 = {i : M λ1 < λi ≤ M λN (1) }
= {N (1), ..., N (2) − 1} ,
where N (2) ≥ N (1) + 1 ,
...
Iℓ
= {i : M λN (ℓ−1) < λi ≤ M λN (ℓ) }
= {N (ℓ), ..., m} ,
where m ≥ N (ℓ) + 1 ,
where 0 ≤ ℓ ≤ m − 1. The idea here is that two scaling parameters λi and λj are comparable
if i, j ∈ Ik for the same k ∈ {0, ..., ℓ} (and in this case the regions of concentration BRi (zi ) and
BRj (zj ) of the bubbles Ui and Uj are well-separated); otherwise, if i < j, then λj is much larger
that λi , and then Uj is concentrated in a region BRj (zj ) ⊂ Bri (zj ) which is negligible for the
bubble Ui .
In order to make this idea precise we now introduce for each bubble Ui two cut-off functions
ψi and ζi . The first one will be used to localize where Ui lives, the second one to carve out a
fixed neighborhood of such region. We start by noticing that for each k = 0, ..., ℓ,
|zi − zj | ≥
M
λi
if i < j, i, j ∈ Ik .
(2.11)
Indeed by choosing ε0 small enough in terms of M (say ε0 ≤ 1/3M 2 ) and by exploiting (2.2),
(2.10), and the definition of Ik , we see that if i < j and i, j ∈ Ik then
λj ≤ M λi ,
λi λj |zi − zj |2 ≥ M 2 ,
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
9
which immediately implies (2.11). For a parameter R∗ to be chosen later on, let us consider a
Lipschitz function ψ : Rn → [0, 1] defined by
|x| < R ,
ψ(x) = 1 ,
log(R∗ + R) − log(|x|)
,
log(R∗ + R) − log(R)
ψ(x) = 0 ,
ψ(x) =
R ≤ |x| ≤ R + R∗ ,
|x| > R + R∗ ,
and then set
ψi (x) = ψ(λi (x − zi )) ,
The cut-off function ψi satisfies
ψi = 1 on BRi (zi ) ,
∀i = 1, ..., m .
ψi = 0 on B(R+R∗ )/λi (zi )c ,
(2.12)
∫
C(n)
n
|∇ψi | ≤
≤ η,
log(1 + (R∗ /R))n−1
Rn
where the last condition is enforced by requiring R∗ to be large enough with respect to R and
η. Moreover, thanks to (2.11), if M > 2(R∗ + R) then we can definitely entail that for each
k = 0, ..., ℓ,
sptψi ∩ sptψj = ∅
∀i, j ∈ Ik .
(2.13)
By using the cut-off functions ψi we can efficiently separate the bubbles with comparable concentration parameters. However, it is still possible that a bubble may sit in the radius of
concentration of a less concentrated bubble, or it may host inside its radius of concentration
some non-comparably concentrated bubbles. In order to wipe out these effects we notice that
R + R∗ √
ri
≤ M rj ≤ √ < r i ,
if i < j do not belong to a same Ik .
(2.14)
λj
M
Indeed, if i and j are as in (2.14), then M λi < λj , thus rj ≤ ri /M and
√
R + R∗
R + R∗
ri
=
rj ≤ M rj ≤ √ ,
λj
r
M
provided M ≥ ((R + R∗ )/r)2 . Having in mind (2.14), we define a Lipschitz function ζ : Rn →
[0, 1] by taking
r
ζ(x) = 0 ,
|x| < √ ,
M
√
log(|x|) − log(r/ M )
r
√ ≤ |x| ≤ r ,
√
ζ(x) =
,
log(r) − log(r/ M )
M
ζ(x) = 1 ,
|x| > r .
and then set
ζij (x) = ζ(λi (x − zj )) ,
if i < j do not belong to a same Ik .
Notice that,
ζij = 1 on Bri (zj )c ,
ζij = 0 on Bri /√M (zj ) ,
∫
C(n)
√
|∇ζij |n ≤
≤ η,
log( M )n−1
Rn
(2.15)
with (by (2.14))
sptψj ⊂ {ζij = 0} ,
if i < j do not belong to a same Ik ,
(2.16)
10
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
so that ζij can be used to “carve Uj out of Ui ” for such values of i and j. We finally introduce
the family {φi }m
i=1 of Lipschitz cut-off functions with values in [0, 1] needed in our argument.
We start by defining φi for the most concentrated bubbles, the ones indexed in Iℓ . In this case
we simply set
φi = ψi ,
i ∈ Iℓ .
Now pick i ∈ Iℓ−1 : we cannot just set φi = ψi , because the support of ψi may intersect the
support of ψj for some j ∈ Iℓ . To solve this problem we exploit the cut-off functions ζij with
j ∈ Iℓ , and set
∏
φi = ψi
ζij ,
i ∈ Iℓ−1 .
j∈Iℓ
More generally, for every k = 0, ..., ℓ − 1 we set
∏
φi = ψi
ζij ,
∀i ∈ Ik .
(2.17)
j∈Iℓ ∪Iℓ−1 ∪···∪Ik+1
We now prove the key properties (2.8) of the functions φi . First of all, these are clearly Lipschitz
functions taking values in [0, 1] with
∫
|∇φi |n ≤ C(L, n) η ,
(2.18)
Rn
where (2.18) is a trivial consequence of the inequalities in (2.12) and (2.15) (recall that m ≤ L).
Next we notice that
{sptφi }m
(2.19)
i=1 is a disjoint family of sets .
Indeed, let us fix i < j: if i, j belongs to a same Ik , then sptφi ∩ sptφj = ∅ thanks to (2.13);
otherwise, i ∈ Ik with k ∈ {0, ..., ℓ − 1} and j ∈ Ih with h ∈ {k + 1, ..., ℓ}, so that ζij is a factor
in the definition (2.17) of φi , and thus
{ζij = 0} ⊂ {φi = 0}
where sptψj ⊂ {ζij = 0} by (2.16) and where, trivially, sptφj ⊂ sptψj . This proves (2.19).
Finally,
∫
⋆
Ui2 + |∇Ui |2 + |∇Vi |2 + |∇Wiα |2 ≤ C(L, n) η 2 .
(2.20)
{φi <1}
Indeed, for example in the case when i ∈ Ik for some k < ℓ, since φi (x) = 1 if and only if
ψi (x) = 1 and ζij (x) = 1 for each j ∈ Ik+1 ∪ · · · ∪ Iℓ , then by (2.12) and (2.15) we find
∩
{φi = 1} = BRi (zi ) ∩
Bri (zj )c ,
j∈Ik+1 ∪···∪Iℓ
so that
∪
{φi < 1} = BRi (zi )c ∪
Bri (zj ) ,
j∈Ik+1 ∪···∪Iℓ
and thus (2.20) follows by exploiting (2.9) applied with Ui , Vi , Wiα , Ri and ri in place of U , V ,
W α , R and r. We have thus completed the proof of step two.
Step three: We introduce a parameter τ = τ (L, n) > 0 (so that η will need to be small with
respect to τ ) and prove that
∫
m ∫
∑
1 + C(L, n) τ
p−1 2
|∇ρ|2 .
(2.21)
Ui ρ ≤
Λ
n
n
R
R
i=1
(This lower bound largely improves (2.6) of course.) To this end, we set
ρi = φ i ρ ,
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
and first show that
m ∫
∑
Rn
i=1
11
∫
|∇ρi |2 ≤ (1 + τ )
Indeed, for every i = 1, ..., m we have
∫
∫
2
|∇ρi | ≤ (1 + τ )
Rn
Rn
φ2i |∇ρ|2
Rn
|∇ρ|2 .
+ (1 + τ
−1
(2.22)
∫
)
Rn
ρ2 |∇φ|2 .
Also, by Hölder and Sobolev inequalities, and by (2.18), we have
∫
∫
(∫
)2/n
n
2/n
2
2
2
≤ C(L, n) η
ρ |∇φ| ≤ ∥ρ∥L2⋆ (Rn )
|∇φi |
Rn
Rn
Rn
|∇ρ|2 .
(2.23)
Hence by adding up over i = 1, ..., m and by taking into account (2.19) we find
m ∫
(
)∫
∑
2
−1 2/n
|∇ρi | ≤ (1 + τ ) + C(L, n) (1 + τ )η
|∇ρ|2
i=1
Rn
Rn
and thus conclude the proof of (2.22) by taking η small enough with respect to τ . Next we claim
that
∫
∫
∫
∇ρi · ∇Ui + ∇ρi · ∇Vi + ∇ρi · ∇Wiα ≤ C(L, n) ∥∇ρ∥L2 (Rn ) η 1/n . (2.24)
n
n
n
R
R
R
∫
Indeed, by Rn ∇ρ · ∇Ui = 0 (recall (2.4)), we have
∫
∫
∫
∇ρi · ∇Ui =
(φi − 1)∇ρ · ∇Ui +
ρ∇φi · ∇Ui
Rn
Rn
where thanks to (2.20),
∫
2 ∫
(φi − 1)∇ρ · ∇Ui ≤
Rn
Rn
∫
|∇ρ|
Rn
while by arguing exactly as in (2.23)
∫
∫
2 ∫
2
ρ∇φ
·
∇U
≤
|∇U
|
i
i
i
Rn
∫
2
Rn
Rn
|∇Ui | ≤ C(L, n) η
2
{φi <1}
2
Rn
|∇ρ|2
∫
ρ2 |∇φi |2 ≤ C(L, n) η 2/n
Rn
|∇ρ|2 ;
this proves (2.24).
∫
∫
We now prove that each Rn |∇ρi |2 is close to bound from above Λ Rn Uip−1 ρ2 . Indeed, let
us set
T
ρi = ρ⊥
i + ρi
where ρTi is the projection of ρi over the subspace of H̊ 1 (Rn ) spanned by {Ui , Vi , Wiα : 1 ≤ α ≤ n}
(recall that that the scalar product on H̊ 1 (Rn ) is given by the L2 -scalar product of the gradients).
By using the spectral analysis argument leading also to (2.6), we find that
∫
∫
2
2
|∇ρ⊥
|
≥
Λ
Uip−1 (ρ⊥
i
i ) .
Rn
By (2.24)
Rn
∫
Rn
∫
|∇ρTi |2
≤ C(n) η
so that by the Sobolev inequality
∫
(∫
p−1
T 2
Ui (ρi ) ≤ C(n)
Rn
Rn
⋆
(ρTi )2
2/n
)2/2⋆
Rn
|∇ρ|2
∫
≤ C(n) η
2/n
Rn
Again with η small enough with respect to τ , we thus find
∫
∫
∫
2
−1
Uip−1 ρ2i ≤ (1 + τ )
Uip−1 (ρ⊥
)
+
(1
+
τ
)
i
Rn
Rn
Rn
|∇ρ|2 .
Uip−1 (ρTi )2
12
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
≤
1+τ
Λ
∫
Rn
2
|∇ρ⊥
i |
+ (1 + τ
−1
∫
)C(L, n) η
2/n
Rn
|∇ρ|2 .
Finally we notice that by Hölder inequality and by (2.20)
∫
∫
∫
(∫
)
⋆ 2/n
Uip−1 ρ2i −
Uip−1 ρ2 ≤
Uip−1 ρ2 ≤ ∥ρ∥2L2⋆ (Rn )
Ui2
Rn
Rn
{φi <1}
{φi <1}
∫
≤ C(L, n) η 4/n
|∇ρ|2 ,
∫
Rn
∫
2
2
to conclude that, thanks also to Rn |∇ρ⊥
i | ≤ Rn |∇ρi | ,
∫
∫
( η 2/n
)∫
1+τ
p−1 2
2
4/n
|∇ρi | + C(L, n)
+η
|∇ρ|2 .
Ui ρ ≤
Λ
τ
Rn
Rn
Rn
(2.25)
By combining (2.25) with (2.22) and by taking η small enough with respect to τ we deduce
(2.21).
Step four: Having in mind to exploit Λ > p and (2.21) we now prove that
∫
m ∫
(∫
∑
p−1 2
2
|∇ρ| ≤ C(L, n) ∥∇ρ∥L2 (Rn ) Φn (m, δ(u)) + p
Ui ρ + o
Rn
i=1
Rn
Rn
)
|∇ρ|2 ,
(2.26)
∫
where o( Rn |∇ρ|2 ) is controlled in terms of the smallness of δ0 , and where Φn was defined in
(1.11). To this end let us set K = −u−p ∆u ∈ C ∞ (Rn ), so that by (2.1)
(∫
) ⋆
⋆ 1/2
δ(u) =
|K − 1|2n/(n+2) u2
≤ δ0 .
Rn
We shall first test the equation −∆u = K up with ρ and then with σ =
Testing with ρ: We are going to prove the preliminary bound
∫
∫
∫
(∫
|∇ρ|2 ≤ S ∥∇ρ∥L2 (Rn ) δ(u) +
σpρ + p
σ p−1 ρ2 + O
Rn
Rn
Rn
∑m
Rn
i=1 Ui .
|∇ρ|2
)1+γ
,
(2.27)
where
O is meant in∫ terms of a∫ constant depending on L and n only (in particular
∫
O( Rn |∇ρ|2 )1+γ = o( Rn |∇ρ|2 ) as Rn |∇ρ|2 ≤ ε0 and we can pick ε0 as small as we wish
provided δ0 is small enough), and where
{1
2 }
γ = min
,
.
2 n−2
∫
Indeed, by testing −∆u = K up with ρ and using Rn ∇ρ · ∇Ui = 0, see (2.4), we find
∫
∫
∫
∫
2
p
p
|∇ρ| =
Ku ρ=
u ρ+
(K − 1) up ρ ,
(2.28)
Rn
Rn
Rn
⋆
⋆
2 /(2 )′ and
Rn
⋆
(2 )′ =
where by Hölder inequality and since p =
2n/(n + 2)
∫
(K − 1) up ρ ≤ δ(u) ∥ρ∥L2⋆ (Rn ) ≤ S ∥∇ρ∥L2 (Rn ) δ(u) .
(2.29)
We are thus left to prove that
∫
∫
p
u ρ=
(2.30)
Rn
Rn
Rn
∫
p
σ ρ+p
σ
p−1 2
(∫
ρ +O
Rn
Rn
|∇ρ|2
)1+γ
.
To this end we notice that if n ≥ 6, then p = 1 + s with s = 4/(n − 2) ∈ (0, 1], therefore
|ap − bp − p bp−1 (a − b)| ≤ C(n) |a − b|p
∀a, b ≥ 0 .
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
Applying this bound with a = u(x) = σ(x) + ρ(x) and b = σ(x), we find
p
u − σ p − p σ p−1 ρ ≤ C(n) |ρ|p ,
13
(2.31)
and then (2.30) follows by the Sobolev inequality as the error term is
∫
∫
(∫
)2⋆ /2
p 2⋆
|ρ| ρ ≤
|ρ| ≤ C(n)
|∇ρ|2
.
Rn
Rn
Rn
2⋆
In the cases n = 3, 4, 5 one has
> 3, thus p > 2 and
p
a − bp − pbp−1 (a − b) ≤ C(n) max{a, b}p−2 |a − b|2
∀a, b ≥ 0 ,
{
}p−2 2
p
ρ .
u − σ p − p σ p−1 ρ ≤ C(n) max u, σ
so that now
(2.33)
Since p − 2 = 2⋆ − 3 and 2⋆ /(2⋆ − 3) = (2⋆ /3)′ we have that
⋆( ∫
(∫
) ⋆
∫
{
}2⋆ (2 −3)/2
{
}p−2 3
max u, σ
|ρ| ≤
max u, σ
Rn
Rn
(2.32)
Rn
2⋆
)3/2⋆
|ρ|
≤ C(L, n) ∥∇ρ∥32 ,
where we have also used (2.1) and the Sobolev inequality. We have thus proved (2.30), and
therefore (2.27).
In order to pass from (2.27) to (2.26) we need to estimate the first order term in ρ appearing
in (2.27). Before discussing this point it is convenient to notice that, since by Hölder and Sobolev
inequalities
∫
(∫
)
p−1 2 σ ρ =O
|∇ρ|2 ,
p
Rn
Rn
(2.27) implies, in particular, that
∫
(
)
(
)
σ p ρ = O ∥∇ρ∥2L2 (Rn ) + O ∥∇ρ∥L2 (Rn ) δ(u) .
(2.34)
Testing with σ: We are now going to prove that
∫
(∫
p σ ρ ≤ C(L, n) ∥∇ρ∥L2 (Rn ) Φn (m, δ(u)) + o
(2.35)
Rn
Rn
Rn
)
|∇ρ|2 ,
which combined with (2.27) gives (2.26). As an intermediate step towards (2.35) we prove that
∫ (
m
)
(
)
∑
∑
⋆
⋆
min{p,2}
σ2 −
Ui2 −
(Uip Uj + Ui Ujp ) ≤ C(L, n) δ(u) + ∥∇ρ∥L2 (Rn ) .
(2.36)
Rn
i=1
By testing −∆u = K
1≤i<j≤m
up
with σ and exploiting again the fact that
∫
m ∫
∑
∇Ui · ∇Uj =
K up σ .
i,j=1
Rn
∫
Rn
∇Ui · ∇ρ = 0, we find
Rn
∫
⋆
Also, by Hölder inequality, since p = 2⋆ /(2⋆ )′ and Rn Ui2 ≤ C(n), we also have
∫
|K − 1| up σ ≤ C(L, n) δ(u) .
Rn
Hence
m ∫
∑
i,j=1
Rn
∫
∇Ui · ∇Uj =
up σ + O(δ(u)) .
Rn
(2.37)
14
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
We now replace up by σ p + p σ p−1 ρ up to an error that can be estimated as follows: if n ≥ 6, by
applying Hölder inequality with exponent (2⋆ )′ /2⋆ we can estimate the error from (2.31) as
∫
(∫
) ⋆′
(
)p
⋆ ′ 1/(2 )
p
σ |ρ| ≤ C(n, L)
|ρ|p (2 )
= C(n, L) ∥ρ∥pL2⋆ (Rn ) = O ∥∇ρ∥L2 (Rn ) ;
Rn
Rn
if instead 3 ≤ n ≤ 5, then by applying Hölder inequality first with exponent 2⋆ /2 and then with
exponent s = 4/(n − 2), we can estimate the error from (2.33) as
(∫
)
∫
{
}p−2
{
}(p−2)n/2 n/2 2/n
2
max u, σ
σ|ρ| ≤
max u, σ
σ
∥ρ∥2L2⋆ (Rn )
Rn
Rn
(
)2
≤ C(n, L) ∥ρ∥2L2⋆ (Rn ) = O ∥∇ρ∥L2 (Rn ) .
Combining the last two estimates with (2.37) we conclude that
∫
m ∫
(
)
∑
⋆
min{p,2}
∇Ui · ∇Uj =
σ 2 + p σ p ρ + O δ(u) + ∥∇ρ∥L2 (Rn ) ,
i,j=1
Rn
Rn
which, thanks to (2.34), gives us
∫
m ∫
∑
∇Ui · ∇Uj =
Rn
i,j=1
(
)
⋆
min{p,2}
σ 2 + O δ(u) + ∥∇ρ∥L2 (Rn ) .
Rn
(2.38)
We now recall that by testing −∆Ui = Uip with Ui and Uj , j =
̸ i, one finds
∫
∫
∫
∫
∫
p
2
2⋆
|∇Ui | =
Ui ,
∇Ui · ∇Uj =
Ui Uj =
Ujp Ui ,
Rn
so that
Rn
m ∫
∑
i,j=1
Rn
Rn
∇Ui · ∇Uj =
Rn
m
∑
∑
⋆
Ui2 +
i=1
Rn
Uip Uj + Ui Ujp .
1≤i<j≤m
Hence (2.38) leads to (2.36). We next notice that since 2⋆ > 2 one has, for some c(n, m) > 0,
m
m
(∑
)2⋆ ∑
∑
⋆
−
a2i −
(api aj + ai apj )
ai
i=1
i=1
≥ c(n, m)
1≤i<j≤m
∑
api aj + ap−1
a2j + ai apj + a2i ajp−1 ,
i
1≤i<j≤m
whenever ai ≥ 0. By combining this inequality with (2.36), and setting
∫
Aij =
Uip Uj + Uip−1 Uj2 + Ui Ujp + Ui2 Ujp−1 ,
Rn
we find that
∑
(
)
min{p,2}
Aij ≤ C(L, n) δ(u) + ∥∇ρ∥L2 (Rn ) .
(2.39)
1≤i<j≤m
We∫ now come to ∫the linear term in ρ appearing in (2.27). By exploiting once again 0 =
− Rn ∇Ui · ∇ρ = Rn Uip ρ we find
m
∫
∫ p ∑ p
p σ
ρ
≤
σ
−
U
i |ρ| .
Rn
Rn
i=1
Taking into account
m
m
( ∑
)p ∑
∑ p−1
ai −
api ≤ C(m)
ai aj
i=1
i=1
i̸=j
∀ai ≥ 0 ,
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
15
and exploiting Hölder and Sobolev inequalities, we thus find
∫
) ⋆′
∑(∫
(p−1)(2⋆ )′ (2⋆ )′ 1/(2 )
σ p ρ ≤ C(L, n) ∥∇ρ∥L2 (Rn )
Ui
Uj
.
Rn
i̸=j
Rn
As shown in Appendix A, for every i ̸= j we have
{
(∫
) ⋆′
Aij log2/3 (1 + Aij ) ,
(p−1)(2⋆ )′ (2⋆ )′ 1/(2 )
Ui
Uj
≤ C(n)
Aij ,
Rn
if n = 6 ,
if n =
̸ 6.
If n ̸= 6 we thus find
∫
(
)
(∫
p σ ρ ≤ O ∥∇ρ∥L2 (Rn ) δ(u) + O
.
Rn
Rn
|∇ρ|2
)(1+min{p,2})/2
(2.40)
If n = 6, by taking into account that
a log2/3 (1 + a) + b log2/3 (1 + b) ≤ (a + b) log2/3 (1 + a + b)
∀a, b ≥ 0 ,
we get
∫
(
)
(
)
min{p,2}
min{p,2}
σ p ρ ≤ C(L, n)∥∇ρ∥L2 (Rn ) δ(u) + ∥∇ρ∥L2 (Rn )
log2/3 1 + δ(u) + ∥∇ρ∥L2 (Rn ) .
Rn
Since
(
)
(a + b) log2/3 (1 + a + b) ≤ C a log2/3 (1 + a) + b log2/3 (1 + b)
this yields
∫
Rn
(∫
2/3
σ ρ ≤ C(L, n)∥∇ρ∥L2 (Rn ) δ(u) log (1 + δ(u)) + o
∀a, b ∈ [0, 1] ,
p
Rn
)
|∇ρ|2 ,
completing the proof of (2.35).
Conclusion: We prove (1.12), (1.13), and (1.14).
We multiply (2.21) by (1 − τ ) and then use (2.26) to find that
∫
m ∫
( Λ(1 − τ )
)∑
2
τ
|∇ρ| +
−p
Uip−1 ρ2
1
+
C(L,
n)
τ
Rn
Rn
i=1
≤ C(L, n) ∥∇ρ∥L2 (Rn ) Φn (m, δ(u)) + o
Provided τ is small enough in terms of Λ − p > 0, this gives
∫
(∫
|∇ρ|2 ≤ C(L, n) ∥∇ρ∥L2 (Rn ) Φn (m, δ(u)) + o
Rn
Rn
(∫
Rn
)
|∇ρ|2 .
)
|∇ρ|2 ,
and thus conclude the proof of (1.12).
Then, combining (1.12) with (2.39), we see that
∫
Uip Uj + Uip−1 Uj2 + Ui Ujp + Ui2 Ujp−1 ≤ C(L, n) δ(u)
(2.41)
Rn
for every i ̸= j, that combined with (A.4) below gives immediately (1.13).
We finally prove (1.14): by orthogonality
∫
∫
∫
|∇u|2 =
|∇σ|2 +
|∇ρ|2 ,
Rn
Rn
Rn
∑
2⋆ 2/n (compare with [Bre07b, Proposition 3]) we have
and since
= (Ui )2/n Ui ≤ Ui ( m
j=1 Uj )
∫
∫
m ∫
m
m ∫
(∑
)2/n ( ∑
)2/n ( ∫
)
∑
⋆ (n−2)/n
p
2
2
2⋆
2⋆
|∇σ| =
σ Ui ≤
σ
Uj
≤
Uj
σ2
Uip
Rn
2⋆
i=1
Rn
Rn
j=1
j=1
Rn
Rn
16
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
n 2/n
(∫
= (mS )
⋆
σ2
)2/2⋆
.
Rn
(note that (n − 2)/n = 2/2⋆ ). On the other hand, since −∆Ui = Uip , using (2.39), (1.10), and
(1.12), we have
∫
m ∫
m ∫
∑
∑∫
∑
∑∫
2
2
2
∇Ui · ∇Uj =
|∇Ui | −
Uip Uj
|∇σ| =
|∇Ui | +
Rn
Rn
i=1
i̸=j
Rn
i=1
Rn
i̸=j
Rn
≥ m S n − C(n) δ(u).
In addition, thanks to (2.36), (2.39), (1.10), and (1.12)
m
∫
∫ ( ⋆ ∑
)
⋆ 2⋆
n
2
σ − mS = σ −
Ui2 ≤ C(n, L) δ(u),
Rn
Rn
i=1
that implies, in particular, that
(∫
n
n 2/n
m S − (mS )
Hence, since
deduce that
∫
Rn
⋆
σ2
Rn
|∇ρ|2
=
O(δ(u)2 )
)2/2⋆ ≤ C(n, L) δ(u).
(by (1.12)), combining the above estimates together we
∫
−C(n, L) δ(u) ≤
|∇u| − (mS )
2
Rn
n 2/n
(∫
⋆
σ2
)2/2⋆
Rn
≤ C(n, L) δ(u)2 .
Since σ = u − ρ and 2⋆ > 2, we find
2⋆
σ − u2⋆ − 2⋆ up ρ ≤ C(n, L) max{u, |ρ|}p−1 ρ2 .
Also, thanks to (2.30), (2.34), and (1.12),
∫
∫
∫
(∫
p p p−1 2
u ρ ≤ σ ρ + p
σ ρ + C(L, n)
Rn
Rn
Rn
Rn
|∇ρ|2
)1+γ
≤ C(L, n) Φn (m, δ(u))2 ,
Furthermore, again by Hölder and Sobolev inequalities,
∫
max{u, |ρ|}p−1 ρ2 ≤ C(L, n) Φn (m, δ(u))2 .
Rn
Combining these four estimates together, completes the proof of (1.14).
3. An application to the Yamabe flow on the sphere: proof of Theorem 1.4
This section is devoted to the proof of Theorem 1.4. Let us recall from the introduction
that given u0 with
∫
⋆
u0 ∈ H̊ 1 (Rn ) ∩ C ∞ (Rn )
u0 > 0
u20 = 1 ,
Rn
then one can find (see [Ham89]) u(t) with u(0) = u0 such that g = u(t)p−1 dx evolves according
to the Yamabe flow
d
g = −(Rg − rg ) g
t ≥ 0,
(3.1)
dt
where, in terms of u(t), we have
4(n − 1) ∆u
Rg = −
n − 2 up
∫
∫
Rg dvol g
4(n − 1) Rn |∇u|2
4(n − 1)
∫
=
K0 (u) .
rg =
=
⋆
2
vol g
n−2
n−2
Rn u
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
17
By writing (3.1) in terms of u(t) we find
) ⋆
d 2⋆
2n(n − 1) ( ∆u
u =
+
K
(u)
u2
0
dt
n−2
up
so that
∫
∫
⋆
⋆
u(t)2 =
Rn
Rn
as well as
d
dt
∫
∫
|∇u| = −2(n − 1)
( ∆u
2
Rn
u20 = 1
Rn
up
∀t ≥ 0 ,
∀t ≥ 0
)2 ⋆
+ K0 (u) u2 ≤ −2(n − 1) δ(u)2 .
(3.2)
(3.3)
(3.4)
Having in mind to exploit Theorem 1.1, we now set
(∫
)(n−2)/4
1/(p−1)
|∇u|2
ū = K0 (u)
u=
u.
Rn
Since δ(s w) = s δ(w) and K0 (s w) = s1−p K0 (w) for every function w and every s > 0, we notice
that
∫
(∫
)n/2
(∫
)(n−2)/4
2
2
|∇ū| =
|∇u|
,
K0 (ū) = 1 ,
δ(ū) =
|∇u|2
δ(u) .
(3.5)
Rn
Rn
Rn
Since ∥∇u∥L2 (Rn ) ≤ ∥∇u0 ∥L2 (Rn ) for every t ≥ 0, we can thus find L ∈ N (depending only on
∥∇u0 ∥L2 (Rn ) and n) such that
∫
(
1) n
|∇ū|2 ≤ L +
S
∀t ≥ 0 .
2
Rn
By Theorem 1.1, for each t ≥ 0 we find m = m(t) ∈ {1, ..., L} such that
∫
(∫
) ⋆
⋆ 2/2
−C0 δ(ū) ≤
|∇ū|2 − m2/n S 2
(ū)2
≤ C0 Φn (m, δ(ū))2
Rn
(3.6)
Rn
where m(t) is the number of bubbles v[zi , λi ] one has to use to optimally decompose ū. Exploiting
(3.3) and (3.5), this yields
∫
−C∗ δ(u) ≤
|∇u|2 − m2/n S 2 ≤ C∗ Φn (m, δ(u))2
∀t ≥ 0 ,
(3.7)
Rn
where here and in the following C∗ denotes a constant depending only on n and ∥∇u0 ∥L2 (Rn ) .
We now observe that, thanks to (3.4),
∫ ∞
∫
1
δ(u(t))2 dt ≤
|∇u0 |2 < ∞ ,
c(n)
n
0
R
so that we can find a sequence tj → ∞ such that δ(u(tj )) → 0. By Remark 1.3, there exists an
integer m0 ∈ {1, . . . , L} such that
(
)
1 ∥∇u(tj )∥L2 (Rn ) n
→ m0
as j → ∞.
(3.8)
S ∥u(tj )∥L2∗ (Rn )
∫
Recalling (3.3) and the fact that Rn |∇u(t)|2 is decreasing in time, this yields
(1
)n ( 1
)n
K0 (u(t))1/2 =
∥∇u(t)∥L2 (Rn ) ↓ m
as t → ∞.
(3.9)
S
S
Hence, again by Remark 1.3, this implies that there exists a time t0 > 0 so that we can assume
that m(t) = m0 for every t ≥ t0 . (More precisely, if δ(u(t)) is small enough, say δ(u(t)) ≤ δ0 for
some δ0 = δ0 (L, n), then necessarily m(t) = m0 . On the other hand, if δ(u(t)) is larger than δ0
18
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
then Theorem 1.1 is true with any m ∈ {1, . . . , L}, see the discussion at the beginning of section
2. In particular, with no loss of generality, in this case we can assume that m(t) = m0 .) Setting
∫
2/n
2/n
p(t) =
|∇u(t)|2 − m0 S 2 = K0 (u(t)) − m0 S 2
(3.10)
Rn
it follows by (3.9) that
p(t) ≥ 0
Also, thanks to (3.4) and (3.7),
p′ (t) ≤ −2(n − 1) δ(u(t))2
p(t) ↓ 0 as t → ∞.
and
(
)2
p(t) ≤ C∗ Φn m0 , δ(u(t))
and
∀t ≥ t0 .
(3.11)
By arguing as in [SS03] (see, in particular, the argument after Lemma 4.4 therein), we can
now quickly rule out concentration of volume (that is the possibility that m0 > 1). Indeed, it
follows by (3.11) and the definition of Φn (see (1.11)) that
p(t)
p′ (t) ≤ −C∗
log
4/3
(p(t))
∀t ≥ t0 .
In particular 0 ≤ p(t) ≤ C∗ e−t , and using (3.11) again we deduce that, for every k ∈ N and
k ≥ t0 ,
∫ k+1
∫ k+1
(
)1/2
3/7
δ(u) ≤ C∗
(−p′ )1/2 ≤ C∗ p(k) − p(k + 1)
≤ C∗ e−k /2 ,
(3.12)
3/7
k
k
therefore
∫
∞
δ(u) < ∞ .
(3.13)
t0
Concentration of volume is then excluded as follows. For every measurable set E ⊂ Rn , using
Hölder inequality and (3.2) we have
∫
∫ (
) ⋆
d
∆u
⋆
u2 = c(n)
+
K
(u)
u2 ≤ C(n) δ(u) .
(3.14)
0
p
dt E
u
E
Now, let ε = ε(n, m0 ) be a small constant to be chosen later. Up to increase the value of t0 we
can assume that (recall (3.9))
∫ ∞
K0 (u(t)) − m2/n S 2 ≤ ε ∀t ≥ t0 .
C(n)
δ(u) < ε
and
0
t0
Now, fix η > 0 small enough so that
∫
∫
⋆
2⋆
u(t0 ) +
u(t0 )2 ≤ ε
Rn \B1/η
whenever |E| < η.
E
Then, it follows by (3.14) that for every t ≥ t0 and every |E| < η,
∫
∫
∫ ∞
2⋆
2⋆
u(t) +
u(t) ≤ ε + C(n)
δ(u) ≤ 2 ε .
Rn \B1/η
E
(3.15)
t0
2/n
On the other hand, since K(u(t)) − m0 S 2 ≤ ε, it follows by (1.10) that each bubble Ui (t) in
the decomposition of u(t) satisfies
∫
1
Sn
⋆
− C(m0 , n) ε.
Ui (t)2 =
≥
m0
K(u(t))n/2
Rn
We now fix ε small enough so that
1
− C(m0 , n) ε ≥ 3 ε .
m0
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
19
Then, it follows by Theorem 1.1 and (3.15) applied to the sequence of times tj in (3.8), that u(tj )
can have at most one bubble for j large enough (since, if m0 ≥ 2, then at least one bubble should
concentrate its mass either in a small region or at infinity, but this is prevented by (3.15)).
Thus, we have proved m0 = 1, which implies that Φn (m0 , s) = s and that (up to choosing
t0 larger)
∫
3
|∇u(t)|2 ≤ S 2
∀t ≥ t0 .
2
Rn
Hence we can apply Theorem 1.1 with L = m = 1, and (3.11) takes the form
∫
′
2
2
p (t) ≤ −2(n − 1) δ(u(t)) ,
p(t) ≤ C(n) δ(u(t)) ,
p(t) =
|∇u(t)|2 − S 2
∀t ≥ t0 ,
Rn
which yields
p(t) ≤ p(t0 ) e−c(n) (t−t0 ) ≤
(3
)
S 2 ec(n) t0 e−c(n) t
∀t ≥ t0 .
2
In particular, arguing as in (3.12), we deduce that for any k ≥ t0
∫ k+1
δ(u) ≤ C∗ e−c(n) k/2 ,
k
therefore, up to replacing c(n) with c(n)/2,
∫ ∞
δ(u) ≤ e−c(n) t
∀t ≥ t0 .
(3.16)
t
Also, noticing that p(t) = δSobolev (u(t)) (see (1.20)), we have
δSobolev (u(t)) ≤ C∗ e−c(n) t
∀t ≥ t0 .
So, applying the quantitative Sobolev inequality (1.21), we find that for every t ≥ t0 there exist
κ(t), λ(t) > 0 and z(t) ∈ Rn such that, if we set U (t) = vκ(t) [z(t), λ(t)], then
∥∇u(t) − ∇U (t)∥2L2 (Rn ) ≤ CBE (n) δSobolev (u(t))
≤ C∗ e−c(n) t
∀t ≥ t0 .
(3.17)
The exponential convergence of u(t) to the standard metric of the sphere is now easy, and we
provide the details just for the sake of completeness. By (3.2) we see that if t0 ≤ t < s then
∫ s ( ∆u(r)
)
⋆
⋆
⋆
2⋆
|u(s) − u(t)|2 ≤ |u(s)2 − u(t)2 | ≤ +
K
(u(r))
u(r)
dr
0
p
u(r)
t
so (3.16) yields
∫
Rn
2⋆
|u(s) − u(t)|
∫
≤ C(n)
t
s
δ(u(r)) dr ≤ C∗ e−c(n) t
∀s > t ≥ t0 .
In particular, by the Sobolev inequality and (3.17) it follows that {U (t)}t≥t0 is a Cauchy sequence
⋆
in L2 (Rn ) with an exponential bound
∥U (t) − U (s)∥2L2⋆ (Rn ) ≤ C∗ e−c(n) t
⋆
∀s > t ≥ t0 .
This implies that z(t) → z∞ ∈ Rn , λ(t) → λ∞ ∈ (0, ∞) and κ(t) → κ∞ = S 2 exponentially fast
as t → ∞ (note that κ∞ = S 2 follows by (3.9)). Thus, if we set U∞ = vS 2 [z∞ , λ∞ ],
∥∇U (t) − ∇U∞ ∥2L2 (Rn ) ≤ C∗ e−c(n) t
∀t ≥ t0 ,
that combined with (3.17) yields
∥∇u(t) − ∇U∞ ∥2L2 (Rn ) ≤ C∗ e−c(n) t
∀t ≥ t0 .
Since the above inequality is trivially true on [0, t0 ] simply by choosing C∗ sufficiently large, we
conclude the validity of (1.22).
20
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
We also note that if
∫
Rn
(
|∇u0 |2 ≤ 2 − τ )2/n S 2
(3.18)
for some τ ∈ (0, 1), then we have m(t) ≡ 1 for every t ≥ 0 and the above argument works with
t0 = 0. In particular we get
(∫
)
2
∥∇u(t) − ∇U (t)∥L2 (Rn ) ≤ C(n)
|∇u0 |2 − S 2 e−c(n) t
∀t > 0 ,
(3.19)
Rn
and the last argument leads to
∥∇u(t) −
∇U ∥2L2 (Rn )
≤ C(n)
(∫
Rn
)
|∇u0 |2 − S 2 e−c(n) t
∀t ≥ 0 .
This is (1.23).
The proof of Theorem 1.4 is thus complete.
Appendix A. Interactions between bubbles
A key point in the the proof of Theorem 1.1 was controlling the interaction integral between
two different bubbles Ui and Uj
(∫
) ⋆′
(p−1)(2⋆ )′ (2⋆ )′ 1/(2 )
Ui
Uj
Rn
by means of the interaction integral
∫
Uip Uj + Uip−1 Uj2 + Ujp Ui + Ujp−1 Ui2 .
Rn
The aim of this appendix is providing a detailed justifications of this estimate, using explicit
formulas for the interaction of bubbles. More precisely, we aim to prove the following proposition.
We recall the definition (1.4) of vκ , so that
v1 (x) =
K(n)
(1 + |x|2 )(n−2)/2
∀x ∈ Rn ,
and v[z, λ](x) = λ(n−2)/2 v(λ(x − z)) for x, z ∈ Rn , λ > 0.
Proposition A.1. There exists a constant C0 (n) with the following property. Given z1 , z2 ∈ Rn
and λ2 ≥ λ1 > 0, set U1 = v1 [z1 , λ1 ], U2 = v1 [z2 , λ2 ], and
(∫
) ⋆ ′ (∫
) ⋆′
(p−1)(2⋆ )′ (2⋆ )′ 1/(2 )
(p−1)(2⋆ )′ (2⋆ )′ 1/(2 )
A=
U1
U2
+
U2
U1
(A.1)
n
Rn
∫ R
U1p U2 + U1p−1 U22 + U2p U1 + U2p−1 U12 .
B=
(A.2)
Rn
If B ≤ 1/9, then
{
A ≤ C0 (n)
and
C0 (n) B ≥
(




(




B
if n ̸= 6 ,
B log2/3 (1/B)
if n = 6 ,
λ1 /λ2
1 + λ21 |z1 − z2 |2
λ1 /λ2
1 + λ21 |z1 − z2 |2
(A.3)
)2
if n ≥ 6 ,
)(n−2)/2
(A.4)
if n < 6 .
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
21
1+ℓ
W−
W = B1+ℓ (z̄)
W+
0
z
z̄
Bℓ/2 (0)
ℓ
Figure 1. The sets W , W + , and W − when ℓ/2 ≥ 1.
Proof. By a translation and a scaling we may then assume that
U1 (x) =
K(n)
(1 + |x|2 )(n−2)/2
U2 (x) =
K(n) λ(n−2)/2
(1 + λ2 |x − z|2 )(n−2)/2
(A.5)
where z ∈ Rn and λ ≥ 1. We set z̄ = z/2 and then consider the sets
{
}
{
}
W = B1+ℓ (z̄) ,
W + = x ∈ W : (x − z̄) · z ≥ 0 , W − = x ∈ W : (x − z̄) · z ≤ 0 ,
(where, say, W + = W ∩ {x · e1 ≥ 0} and W − = W ∩ {x · e1 ≤ 0} if ℓ = 0); see Figure 1. Since
U2 is more concentrated than U1 (as λ ≥ 1), we see that given α, β > 0 with α + β = 2⋆ , the
∫
∫
largest part of the interaction integral Rn U1α U2β is going to be W + U1α U2β . Hence, in order to
∫
∫
prove (A.3), we are going to use an upper bound on Rn U1α U2β and a lower bound on W + U1α U2β
in terms of λ and ℓ.
To simplify the exposition, given a, b > 0 we shall write a . b (resp. a ≈ b) if there exists
a positive dimensional constant C(n) such that a ≤ C(n) b (resp. a/C(n) ≤ b ≤ C(n) a). For
example, since 1 + |w| ≈ (1 + |w|2 )1/2 for every w ∈ Rn , then we have
U1 (x) ≈
1
(1 + |x|)n−2
U2 (x) ≈
λ(n−2)/2
(1 + λ|x − z|)n−2
We now obtain various bounds on
∫
U1α U2β ,
when β ∈ (0, 2⋆ )
and
∀x ∈ Rn .
(A.6)
α = 2⋆ − β ,
on the domains of integration W + , Rn \ W , and W − .
Sharp estimate on W + : We prove that
∫
W+
U1α U2β ≈
(n−2)β
−n
2
λ
(1 + ℓ)2n−(n−2) β

if (n − 2)β > n
 1 (
)
log 1 + λ(1 + ℓ)
if (n − 2)β = n
)n−(n−2)β
 (
λ(1 + ℓ)
if (n − 2)β < n .
(A.7)
Since W + ⊂ B2(1+ℓ) (0) \ Bℓ/2 (0), thanks to (A.6) we find
U1 ≈
1
(1 + ℓ)n−2
on W + .
(A.8)
22
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
By using Bℓ/2 (z) ⊂ W + ⊂ B2(1+ℓ) (z) and the estimate for U2 in (A.6), we get
∫
∫
(n−2)β
dy
β
−n
U2 ≈ λ 2
.
β(n−2)
W+
Bλ(1+ℓ) (0) (1 + |y|)
Combining (A.8) and this last estimate, we prove (A.7).
Upper bound on Rn \ W : We claim that
∫
Rn \W
∫
U1α U2β
.
W+
U1α U2β .
(A.9)
First, one checks the following facts:
U1 (x) ≈
U2 (x) ≈
1
|x|n−2
for x ∈ Rn \ B1 (0) ,
1
λ(n−2)/2 |x
for x ∈ Rn \ B1 (z) ,
−
|x| ≈ |x − z̄| ≈ |x − z|
z|n−2
(A.10)
for x ∈ Rn \ W .
By the inclusion B1 (0) ∪ B1 (z) ⊂ W we thus find that
∫
∫
1
dx
α β
≈ β(n−2)/2
.
U1 U2 ≈
β(n−2)/2
2n
|x − z̄|
λ
(1 + ℓ)n
Rn \W
Rn \W λ
(A.11)
Then (A.9) follows immediately by comparison with (A.7) (and actually the integral over W + is
equivalent to that over Rn \W when (n−2)β < n, is larger by a logarithmic factor if (n−2)β = n,
and is larger by power factors if (n − 2)β > n).
Upper bound on W − : We are going to prove that
∫
∫
{∫
α β
α β
U1 U2 . max
U1 U2 ,
W−
W+
W+
}
U1β U2α .
(A.12)
Let g be the reflection of Rn with respect to the hyperplane dividing W + and W − . Then
W − = g(W + ) with det ∇g = 1, thus
∫
∫
∫
α β
α
β
U1 U2 =
U1 (g(x)) U2 (g(x)) dx .
U2β
W−
W+
W+
where have used the geometric fact that U2 (g(x)) ≤ U2 (x) for each x ∈ W + , and the trivial
estimate U1 . 1 on Rn . If ℓ ≤ 1, then 1 ≈ (1 + ℓ)−(n−2)α , and thus we have proved
∫
∫
∫
U1α U2β . (1 + ℓ)−(n−2)α
U2β ≈
U1α U2β
W−
W+
W+
accordingly to (A.8). This shows (A.12) in the case ℓ ≤ 1. Now let us assume that ℓ ≥ 1. Since
ℓ + |x| . |x − z| for x ∈ Rn \ Bℓ/2 (z) and W − ⊂ Rn \ Bℓ/2 (z), by (A.6) we find
U2 (x) .
1
λ(n−2)/2 |x
−
z|n−2
.
1
+ |x|)n−2
λ(n−2)/2 (ℓ
∀x ∈ W − .
Since ℓ ≥ 1 gives ℓ + |x| ≈ ℓ on B2(1+ℓ) (0), by using W − ⊂ B2(1+ℓ) (0) and (A.6)
find, setting for brevity α(n − 2) = γ, so that β(n − 2) = 2n − γ,
 −n
ℓ

∫
∫

(n−2)β
dx
α β
− 2
ℓ−n (1 + log ℓ)
U1 U2 .
≈λ
(n−2)β/2 (1 + |x|)γ ℓ2n−γ

−
λ
W
Bℓ (0)
 γ−2n
ℓ
for U1 we just
if γ < n ,
if γ = n ,
if γ > n ,
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
= λ−
(n−2)β
2
 −n
ℓ
if (n − 2)β > n ,


ℓ−n (1 + log ℓ) if (n − 2)β = n ,

 −(n−2)β
ℓ
if (n − 2)β < n .
23
(A.13)
By combining (A.7) with this last estimate we see that
∫
∫
α β
U1 U2 ≤
U1α U2β
if (n − 2)β ≥ n .
W−
W+
∫
In the case (n − 2)β < n we need the term W + U1β U2α . By arguing exactly as in the proof of
(A.7) and by taking into account that (n − 2)β − n = n − (n − 2)α, we find

(n−2)α
∫
if (n − 2)β < n
 1 (
−n
)
λ 2
β α
log
1
+
λ(1
+
ℓ)
if (n − 2)β = n
U1 U2 ≈
)n−(n−2)α
(1 + ℓ)2n−(n−2) α  (
W+
λ(1 + ℓ)
if (n − 2)β > n ,
and hence, again by (n − 2)α = 2n − (n − 2)β, we see that, in the case (n − 2)β < n,
(n−2)β
∫
∫
λ− 2
β α
U1 U2 ≈
&
U1α U2β
(n−2)β
+
−
(1
+
ℓ)
W
W
thanks to (A.13).
Conclusion: Let us set
{( ∫
) ⋆′ (
(p−1)(2⋆ )′ (2⋆ )′ 1/(2 )
U1
U2
,
∫
(2⋆ )′
A0 = max
U1
W+
W+
∫
B0 =
U1p U2 + U1p−1 U22 + U1 U2p + U12 U2p−1 .
) ⋆ ′}
(p−1)(2⋆ )′ 1/(2 )
U2
(A.14)
(A.15)
W+
By (A.7), (A.9) and (A.12) we have A . A0 . Since B0 ≤ B and x log2/3 (1/x) is increasing
for x ∈ (0, e−2/3 ) with e−2/3 > 1/9 ≥ B, we conclude that in order to complete the proof it is
enough to show

if n ̸= 6 ,
 B0
(
)
A0 .
(A.16)
1
 B0 log2/3
if n = 6 ,
B0
We now repeatedly apply (A.7) with different values of β.
Setting β = (2⋆ )′ = 2n/(n + 2) we find that (n − 2)β > n if and only if n > 6, so that

n−2
n
−
 1
⋆
′
(∫
)
(
) if n > 6
λ 2 β
(p−1)(2⋆ )′ (2⋆ )′ 1/(2 )
1/β
log
1 + λ(1 + ℓ) if n = 6
U1
U2
≈
2n
−(n−2)  (
) n −(n−2)
W+
(1 + ℓ) β
λ(1 + ℓ) β
if n < 6 .


 1
(
) if n > 6
1
2/3
log
1 + λ(1 + ℓ) if n = 6
≈
(A.17)
λ2 (1 + ℓ)4 
 (λ(1 + ℓ)) 6−n
2
if n < 6 .
Setting β = (p − 1)(2⋆ )′ = 8n/[(n − 2)(n + 2)] we have (n − 2)β > n if and only if n < 6
 (
) n −(n−2)
n−2
β
−n

λ(1
+
ℓ)
(∫
)1/(2⋆ )′
2
β
⋆
′
⋆
′
λ
(
) if n > 6
(2 ) (p−1)(2 )
1/β
U1 U2
≈
log
1
+
λ(1
+
ℓ)
if n = 6 (A.18)
2n
−(n−2) 
W+
(1 + ℓ) β
1
if n < 6 .

(
) (n−2)(n−6)

8
 λ(1 + ℓ)
1
(
) if n > 6
2/3
≈
2
2
log
1 + λ(1 + ℓ) if n = 6
(n−2)
(n−2)


λ 8 (1 + ℓ) 4
1
if n < 6 .
24
G. CIRAOLO, A. FIGALLI, AND F. MAGGI
Estimating A0 : If n > 6 then by (A.17) and (A.18) we have
A0
(
) (n−2)(n−6)
8
λ(1 + ℓ)
1
1
1
≈
+
= 2
+ n−2
(n−2)(n+2)
2
4
4
(n−2)2
(n−2)2
λ (1 + ℓ)
λ (1 + ℓ)
8
λ 2 (1 + ℓ)
λ 8 (1 + ℓ) 4
1
≈ 2
.
λ (1 + ℓ)4
In the case n = 6, (A.17) and (A.18) give us
A0 ≈
log2/3 (1 + λ(1 + ℓ))
λ2 (1 + ℓ)4
If n < 6 by (A.17) and (A.18) we find
A0 ≈
Summarizing
1
λ
n−2
2
(1 + ℓ)
n+2
2
1
+
λ
(n−2)2
8
(1 + ℓ)
(n−2)2
4
 −2
λ (1 + ℓ)−4



A0 ≈ λ−2 (1 + ℓ)−4 log2/3 (1 + λ(1 + ℓ))


n+2
 − n−2
λ 2 (1 + ℓ)− 2
≈
1
λ
n−2
2
(1 + ℓ)
n+2
2
,
if n > 6 ,
if n = 6 ,
(A.19)
if n < 6 .
We now estimate B0 . To this end we plug more values of β in (A.7), as follows:
Setting β = 1 we find (n − 2)β < n for every n, and thus (A.7) gives
(
)n−(n−2)β
(n−2)β
∫
λ 2 −n λ(1 + ℓ)
n−2
p
U1 U2 ≈
= λ− 2 (1 + ℓ)−n .
2n−β(n−2)
(1 + ℓ)
W+
(A.20)
Setting β = p we enforce (n − 2)β > n for every n, and thus (A.7) gives
∫
W+
(n−2)β
U1 U2p
n−2
λ 2 −n
= λ− 2 (1 + ℓ)−(n−2)
≈
2n−β(n−2)
(1 + ℓ)
Setting β = 2 we find (n − 2)β > n if and only if n > 4 and thus

(n−2)β
∫
if n > 4
 1 (
−n
)
λ 2
p−1 2
log
1
+
λ(1
+
ℓ)
if n = 4
U1 U2 ≈
)n−(n−2)β
(1 + ℓ)2n−β(n−2)  (
W+
λ(1 + ℓ)
if n = 3 .

 1 (
) if n > 4
1
log
1
+
λ(1
+
ℓ)
if n = 4
≈ 2
λ (1 + ℓ)4  λ(1 + ℓ)
if n = 3 .
Setting β = p − 1 = 4/(n − 2), we have (n − 2)β = 4 so that by (A.7)
 (
)n−(n−2)β
(n−2)β
∫

−n
λ(1
+
ℓ)
2
λ
(
)
U12 U2p−1 ≈
2n−β(n−2)  log 1 + λ(1 + ℓ)
+
(1
+
ℓ)
W
1

n−4
 (λ(1
( + ℓ))
)
1
log
1
+
λ(1
+
ℓ)
≈ n−2
λ
(1 + ℓ)2n−4  1
if n > 4
if n = 4
if n = 3 .
if n > 4
if n = 4
if n = 3 .
(A.21)
(A.22)
(A.23)
METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
Estimating B0 : by combining (A.20), (A.21), (A.22) and (A.23) we find
 −2
 λ (1 + ℓ)−4 + λ(−2 (1 + ℓ)−n ) if
− n−2
−(n−2)
λ−2 (1 + ℓ)−4 log 1 + λ(1 + ℓ) if
B0 ≈ λ 2 (1 + ℓ)
+
 −1
λ (1 + ℓ)−3 + λ−1 (1 + ℓ)−2
if
 −2
−4
 λ (1 + ℓ)
(
)
n−2
λ−2 (1 + ℓ)−4 log 1 + λ(1 + ℓ)
≈ λ− 2 (1 + ℓ)−(n−2) +
 −1
λ (1 + ℓ)−2
and thus
{
B0 ≈
λ−2 (1 + ℓ)−4
if n ≥ 6
− n−2
−(n−2)
λ 2 (1 + ℓ)
if n < 6 .
25
n>4
n=4
n=3
if n > 4
if n = 4
if n = 3
(A.24)
By (A.19) and (A.24) we see that A0 ≈ B0 if n > 6, while A0 . B0 if n < 6 (as for n < 6 the
power (n + 2)/2 of (1 + ℓ) appearing in A0 is larger than the power (n − 2) of (1 + ℓ) appearing
in B0 ). When n = 6, B0 ≈ λ−2 (1 + ℓ)−4 gives us
(
1 ) log2/3 (1 + λ(1 + ℓ)2 )
log2/3 (1 + λ(1 + ℓ))
B0 log2/3 1 + √
≈
≥
≈ A0 .
λ2 (1 + ℓ)4
λ2 (1 + ℓ)4
B0
Since log(1 + x) ≤ 2 log(x) for x > 3 and since B0 ≤ B ≤ 1/9, we deduce that A0 .
B0 log2/3 (1/B0 ) when n = 6. Finally, we immediately get (A.4) from (A.24) using that
(1 + ℓ)2 ≈ (1 + ℓ2 ) and noticing that the above estimates have been obtained by rescaling
the parameters λi and the distance between the centers as follows:
λ :=
λ2
,
λ1
ℓ := λ1 |z1 − z2 | .
Thus the proof of the proposition is complete.
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METRICS WITH ALMOST CONSTANT SCALAR CURVATURE ON THE SPHERE
27
Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123
Palermo, Italy
E-mail address: giulio.ciraolo@unipa.it
Mathematics Department, The University of Texas at Austin, 2515 Speedway Stop C1200, RLM
8.100 Austin, TX 78712, USA
E-mail address: figalli@math.utexas.edu
Mathematics Department, The University of Texas at Austin, 2515 Speedway Stop C1200, RLM
8.100 Austin, TX 78712, USA
E-mail address: maggi@math.utexas.edu