II

# Quantitative stability of Sobolev inequalities on compact Riemannian
manifolds

Francesco Nobili  Universitá di Pisa, Dipartimento di Matematica, Largo Bruno
Pontecorvo 5, 56127 Pisa, Italy<EMAIL_ADDRESS>and  Davide
Parise  University of California San Diego, Department of Mathematics, 9500
Gilman Drive #0112, La Jolla, CA 92093-0112, United States of America
<EMAIL_ADDRESS>
(Date: .
MSC(2020): 26D10, 58K05 (primary), 46E35 (secondary).
_Keywords_ : Sobolev inequality, Stability, Riemannian manifolds)

###### Abstract.

We study quantitative stability results for different classes of Sobolev
inequalities on general compact Riemannian manifolds. We prove that, up to
constants depending on the manifold, a function that nearly saturates a
critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty
set of extremal functions, provided that the corresponding optimal Sobolev
constant satisfies a suitable strict bound. The case of sub-critical Sobolev
inequalities is also covered. Finally, we discuss degenerate phenomena in our
quantitative controls.

###### Contents

  1. 1 Introduction
  2. 2 Properties of Sobolev inequalities
    1. 2.1 Optimal constants and extremal functions
    2. 2.2 First and second-order variations
    3. 2.3 Lyapunov-Schmidt reduction
  3. 3 Quantitative stability
    1. 3.1 Local quantitative stability
    2. 3.2 Proof of Theorem 1.1 and Theorem 1.2
  4. 4 Degenerate stability
    1. 4.1 Proof of Theorem 1.3
    2. 4.2 Proof of Corollary 1.4

##  1\. Introduction

In the standard round sphere $\mathbb{S}^{d}$, for $d>2$, the sharp Sobolev
inequality reads:

(1.1) $\|u\|_{L^{2^{*}}}^{2}\leq{\sf S}_{d}^{2}\|\nabla u\|_{L^{2}}^{2}+{\rm
Vol}(\mathbb{S}^{d})^{-2/d}\|u\|_{L^{2}}^{2},\qquad\forall u\in
W^{1,2}(\mathbb{S}^{d}),$

where $2^{*}:=2d/(d-2)$ is the Sobolev conjugate exponent, and ${\sf S}_{d}>0$
is the optimal constant in the sharp Euclidean Sobolev inequality

(1.2) $\|u\|_{L^{2^{*}}}\leq{\sf S}_{d}\|\nabla u\|_{L^{2}},\qquad\forall
u\in\dot{W}^{1,2}(\mathbb{R}^{d}).$

The value

${\sf S}_{d}=\left(\frac{2^{*}-2}{d}\right)^{\frac{1}{2}}{\rm
Vol}(\mathbb{S}^{d})^{-1/d},$

was found by Aubin [Aub76b] and Talenti [Tal76], while the validity of (1.1)
goes back to [Aub76a] where also non-constant _extremal functions_ were
classified (see [Heb99, Chapter 5]):

(1.3)
$u_{a,b,z_{0}}:=a(1-b\cos(d(\cdot,z_{0}))^{\frac{2-d}{2}},\qquad\text{with
}a\in\mathbb{R},\,b\in(0,1),\,z_{0}\in\mathbb{S}^{d}.$

Here $d$ is the geodesic distance on $\mathbb{S}^{d}$. In light of this, a
natural question is the one of stability:

If $u\neq cst$ nearly _saturates_ (1.1), is $u$ close to some $u_{a,b,z_{0}}$?

This question is equivalent (up to a change of coordinates via the
stereographic projection, see e.g. [FL12, Appendix A]) to the analogous
question raised in [BL85] for the Euclidean Sobolev inequality (1.2) and was
addressed in Bianchi and Egnell’s work [BE91]. The _quantitative stability_
for (1.2) initiated in loc. cit. has been then studied thoroughly in
subsequent works [CFMP09, FN19, Neu20, FZ22] dealing with $p$-Sobolev
inequalities for $p\neq 2$, and recently in [DEF+23, DEF+24] with explicit
constant (see also to the related [Kö23] and [Fra23]).

Aim of this manuscript is to study quantitative stability of optimal Sobolev
inequalities on a general closed Riemannian manifold in connection with the
existence of extremal functions. We now start by recalling the notions of
optimal constants appearing in the celebrated AB-program referring to [Heb99],
see also [Aub98, DH02a] and references therein. Given $d>2$ and $(M,g)$ a
closed, i.e. compact and boundaryless, $d$-dimensional smooth Riemannian
manifolds, we can consider for constants $A,B\geq 0$ the following type of
Sobolev inequalities

($\ast$) $\|u\|_{L^{2^{*}}}^{2}\leq A\|\nabla
u\|_{L^{2}}^{2}+B\|u\|_{L^{2}}^{2},\qquad\forall u\in W^{1,2}(M).$

Notice that an equation as (1.2) cannot hold due to the presence of constant
functions and, differently from the Euclidean case, it is not straightforward
to consider a single sharp Sobolev inequality due to the presence of two
constants. The AB-program then starts with the definition of the following
Sobolev constants:

$\begin{array}[]{cc}\alpha(M):=\inf\\{A\ :\ \eqref{eq:SobAB}\text{ holds for
some $B$}\\},&\quad\beta(M):=\inf\\{B\ :\ \eqref{eq:SobAB}\text{ holds for
some $A$}\\},\end{array}$

where we understand the infimum over an empty set as being $+\infty$. The
first natural problem is then to determine the value of $\alpha(M)$ and
$\beta(M)$. It is rather straightforward to see (e.g. [Heb99, Sec 4.1]) that
the latter satisfies

$\beta(M)={\rm Vol}_{g}(M)^{-2/d}.$

On the other hand, the value of $\alpha(M)$ turns out to be linked to the
sharp Euclidean Sobolev constant. More precisely, in [Aub76b] (see also
[Heb99, Theorem 4.5]) it is shown that

(1.4) $\alpha(M)={\sf S}_{d}^{2}.$

In particular, $\alpha(M)$ does not depend on $M$ but only on its dimension
$d$. A more subtle question is whether these two constants are actually
attained, i.e. if they are minima. For instance, for $M=\mathbb{S}^{d}$ we
have in (1.1) the validity of a Sobolev inequality with
$A=\alpha(\mathbb{S}^{d})$ and $B=\beta(\mathbb{S}^{d})$ and they are attained
by the family (1.3).

More generally, and starting from the easiest one, it was shown in [Bak94]
that $\beta(M)$ is a minimum (see also [Heb99, Theorem 4.2]), i.e. there is
$\overline{A}>0$ so that ($\ast$ ‣ 1) holds with $\overline{A}$ and
$\beta(M)$. It is instead a deep result of [HV96] that $\alpha(M)$ is
attained, i.e. there is $\overline{B}>0$ so that ($\ast$ ‣ 1) holds with
$\alpha(M)$ and $\overline{B}$. Thanks to these results, we can define two
further notions of Sobolev constants. We can fix $B=\beta(M)$ and proceed with
the $A$-part of this program, i.e. minimizing over the admissible $A>0$ and
consider more generally all sub-critical ranges

($\ast_{A}$) $\|u\|_{L^{q}}^{2}\leq A\|\nabla u\|_{L^{2}}^{2}+{\rm
Vol}_{g}(M)^{2/q-1}\|u\|_{L^{2}}^{2},$

for $q\in(2,2^{*}]$. Conversely, we can fix $A=\alpha(M)$ and proceed with the
$B$-part of this program, i.e. minimizing over the admissible $B>0$ and
consider

($\ast_{B}$) $\|u\|_{L^{2^{*}}}^{2}\leq{\sf S}_{d}^{2}\|\nabla
u\|_{L^{2}}^{2}+B\|u\|_{L^{2}}^{2}.$

Then, we can define the corresponding notions of second-best optimal constants

$A_{q}^{\rm opt}(M):=\inf\\{A>0\colon\eqref{eq:Sob_qA}\text{ holds with
}A\\},\quad B_{2^{*}}^{\rm opt}(M):=\inf\\{B>0\colon\eqref{eq:Sob_qB}\text{
holds with }B\\},$

Differently from $\alpha(M),\beta(M)$ these two constants are automatically
minima. Moreover, universal bounds depending on the geometry of $M$ are given
for $A_{q}^{\rm opt}(M)$ in [DH02b, Theorem 4.4] (see also [NV22, Proposition
5.1] for general $q$), and for $B_{2^{*}}^{\rm opt}(M)$ in [HV95].
Having established two classes of optimal Sobolev inequalities, it is natural
to investigate the existence of extremal functions. Let us consider the
following sets:

$\displaystyle\mathcal{M}_{q}(A)$ $\displaystyle:=\\{u\in
W^{1,2}\setminus\\{0\\}\colon\text{equality holds with }u\text{ in
\eqref{eq:Sob_qA} for }A=A^{\rm opt}_{q}(M)\\},$
$\displaystyle\mathcal{M}_{2^{*}}(B)$ $\displaystyle:=\\{u\in
W^{1,2}\setminus\\{0\\}\colon\text{equality holds with }u\text{ in
\eqref{eq:Sob_qB} for }B=B^{\rm opt}_{2^{*}}(M)\\}.$

The existence and compactness properties of extremal functions become
challenging in the critical case due to the apriori lack of compactness of the
embedding $W^{1,2}\hookrightarrow L^{2^{*}}$. We will see that compactness
properties are related to the values of the constants $A^{\rm
opt}_{2^{*}}(M),B^{\rm opt}_{2^{*}}(M)$. Notice however that
$\mathcal{M}_{q}(A)$ always contains constant functions thanks to the value of
the $B$-constant in ($\ast_{A}$). In particular, it is never empty as defined.
However, we do not restrict our analysis to _non-constant_ extremal functions
in this case, as these could be the only extremal functions (see [Fra22] and
below). Finally, we refer the reader to [HV01, DD01, BM07, BM09, Bar10, BM12],
where extremal functions in the AB-program have been investigated.

Statement of the main results. Here we present our main quantitative stability
results. As already observed, $\mathcal{M}_{q}(A)$ is never empty as it
contains constant functions. Also, by definition of $\alpha(M)$, we always
have $A^{\rm opt}_{2^{*}}(M)\geq{\sf S}_{d}^{2}$. However, assuming this
inequality to be strict, we gain pre-compactness of (normalized) extremizing
sequences via concentration-compactness methods in the spirit of [Lio84,
Lio85], cf. Proposition 2.1. Under this assumption, or more easily in the sub-
critical range, we can prove the following.

######  Theorem 1.1.

Let $(M,g)$ be a closed $d$-dimensional Riemannian manifolds, $d>2$. Assume
that either $2<q<2^{*}$, or $q=2^{*}$ and $A^{\rm opt}_{2^{*}}(M)>{\sf
S}_{d}^{2}$. Then, there are non-negative constants $C,\gamma$ depending on
$(M,g)$ and on $q$ so that, for every $u\in W^{1,2}(M)\setminus\\{0\\}$, it
holds

$\frac{A^{\rm opt}_{q}(M)\|\nabla u\|^{2}_{L^{2}}+{\rm
Vol}_{g}(M)^{2/q-1}\|u\|_{L^{2}}^{2}}{\|u\|_{L^{q}}^{2}}-1\geq
C\left(\inf_{v\in{\mathcal{M}}_{q}(A)}\frac{\|u-v\|_{W^{1,2}}}{\|u\|_{W^{1,2}}}\right)^{2+\gamma}.$

The above stability result in Theorem 1.1 is _strong_ , as we are able to
control the full $W^{1,2}$-distance from a set a non empty set of extremal
functions with the related Sobolev deficit. On the other hand, the constant
$C$ and the exponent $\gamma$ crucially depend on the manifold $M$ and on the
value $q$ as an outcome of the proof-strategy. A natural question at this
point is whether, on a manifold $(M,g)$, we can expect the strict binding
inequality to hold and, therefore, a critical stability result. This heavily
depends on the metric $g$, as it is known from [DD01, Corollary 2] that we
always have $A^{\rm opt}_{2^{*}}(M,h)={\sf S}_{d}^{2}$ (adding the dependence
on the metric) for a suitable metric $h$ in the conformal class of $g$.
However, for certain manifolds this scenario is very rigid (see Proposition
5.9 and Proposition 5.12 in [Heb99]). More generally, the continuity
properties studied in [NV22, Theorem 6.2] suggests that, as soon as there is
$g$ satisfying $A^{\rm opt}_{2^{*}}(M,g)>{\sf S}_{d}^{2}$, there are
infinitely many others metrics.

We pass now to our second main result, which is a weaker stability results
only around minimizer for $B^{\rm opt}_{2^{*}}(M)$. Let us first comment on
the non-emptyness of the set of extremal functions for $B^{\rm
opt}_{2^{*}}(M)$. When $d\geq 4$, work of Hebey [Heb99, Proposition 5.1]
implies

(1.5) $B^{\rm opt}_{2^{*}}(M)\geq\frac{d-2}{4(d-1)}{\sf S}_{d}^{2}\max_{M}{\rm
R}_{g},$

where ${\rm R}_{g}$ denotes the scalar curvature of $M$. When this inequality
is strict, the deep analysis of [DD01] guarantees that
$\mathcal{M}_{2^{*}}(B)\neq\varnothing$. See also Theorem 2.3 below for
further details. As constant functions might not be extremals, unless $B^{\rm
opt}_{2^{*}}(M)=\beta(M)$, this result is striking. The dimensional
restriction $d\geq 4$ is due to the need to consider a more general class of
$p$-Sobolev inequalities with $p\neq 2$ when $d=3$, see [DD01]. Under this
assumption, we prove the following.

######  Theorem 1.2.

Let $(M,g)$ be a closed $d$-dimensional Riemannian manifolds, $d\geq 4$.
Assume $B^{\rm opt}_{2^{*}}(M)>\frac{d-2}{4(d-1)}{\sf S}_{d}^{2}\max_{M}{\rm
R}_{g}$. Then, there are non-negative constants $C,\gamma,\delta$ depending on
$(M,g)$ so that it holds

${\sf S}_{d}^{2}\|\nabla u\|^{2}_{L^{2}}+B^{\rm
opt}_{2^{*}}(M)\|u\|_{L^{2}}^{2}-1\geq
C\left(\inf_{v\in{\mathcal{M}}_{2^{*}}(B)}\frac{\|u-v\|_{W^{1,2}}}{\|u\|_{W^{1,2}}}\right)^{2+\gamma},$

for all $u\in W^{1,2}(M)$ so that $\|u\|_{L^{2^{*}}}=1$ and
$\|u-v\|_{W^{1,2}}<\delta\|u\|_{W^{1,2}}$ for all
$v\in{\mathcal{M}}_{2^{*}}(B)$ with $\|v\|_{L^{2^{*}}}=1$.

The above is a weaker form of stability due to the assumption forcing $u$ to
be close to the set of extremizers. Indeed, contrary to Theorem 1.1, here we
cannot rely on a concentration-compactness principle (see Remark 3.3 for
details). Besides, the continuous dependence $g\mapsto B^{\rm
opt}_{2^{*}}(M,g)$ studied in [BM12] implies the abundance of metrics for
which (1.5) holds strict.

Given the above results, it is natural to ask if our results hold _sharp_ with
quadratic exponent, i.e. with $\gamma=0$. This is the typical desirable
feature in stability problems. For instance, in [CES23], respectively [ENS22],
the authors show that quadratic stability might fail for the quantitative
isoperimetric inequality, respectively the quantitative stability of
minimizing Yamabe metrics, on specific manifolds. This degenerate phenomenon
has been later analyzed by [Fra22] (see Corollary 1.4 below), and subsequently
in [FP24], and [BDS23, BDS24b, BDS24a] for different functional inequalities.
Our next results deal with degenerate phenomena.

######  Theorem 1.3.

Let $q\in(2,2^{*}]$ and let $(M,g)$ be satisfying the hypothesis of Theorem
1.1. Assume further that there are no non-constant extremal functions, i.e.
$\mathcal{M}_{q}(A)=\\{c\colon c\in\mathbb{R}\setminus\\{0\\}\\}.$ Then, the
stability in Theorem 1.1 is degenerate, i.e. it must hold with some
$\gamma>0$.

We point out that this result is in line with the examples studied by Frank in
[Fra22], as we prove in the next corollary. The exponent $\gamma=2$ follows
from the analysis in loc. cit.

######  Corollary 1.4.

For $d>2$, we have:

  * i)

Theorem 1.1 does not hold with $\gamma=0$ for $q\in(2,2^{*})$ and
$M=\mathbb{S}^{d}$;

  * ii)

Theorem 1.1 does not hold with $\gamma=0$ for $q=2^{*}$ and
$M=\mathbb{S}^{1}\left(\tfrac{1}{\sqrt{d-2}}\right)\times\mathbb{S}^{d-1}$.

Finally, in all of the above, $\gamma=2$ is sharp.

It is unclear whether this degenerate phenomenon can happen also in Theorem
1.2, and if $M_{*}=\mathbb{S}^{1}(1/\sqrt{d-2})\times\mathbb{S}^{d-1}$
provides again a counterexample. Moreover, we are only aware of estimates for
$B^{\rm opt}_{2^{*}}(M_{*})$ ([HV92] and [Heb99, Proposition 5.4]) and not its
explicit value.

Comparison with related works. Quantitative stability results are ubiquitous
in the literature. In the classical Euclidean setting, several geometric and
functional inequalities have been analyzed from the viewpoint of stability
with different techniques. We refer to the aforementioned references regarding
the Sobolev inequality, to the surveys [Fig13, Fus15], and the references
therein.

On the contrary, in the non-linear framework of Riemannian manifolds, we
cannot rely on underlying symmetries of any kind as, for instance, in (1.3)
for $\mathbb{S}^{d}$. In fact, our main argument follows a well-established
approach, that is to argue via a Lyapunov-Schmidt reduction argument [Sim83]
and Łojasievicz inequality [Łoj65] to produce quantitative stability
estimates. This fruitful interaction has been pionereed in [CES23] for the
quantitative stability of the Riemannian isoperimetric inequality, and it was
later employed to study stability properties of minimizing Yamabe metrics in
[ENS22] (see also the recent [CK24], we refer to [Yam60] and the surveys
[LP87, BM11] regarding the Yamabe problem). In the present work, we exploit
the similarities between the latter and our setting.

Besides degenerate phenomena, it is natural to investigate sharp stability
results with exponent $\gamma=0$ holding _generically_ in the space of
Riemannian metrics. However, in contrast with [ENS22] where this follows by
work Schoen [Sch89], see also [And05], the picture here seems to be more
delicate. First, to formulate a generic statement, one needs to check that the
strict inequalities on our optimal constants are open conditions in the set of
$C^{3}$-metrics. In this direction, recall the continuity properties of
$g\mapsto A^{\rm opt}_{q}(M,g),g\mapsto B^{\rm opt}_{2^{*}}(M,g)$ studied in
[NV22] and [BM12], respectively. Then, the key point would be to prove that
generically inside the relevant open sets of metrics, minimizers (or, more
generally, non-negative critical points) of

$\frac{A^{\rm opt}_{q}(M,g)\|\nabla u\|^{2}_{L^{2}}+{\rm
Vol}_{g}(M)^{2/q-1}\|u\|_{L^{2}}^{2}}{\|u\|_{L^{q}}^{2}}\quad\text{and}\quad\frac{{\sf
S}_{d}^{2}\|\nabla u\|^{2}_{L^{2}}+B^{\rm
opt}_{2^{*}}(M,g)\|u\|_{L^{2}}^{2}}{\|u\|_{L^{2^{*}}}^{2}}$

are non-degenerate. Notice again that the first always admits constant
minimizers while, the second does not in general, unless $B^{\rm
opt}_{2^{*}}(M,g)={\rm Vol}_{g}(M)^{2/2^{*}-1}$ (this, however, can always
happen in the conformal class of a given metric again by [DD01, Corollary 2]).
Constant critical points are typically excluded from this kind of analysis,
see e.g. [GM13, MP09]. Recall also that they can be pathological in this
regard, as discussed in Theorem 1.3 and Corollary 1.4. Finally, even
restricting our investigation to nonconstant extremal functions, it seems that
standard transversality arguments [Hen05, Whi91] would require extra
regularity on the dependence $g\mapsto A_{q}(M,g),g\mapsto B^{\rm
opt}_{2^{*}}(M,g)$.

We conclude by mentioning that assuming Ricci curvature constraints makes it
possible to investigate more sophisticated stability statements in comparison
geometry. In this fashion, we recall [CMM19] for the Lévy-Gromov isoperimetric
inequality, and [BBG85, Ber07, CMS23, FGS22] for the Lichnerowicz spectral
gap. Concerning Sobolev inequalities, _qualitative_ stability results were
deduced in [NV22, NV24] under Ricci lower bounds. The main difference with
this note (besides the qualitative/quantitative analysis and curvature bounds)
is that in [NV22, NV24] the stability properties are studied facing lack of
compactness and with explicit classes of -a priori non-extremal- functions
having the radial profile of bubbles. Besides, quantitative stability results
in the Hyperbolic space have been obtained for the isoperimetric problem in
[BDS15], and for the sharp Poincaré-Sobolev inequality in [BGKM22a, BGKM22b].

##  2\. Properties of Sobolev inequalities

###  2.1. Optimal constants and extremal functions

In this part, we prove a key pre-compactness result for normalized extremizing
sequences of $A^{\rm opt}_{2^{*}}(M)$, and mention analogous compactness
properties for extremal functions of $B^{\rm opt}_{2^{*}}(M)$. We start with
the former.

######  Proposition 2.1.

Let $(M,g)$ be a compact $d$-dimensional Riemannian manifolds, $d>2$. Assume
that either $q<2^{*}$ or

$q=2^{*}\qquad\text{and}\qquad A^{\rm opt}_{2^{*}}(M)>{\sf S}_{d}^{2}.$

Then, for every $(u_{n})\subseteq W^{1,2}(M)$ non-constant with
$\|u_{n}\|_{L^{q}}\equiv 1$ satisfying

$\frac{\|u_{n}\|_{L^{q}}-{\rm Vol}_{g}(M)^{2/q-1}\|u_{n}\|_{L^{2}}}{\|\nabla
u_{n}\|_{L^{2}}}\to A^{\rm opt}_{q}(M),\qquad\text{as }n\uparrow\infty,$

it holds up to subsequence that $u_{n}$ converges strongly in $L^{q}$ and in
$W^{1,2}$ to some $u\in\mathcal{M}_{q}(A)$ with $\|u\|_{L^{2^{*}}}=1$. In
particular, $\\{u\in\mathcal{M}_{q}(A)\colon\|u\|_{L^{q}}=1\\}$ is pre-compact
in the $W^{1,2}$-topology.

###### Proof.

Up to a not relabelled subsequence, we can assume by Rellich pre-compactness
that $u_{n}\to u$ in $L^{q}$ for every $q<2^{*}$. Moreover, by assumption, we
have for some $\delta_{n}\downarrow 0$ that

(2.1) $1=\|u_{n}\|_{L^{q}}^{2}\geq(A^{\rm opt}_{q}(M)-\delta_{n})\|\nabla
u_{n}\|_{L^{2}}^{2}+{\rm Vol}_{g}(M)^{2/q-1}\|u_{n}\|_{L^{2}}^{2},$

In particular, $u_{n}$ is $W^{1,2}$-bounded, and, up to a further subsequence,
we also have $u_{n}\rightharpoonup u$ weakly in $W^{1,2}(M)$ to some $u\in
W^{1,2}(M)$.

If $q<2^{*}$, by lower-semicontinuity of the gradient term and
$L^{2}$-convergence of $u_{n}$ to $u$, from (2.1) we reach

$1\geq A^{\rm opt}_{q}(M)\|\nabla u\|_{L^{2}}+{\rm
Vol}_{g}(M)^{2/q-1}\|u\|_{L^{2}}\geq\|u\|_{L^{q}}^{2}.$

However, $\|u\|_{L^{q}}=\lim_{n}\|u_{n}\|_{L^{q}}=1$ so that
$u\in\mathcal{M}_{q}(A)$, giving in turn that $\|\nabla
u_{n}\|_{L^{2}}\to\|\nabla u\|_{L^{2}}.$ In particular, there is convergence
of $\|u_{n}\|_{W^{1,2}}$ to $\|u\|_{W^{1,2}}$. Being $u_{n}$ already weakly
converging, this is equivalent to strong $W^{1,2}$-convergence concluding the
proof in this case.

If $q=2^{*}$ we argue by concentration-compactness. We shall use Proposition
2.2 below and make use of the assumption in this case to rule out
concentration phenomenona. If (a) in Proposition 2.2 occurs, then an analogous
argument as done after (2.1) gives the conclusion. We are left to exclude (b)
in Proposition 2.2 using the assumption in this case. By definition of
$\alpha(M)={\sf S}_{d}^{2}$ (cf. (1.4)), for every $\epsilon>0$ there is
$B_{\epsilon}>0$ so that, for every $n\in\mathbb{N}$ it is possible to write

$1=\|u_{n}\|_{L^{2^{*}}}^{2}\leq({\sf S}_{d}^{2}+\epsilon)\|\nabla
u_{n}\|_{L^{2}}^{2}+B_{\epsilon}\|u_{n}\|_{L^{2}}.$

This and the fact that $\|u_{n}\|_{L^{2}}\to 0$ guarantee that
$\liminf_{n}\|\nabla u_{n}\|_{L^{2}}>0$. Rearranging now the above with (2.1),
taking $n$ to infinity, and recalling that $u_{n}$ is $L^{2}$-converging to
zero in this case, we get

$A^{\rm opt}_{2^{*}}(M)\leq{\sf S}_{d}^{2}+\epsilon.$

We see now that for $\epsilon$ small enough we eventually reach a
contradiction with $A^{\rm opt}_{2^{*}}(M)>{\sf S}_{d}^{2}$. So (b) does not
occur and the proof is concluded also in this case.

Finally, the last statement follows easily from the above analysis. Indeed, a
sequence $(u_{n})\subseteq\\{u\in\mathcal{M}_{q}(A)\colon\|u\|_{L^{q}}=1\\}$
is either eventually constant, or not. In the first situation, the
renormalization guarantees that, up to a non-relabelled subsequence,
$u_{n}\equiv cst.$ and we are done. The second situation is instead a
particular case of what was addressed before. The proof is therefore
concluded. ∎

In what follows we prove a dichotomy result based on concentration compactness
arguments that were needed in the above proof. This was studied in [NV22,
Theorem 6.1] in the spirit of [Lio84, Lio85] but in a different setting
(converging metric measure spaces with synthetic Ricci bounds). We include
here a quick proof with the core of the argument to be self-contained.

######  Proposition 2.2.

Let $(M,g)$ be a compact and complete $d$-dimensional Riemannian manifold,
$d>2$. Suppose that it holds

(2.2) $\|u\|_{L^{2^{*}}}^{2}\leq A\|\nabla
u\|_{L^{2}}^{2}+B\|u\|_{L^{2}}^{2},\qquad\forall u\in W^{1,2}(M),$

for some constants $A,B>0$, and there exist a sequence $(u_{n})\subset
W^{1,2}(M)\setminus\\{0\\}$ satisfying

$\|u_{n}\|^{2}_{L^{2^{*}}}\geq A_{n}\|\nabla
u_{n}\|^{2}_{L^{2}}+B_{n}\|u_{n}\|^{2}_{L^{2}},$

for some sequences $A_{n}\to A$, $B_{n}\to B$. Then, setting
$\tilde{u}_{n}:=u_{n}\|u_{n}\|_{L^{2^{*}}}^{-1}$, there exists a non-relabeled
subsequence such that only one of the following holds:

  * (a)

$\tilde{u}_{n}$ converges $L^{2^{*}}$-strong to a function $u\in W^{1,2}(M)$;

  * (b)

$\|\tilde{u}_{n}\|_{L^{2}}\to 0$ and there exists $p\in M$ so that
$|\tilde{u}_{n}|^{2^{*}}{\rm Vol}_{g}\rightharpoonup\delta_{p}$ in duality
with $C(M)$.

###### Proof.

By a scaling argument, we can assume $\|u_{n}\|_{L^{2^{*}}}\equiv 1$. As
$(A_{n},B_{n})\to(A,B)$, we directly deduce that $u_{n}$ is $W^{1,2}$-bounded.
Hence, up to a not relabeled subsequence, we have that $u_{n}$ converges to
some $u\in W^{1,2}(M)$ weakly in $W^{1,2}$ and in $L^{2^{*}}$, and strongly in
$L^{2}$, to some function $u\in W^{1,2}(M)$ and, by tightness as $M$ is
compact, that $|u_{n}|^{2^{*}}{\rm Vol}_{g}\rightharpoonup\nu,$ as well as
$|\nabla u_{n}|^{2}{\rm Vol}_{g}\rightharpoonup\mu$, in duality with $C(M)$
for a probability measure $\mu$ and for a finite non-negative measure $\nu$.
We have [Lio84, Lio85] (see also the arguments [NV22, Lemma 6.6]) that there
is a countable set of indices $J$, points $(x_{j})_{j\in J}\subseteq M$ and
weights $(\mu_{j})_{j\in J},(\nu_{j})_{j\in J}\subseteq\mathbb{R}^{+}$,
satisfying

$\displaystyle\nu=|u|^{2^{*}}{\rm Vol}_{g}+\sum_{j\in
J}\nu_{j}\delta_{x_{j}},$ $\displaystyle\mu\geq|\nabla u|^{2}{\rm
Vol}_{g}+\sum_{j\in J}\mu_{j}\delta_{x_{j}},$
$\displaystyle\nu_{j}^{2/2^{*}}\leq A\mu_{j},\qquad\sum_{j\in
J}\nu_{j}^{2/2^{*}}<\infty.$

We can therefore estimate

$\displaystyle 1=\lim_{n\uparrow\infty}\|u_{n}\|^{2}_{L^{2^{*}}}$
$\displaystyle\geq\liminf_{n\uparrow\infty}A_{n}\|\nabla
u_{n}\|^{2}_{L^{2}}+B_{n}\|u_{n}\|^{2}_{L^{2}}$ $\displaystyle\geq
A\mu(M)+B\|u\|_{L^{2}}^{2}$ $\displaystyle\geq A\|\nabla
u\|_{L^{2}}^{2}+\sum_{j\in J}A\mu_{j}+B\|u\|_{L^{2}}^{2}$
$\displaystyle\geq\Big{(}\int|u|^{2^{*}}d{\rm
Vol}_{g}\Big{)}^{2/2^{*}}+\sum_{j\in J}\nu_{j}^{2/2^{*}}$
$\displaystyle\geq\Big{(}\int|u|^{2^{*}}d{\rm Vol}_{g}+\sum_{j\in
J}\nu_{j}\Big{)}^{2/2^{*}}=\nu(M)^{2/2^{*}}=1,$

having used that $u$ satisfies (2.2) with $A,B$, the properties of $\mu,\nu$,
as well as the fact that $t\mapsto t^{2/2^{*}}$ is strictly concave. However,
since all inequalities must be equalities, we have that either
$\|u\|_{L^{2^{*}}}=1$ and all $\nu_{j}=0$ for every $j\in J$, or $u=0$ and
there is $j_{0}\in J$ so that $\nu_{j_{0}}=1$ and $\nu_{j}=0$ for all $j\neq
j_{0}$. The first case is precisely conclusion (a), as weak convergence in
$L^{2^{*}}$ together with convergence of the $L^{2^{*}}$ norms implies strong
convergence. If instead $u=0$, then (b) occurred implying at the same time
that $u_{n}$ is $L^{2}$-converging to zero, and $|u_{n}|^{2^{*}}{\rm Vol}_{g}$
is weakly converging to a Dirac mass centred at $p=x_{j_{0}}$. Having analysed
all the possibilities, the proof is now concluded. ∎

We now discuss the case of $B^{\rm opt}_{2^{*}}(M)$. Recall that thanks to
[Heb99, Proposition 5.1], we have the lower bound (1.5). Notice that, even
arguing with Proposition 2.2 for minimizing sequences of $B^{\rm
opt}_{2^{*}}(M)$, we cannot exclude the outcome (b), as this time $A={\sf
S}_{d}^{2}$ and the convergence to zero of the $L^{2}$-norm makes it
impossible to exploit (1.5). However, sufficient criteria guaranteeing that
$\mathcal{M}_{2^{*}}(B)$ is not empty have been analysed in [DD01, Theorem 1
and Corollary 1].

######  Theorem 2.3.

Let $(M,g)$ be a $d$-dimensional compact Riemannian manifold with $d\geq 4$.
Then, at least one of the following is true

  * i)

$\mathcal{M}_{2^{*}}(B)\neq\varnothing$;

  * ii)

$B^{\rm opt}_{2^{*}}(M)=\frac{d-2}{4(d-1)}{\sf S}_{d}^{2}\max_{M}{\rm R}_{g}.$

In particular, if strict inequality holds in (1.5), then
$\mathcal{M}_{2^{*}}(B)\neq\varnothing.$

In lower dimensions, a related result was studied in [DD01] by considering
$p$-Sobolev inequalities with $p\neq 2$. As we stick to $p=2$ in this note, we
do not pursue this any further. The above is a dichotomy result achieved via a
delicate study of blow-up phenomena of concentrating sequences of solutions of
partial differential equations. In particular, strict inequality in (1.5)
guarantees compactness. More generally, we recall that
$\\{u\in\mathcal{M}_{2^{*}}(B)\colon\|u\|_{L^{2^{*}}}=1\\}$ is pre-compact in
$L^{2^{*}}$-topology if strict inequality holds in (1.5). This result is
commonly attributed to the analysis contained in [DD01], see the introduction
in [BM12], as well as [BM12, Corollary 1.4] for generalizations that, as a by-
product, imply this fact. Therefore, arguing exactly as done in Proposition
2.1, we deduce

(2.3) $\\{u\in\mathcal{M}_{2^{*}}(B)\colon\|u\|_{L^{2^{*}}}=1\\}\text{ is
$W^{1,2}$ pre-compact, if $d\geq 4$ and \eqref{eq:B lower bound} holds
strict}.$

Let us discuss a relevant class of manifolds when the above discussion
applies. Notice that i) must occur in Theorem 2.3 if either ${\rm R}_{g}\leq
0$, or ${\rm R}_{g}$ is constant. The first case being clear, we discuss the
second one. The case where $(M,g)$ is conformally equivalent to the round
sphere has been fully investigated in [Heb99, Theorem 5.7]. In the remaining
case, the resolution of the Yamabe problem [Yam60] by combination of [Tru68,
Aub76a, Sch84] guarantees that

$Y(M)<{\sf S}_{d}^{-2},$

having denoted $Y(M)$ the Yamabe constant of $(M,g)$ that is attained by some
$u\in W^{1,2}(M)$

$\|u\|_{L^{2^{*}}}^{2}Y(M)=\|\nabla u\|_{L^{2}}^{2}+\frac{d-2}{4(d-1)}{\rm
R_{g}}\|u\|_{L^{2}},$

having used that ${\rm R}_{g}$ is constant. Thus, there holds

$\displaystyle\|u\|_{L^{2}}^{2}\frac{B_{2^{*}}^{\rm opt}(M)}{{\sf
S}_{d}^{2}}+\|\nabla u\|_{L^{2}}^{2}$ $\displaystyle\geq\|u\|_{L^{2^{*}}}{\sf
S}_{d}^{-2}>\|u\|_{L^{2^{*}}}Y(M)=\|\nabla
u\|_{L^{2}}^{2}+\frac{d-2}{4(d-1)}{\rm R_{g}}\|u\|_{L^{2}}.$

Rearranging, we get that ii) cannot occur.

###  2.2. First and second-order variations

In this part, we study functional properties of abstract Sobolev quotients.
This investigation is analogous to [ENS22, Section 2]. We start setting up
some notation. We fix $q\in(2,2^{*}]$. We omit the dependence on it of various
quantities to lighten the notation. For $\delta>0$, set

$\mathcal{B}:=\\{u\in W^{1,2}(M)\colon u\geq
0,\|u\|_{L^{q}}=1\\},\qquad\mathcal{B}_{\delta}(v):=\\{u\in\mathcal{B}\colon\|u-v\|_{W^{1,2}}\leq\delta\\}.$

Arguing as in [ENS22, Lemma 2.1], $\mathcal{B}\subseteq W^{1,2}$ is a Banach
submanifold and the tangent space at $u\in\mathcal{B}$ is given by

$T_{u}\mathcal{B}:=\Big{\\{}v\in W^{1,2}(M)\colon\int u^{q-1}v\,d{\rm
Vol}_{g}=0\Big{\\}}.$

Since $W^{1,2}(M)$ is a Hilbert space, we denote by $\pi_{T_{u}\mathcal{B}}$
the $L^{2}$-projection operator defined by

$\pi_{T_{u}\mathcal{B}}(\varphi):=\varphi-\Big{(}\int u^{q-1}\varphi\,d{\rm
Vol}_{g}\Big{)}u.$

We compute now the first and second variations of the functional arising by
the Sobolev inequality (cf. [ENS22, Lemma 2.1]).

######  Lemma 2.4.

Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold for $d>2$. Set
for $u\in W^{1,2}(M)\setminus\\{0\\}$ and $A,B>0,q\in(2,2^{*}]$:

$\mathcal{Q}_{q}(u):=\frac{A\|\nabla
u\|_{L^{2}}^{2}+B\|u\|_{L^{2}}^{2}}{\|u\|_{L^{q}}^{2}}.$

For $u\in\mathcal{B}$, the first and second variation of $\mathcal{Q}_{q}$ at
$u$ restricted to $T_{u}\mathcal{B}$ are denoted by
$\nabla_{\mathcal{B}}\mathcal{Q}_{q}(u),\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(u)$
and are respectively given by

$\nabla_{\mathcal{B}}\mathcal{Q}_{q}(u)[\varphi]=2\int A\nabla
u\nabla\varphi+Bu\varphi\,d{\rm Vol}_{g},$

and

$\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(u)[\varphi,\eta]=2\int
A\nabla\varphi\nabla\eta+B\varphi\eta\,d{\rm
Vol}_{g}-2(q-1)\mathcal{Q}_{q}(u)\int u^{q-2}\varphi\eta\,d{\rm Vol}_{g},$

for all $\varphi,\eta\in W^{1,2}(M)$ (omitting the projections
$\pi_{T_{u}\mathcal{B}}$ inside the integrals).

Moreover, the following properties hold:

  * i)

$w\mapsto\frac{\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(w)[\varphi,\cdot]}{\|\varphi\|_{C^{2,\alpha}}}$
is continuous from $C^{2,\alpha}\cap\mathcal{B}\to C^{0,\alpha}$ with uniform
modulus of continuity over $\varphi\in C^{2,\alpha}$;

  * ii)

$w\mapsto\frac{\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(w)[\varphi,\eta]}{\|\varphi\|_{W^{1,2}}\|\eta\|_{W^{1,2}}}$
is continuous from $\mathcal{B}\to\mathbb{R}$ with uniform modulus of
continuity over $\varphi,\eta\in W^{1,2}$.

###### Proof.

We fix any $q\in(2,2^{*}]$ and $u\in\mathcal{B}$, and consider any $\varphi\in
C^{\infty}(M),\epsilon\in(-1,1)$. Set

$u_{\epsilon}:=(u+\epsilon\varphi)/I_{\epsilon},\qquad
I_{\epsilon}:=\|u+\epsilon\varphi\|_{L^{q}}.$

We claim that $u_{\epsilon}$ is eventually for $|\epsilon|$ small, well
defined. Indeed, from the estimate $\big{|}|x+\epsilon
y|^{q}-|x|^{q}\big{|}\leq q|\epsilon y|\big{|}|x+\epsilon
y|^{q-1}+|x|^{q-1}\big{|}$ together with the fact that $u,\varphi\in L^{q}$,
we reach that $I_{\epsilon}\to 1$ as $\epsilon\to 0$. For future use, we
notice

(2.4) $\lim_{\epsilon\to
0}\frac{1-I_{\epsilon}^{2}}{\epsilon}=\frac{2}{q}\Big{(}\int|u|^{q}\,d{\rm
Vol}_{g}\Big{)}^{\frac{2}{q}-1}\lim_{\epsilon}\frac{\int|u|^{q}-|u+\epsilon\varphi|^{q}\,d{\rm
Vol}_{g}}{\epsilon}=-2\int u^{q-1}\varphi\,d{\rm Vol}_{g},$

having used dominated convergence theorem combined with the same estimate as
before for the last equality (recall also that $\|u\|_{L^{q}}=1$). We start
therefore computing the first variation

$\nabla\mathcal{Q}_{q}(u)[\varphi]:=\lim_{\epsilon\to
0}\frac{\mathcal{Q}_{q}(u_{\epsilon})-\mathcal{Q}_{q}(u)}{\epsilon}.$

By polarization, we derive

$\displaystyle\frac{\mathcal{Q}_{q}(u_{\epsilon})-\mathcal{Q}_{q}(u)}{\epsilon}$
$\displaystyle=\frac{\int
A|\nabla(u+\epsilon\varphi)|^{2}+B|u+\epsilon\varphi|^{2}\,d{\rm
Vol}_{g}}{\epsilon I^{2}_{\epsilon}}-\frac{\int A|\nabla
u|^{2}+B|u|^{2}\,d{\rm Vol}_{g}}{\epsilon}$
$\displaystyle=\frac{1}{\epsilon}\Big{[}\frac{1}{I_{\epsilon}^{2}}-1\Big{]}\mathcal{Q}_{q}(u)+\frac{2}{I_{\epsilon}^{2}}\int
A\nabla u\nabla\varphi+Bu\varphi\,d{\rm Vol}_{g}$
$\displaystyle\qquad+\frac{\epsilon}{I_{\epsilon}^{2}}\int
A|\nabla\varphi|^{2}+B\varphi^{2}\,d{\rm Vol}_{g}.$

Hence, recalling that $I_{\epsilon}\to 1$ (2.4), we deduce

$\nabla\mathcal{Q}_{q}(u)[\varphi]=2\int A\nabla
u\cdot\nabla\varphi+Bu\varphi\,d{\rm Vol}_{g}-2\mathcal{Q}_{q}(u)\int
u^{q-1}\varphi\,d{\rm Vol}_{g}.$

So, the restriction to $T_{u}\mathcal{B}$ satisfies

$\nabla_{\mathcal{B}}\mathcal{Q}_{q}(u)[\varphi]=2\int A\nabla
u\cdot\nabla\pi_{T_{u}\mathcal{B}}(\varphi)+Bu\pi_{T_{u}\mathcal{B}}(\varphi)\,d{\rm
Vol}_{g}.$

We compute now the second variation for

$\nabla^{2}\mathcal{Q}_{q}(u)[\varphi,\varphi]:=\frac{d^{2}}{d\epsilon^{2}}\Big{|}_{\epsilon=0}\mathcal{Q}_{q}(u_{\epsilon}).$

Starting from the computations already performed for the first variation, we
have

$\displaystyle\frac{\mathcal{Q}_{q}(u_{\epsilon})-2\mathcal{Q}_{q}(u)+\mathcal{Q}_{q}(v_{-\epsilon})}{\epsilon^{2}}$
$\displaystyle=\frac{1}{\epsilon^{2}}\Big{(}\frac{1}{I^{2}_{\epsilon}}-2+\frac{1}{I^{2}_{-\epsilon}}\Big{)}\mathcal{Q}_{q}(u)$
$\displaystyle\qquad+\frac{2}{\epsilon}\Big{(}\frac{1}{I^{2}_{\epsilon}}-\frac{1}{I^{2}_{-\epsilon}}\Big{)}\int
A\nabla u\nabla\varphi+Bu\varphi\,d{\rm Vol}_{g}$
$\displaystyle\qquad+\Big{(}\frac{1}{I^{2}_{\epsilon}}+\frac{1}{I^{2}_{-\epsilon}}\Big{)}\int
A|\nabla\varphi|^{2}+B\varphi^{2}\,d{\rm Vol}_{g}.$

Hence, taking the limit, we reach

$\displaystyle\nabla^{2}\mathcal{Q}_{q}(u)[\varphi,\varphi]$
$\displaystyle=-2(q-1)\mathcal{Q}_{q}(u)\int u^{q-2}\varphi^{2}\,d{\rm
Vol}_{g}$ $\displaystyle\qquad+8\Big{(}\int u^{q-1}\varphi\,d{\rm
Vol}_{g}\Big{)}\int A\nabla u\cdot\nabla\varphi+Bu\varphi\,d{\rm Vol}_{g}$
$\displaystyle\qquad+2\int A|\nabla\varphi|^{2}+B\varphi^{2}\,d{\rm Vol}_{g}.$

So, the restriction to $T_{u}\mathcal{B}$ reads

$\displaystyle\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(u)[\varphi,\varphi]$
$\displaystyle=2\int
A|\nabla\pi_{T_{u}\mathcal{B}}(\varphi)|^{2}+B\pi_{T_{u}\mathcal{B}}(\varphi)^{2}\,d{\rm
Vol}_{g}$ $\displaystyle\qquad-2(q-1)\mathcal{Q}_{q}(u)\int
u^{q-2}\pi_{T_{u}\mathcal{B}}(\varphi)^{2}\,d{\rm Vol}_{g}.$

Recalling that $\mathcal{Q}_{q}(1)=cst.,$ the desired second variation formula
hold for all couples $\varphi,\eta\in W^{1,2}(M)$ by polarization.

We now verify i). For every $w\in C^{2,\alpha}$, we define
$L_{w}(\varphi):=-2A\Delta\varphi+B\varphi-2(q-1)\mathcal{Q}_{q}(w)w^{q-2}\varphi$.
Then, for every $w,v\in C^{2,\alpha}$ we have for some $C>0$

$\|L_{w}(\varphi)-L_{v}(\varphi)\|_{C^{0,\alpha}}\leq
C\big{(}\|w^{q-2}\varphi\|_{C^{0,\alpha}}|\mathcal{Q}_{q}(w)-\mathcal{Q}_{q}(v)|+\|\varphi|w^{q-2}-v^{q-2}|\|_{C^{0,\alpha}}\big{)}.$

Hence, taking into account the continuity of $t\mapsto t^{q-2}$ and that of
$w\mapsto\mathcal{Q}_{q}(w)$ in $C^{2,\alpha}$, it holds

$\|L_{w}(\varphi)-L_{v}(\varphi)\|_{C^{0,\alpha}}\leq
f(\|w-v\|_{C^{2,\alpha}})\|\varphi\|_{C^{2,\alpha}},$

for some modulus of continuity $t\mapsto f(t)$ with $f(t)\downarrow 0$ as
$t\downarrow 0$ uniform on $\varphi$. To prove i), we thus need to prove
continuity properties of $w\mapsto\pi_{T_{w}\mathcal{B}}$ as a map from
$C^{2,\alpha}\cap\mathcal{B}\to\mathcal{L}(C^{2,\alpha},C^{2,\alpha})$. This
is implied by the estimate

$\big{|}\pi_{T_{w}\mathcal{B}}(\varphi)-\pi_{T_{v}\mathcal{B}}(\varphi)\big{|}=\Big{|}w-v+\int(v^{q-1}-w^{q-1})\varphi\,d{\rm
Vol}_{g}\Big{|}\leq C\|w-w\|_{C^{0,\alpha}}\|\varphi\|_{C^{2,\alpha}}.$

We now prove ii) and conclude. For every $\varphi\in W^{1,2}(M)$, and
interpreting by slight abuse of notation $L_{w}(\varphi)$ in distributional
sense, we estimate similarly as before

$\big{\|}L_{w}(\varphi)-L_{v}(\varphi)\big{\|}_{H^{-1}}\leq
C\big{(}\|w^{q-2}\varphi\|_{H^{-1}}|\mathcal{Q}_{q}(w)-\mathcal{Q}_{q}(v)|+\|\varphi|w^{q-2}-v^{q-2}|\|_{H^{-1}}\big{)}.$

Then, by Hölder and the Sobolev inequality, we get

$\|w^{q-2}\varphi\|_{H^{-1}}\leq\|w^{q-2}\|_{L^{q/(q-2)}}\|\varphi\|_{L^{q}}\leq
C\|w\|_{L^{q}}^{q-2}\|\varphi\|_{W^{1,2}},$

for some $C>0$ and, using $|a-b|^{\frac{1}{q-2}}\leq
c_{q}|a^{\frac{1}{q-2}}-b^{\frac{1}{q-2}}|$, we estimate similarly

$\|\varphi|w^{q-2}-v^{q-2}|\|_{H^{-1}}\leq
c_{q}\|w-v\|_{L^{q}}^{q-2}\|\varphi\|_{L^{q}}\leq
f\big{(}\|w-v\|_{W^{1,2}}\big{)}\|\varphi\|_{W^{1,2}},$

for some modulus of continuity $t\mapsto f(t)$ with $f(t)\downarrow 0$ as
$t\downarrow 0$ uniform on $\varphi$. Recall also that
$\mathcal{Q}_{q}(\cdot)$ is continuous in $W^{1,2}$. From here, the conclusion
follows by continuity properties of $w\mapsto\pi_{T_{w}\mathcal{B}}$ as a
mapping $\mathcal{B}\to\mathcal{L}(W^{1,2},W^{1,2})$. ∎

We also have the crucial property that $\mathcal{Q}_{q}(\cdot)$ is analytic in
the sense of [CCR15, Lemma 6] whose proof carries over in our setting.

######  Proposition 2.5.

Under the same assumptions of Lemma 2.4, the map $u\mapsto\mathcal{Q}_{q}(u)$
is analytic on $\\{u\in C^{2,\alpha}(M),u>0\\}$, in the sense that for every
$u_{0}\in C^{2,\alpha}(M)$ with $u_{0}>0$, there is $\epsilon>0$, and bounded
multilinear operators $\mathcal{Q}_{q}^{(k)}\colon C^{2,\alpha}(M)^{\times
k}\rightarrow\mathbb{R}$ for every $k\geq 0$, such that if
$\|u-u_{0}\|_{C^{2,\alpha}(M)}<\epsilon$, then

$\sum_{k=0}^{\infty}\|\mathcal{Q}_{q}^{(k)}\|\cdot\|u-u_{0}\|_{C^{2,\alpha}(M)}^{k}<\infty,$

as well as

$\mathcal{Q}_{q}(u)=\sum_{k=0}^{\infty}\mathcal{Q}_{q}^{(k)}(u-u_{0},u-u_{0},\ldots,u-u_{0}),\qquad\text{in
}C^{2,\alpha}(M).$

###  2.3. Lyapunov-Schmidt reduction

The aim of this section is to discuss the Lyapunov-Schmidt reduction
associated with a general Sobolev quotient that, intuitively, reduces the
infinite dimensional problem to a more manageable finite dimensional one.

Suppose that

(2.5) $\inf_{u\in W^{1,2}(M)\setminus\\{0\\}}\mathcal{Q}_{q}(u)=\inf_{u\in
W^{1,2}(M)\setminus\\{0\\}}\frac{A\|\nabla
u\|_{L^{2}}^{2}+B\|u\|_{L^{2}}^{2}}{\|u\|_{L^{q}}^{2}}=1.$

for $A,B>0$ and $q\in(2,2^{*}]$ either as in Theorem 1.1 or Theorem 1.2. We
shall not specify their value to unify the discussion here. Consider the set
of normalized extremal functions for (2.5)

$\mathcal{M}^{1}:=\\{u\in\mathcal{B}\colon\mathcal{Q}_{q}(u)=1\\},$

and the set of normalized critical points $\mathcal{CP}^{1}\subseteq
W^{1,2}(M)$ as the collection of $u\in\mathcal{B}$ so that

$\int_{M}A\nabla u\cdot\nabla\varphi+Bu\varphi\,d{\rm
Vol}_{g}=\mathcal{Q}_{q}(u)\int_{M}u^{q-1}\varphi\,d{\rm
Vol}_{g},\qquad\forall\varphi\in C^{\infty}(M).$

The sets $\mathcal{M}^{1}$ and $\mathcal{CP}^{1}$ depend on $q,A,B$ and
coincide with those discussed in the Introduction. Later, in the main proofs,
these constants will be chosen. Furthermore, we can regard
$\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(u)[\cdot,\cdot]$ as a mapping
$T_{u}\mathcal{B}\subseteq W^{1,2}(M)\to H^{-1}(M)$. Since the operator
associated with this mapping is elliptic on a compact manifold, we know that

$K:={\rm Ker}\,\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(u),$

has finite dimension, say dim$(K)=l<\infty$. We then denote by $K^{\perp}$ its
$L^{2}$-orthogonal complement in $W^{1,2}(M)$. Given the properties of
$\mathcal{Q}_{q}(\cdot)$, a proof of the following result follows by [Sim96,
Section 3.12], as well as [ENS22, Appendix A] replacing the use of [ENS22,
Lemma 2.1] with Lemma 2.4. Recall also that any $v\in\mathcal{M}^{1}$ is
$C^{2,\alpha}$-regular and it s either strictly positive or negative, by
elliptic regularity (see, e.g., [Dru00]). We also refer the reader to the
instructive finite dimensional version for smooth functions in [Sim96,
Appendix 3.16.1].

######  Proposition 2.6 (Lyapunov-Schmidt reduction).

Let $(M,g)$ be a $d$-dimensional compact Riemannian manifold, $d>2$ with a
$C^{3}$ metric. Suppose that (2.5) holds with $A,B>0$ and $q\in(2,2^{*}]$. Fix
$v\in\mathcal{M}^{1}$. There are an open neighborhood $U$ of $0$, such that
$U\subset K$, and a map $F\colon U\rightarrow K^{\perp}$ satisfying $F(0)=0$,
and $\nabla F(0)=0$ such that the following hold.

  * i)

Let $\mathfrak{q}\colon U\rightarrow\mathbb{R}$ be the analytic function
defined by $\mathfrak{q}(\varphi)=\mathcal{Q}_{q}(v+\varphi+F(\varphi))$. Then

$\mathcal{L}:=\\{v+\varphi+F(\varphi);\;\varphi\in U\\}\subset\mathcal{B},$

and

$\nabla_{\mathcal{B}}\mathcal{Q}_{q}(v+\varphi+F(\varphi))=\pi_{K}\nabla_{\mathcal{B}}\mathcal{Q}_{q}(v+\varphi+F(\varphi))=\nabla\mathfrak{q}(\varphi).$

Furthermore, $\varphi\mapsto\mathfrak{q}(\varphi)$ is real analytic;

  * ii)

There exists $\delta>0$, depending on $v$ such that for any
$u\in\mathcal{B}_{\delta}(v)$, we have $\pi_{K}(u-v)\in U$. Furthermore, if
$u\in\mathcal{CP}^{1}\cap\mathcal{B}_{\delta}(v)$, then

$u=v+\pi_{K}(u-v)+F(\pi_{K}(u-v));$

  * iii)

There exists $C>0$, such that for all $\varphi\in U$ and $\eta\in K$, we have
the bound

$\|\nabla F(\varphi)[\eta]\|_{C^{2,\alpha}}\leq C\|\eta\|_{C^{0,\alpha}}.$

##  3\. Quantitative stability

Aim of this section is to prove first a local quantitative stability, and then
pass to the proof of our main quantitative stability results.

###  3.1. Local quantitative stability

We shall continue assuming here the validity of (2.5), without specifying the
value of $A,B>0$ and $q\in(2,2^{*}]$. Recall now $K,F$ from Proposition 2.6
and the definitions of $\mathcal{M}^{1}$ and $\mathcal{CP}^{1}$ given above.

Associated to the Lyapunov-Schmidt reduction, we have the following notions
([AS88] see also [CCR15]). For $\delta>0,v\in\mathcal{M}^{1}$ given by
Proposition 2.6 and $u\in\mathcal{B}_{\delta}(v)$, we define the _Lyapunov-
Schmidt projection operator_

$u_{\mathcal{L}}:=v+\pi_{K}(u-v)+F(\pi_{K}(u-v)).$

We can thus distinguish between the following notions:

  * •

We say that $v$ is _non-degenerate_ if ${\rm
Ker}\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(v)=\\{0\\}$. Hence,
$u_{\mathcal{L}}=v$ in this case for every $u\in\mathcal{B}_{\delta}(v)$;

  * •

We say that $v\in\mathcal{CP}^{1}$ is _integrable_ , provided that for every
$\varphi\in{\rm Ker}\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(v)$, there is a
one-parameter family of functions $(v_{t})_{t\in(-\delta,\delta)}$ with
$v_{0}=v$, and $\frac{dv_{t}}{dt}|_{t=0}=\varphi$, as well as
$v_{t}\in\mathcal{CP}^{1}$ for all $t$ sufficiently small. Conversely, we say
that $v$ is non _non-integrable_ if this property does not hold.

Arguing exactly as in [ENS22, Lemma 2.4] in the present setting, we have that
if $v$ is integrable, then $\mathfrak{q}$ as given in Proposition 2.6 is
constant in a neighborhood of $\\{0\\}\in K$, and when $v$ is an integrable
minimizer also

(3.1) $\mathcal{M}^{1}\cap\mathcal{B}_{\delta}(v)=\mathcal{L},$

where $\mathcal{L}$ is the set given in Proposition 2.6. We define, for
$v\in\mathcal{M}^{1}$ and $\delta>0$, the local distance

$d_{\delta}(u,\mathcal{M}^{1}):=\inf\left\\{\frac{\|u-\tilde{v}\|_{W^{1,2}}}{\|u\|_{W^{1,2}}}\colon\tilde{v}\in\mathcal{M}^{1}\cap\mathcal{B}_{\delta}(v)\right\\}.$

Recall also the following inequality due to [Łoj65].

######  Proposition 3.1 (Łojasievicz inequality).

Let $\mathfrak{q}\colon\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a real analytic
map and assume $\nabla\mathfrak{q}(0)=0$. Then, there exist $\delta>0$, $C>0$,
and $\gamma>0$ depending on $\mathfrak{q}$ and the critical point $0$, such
that for all $z\in B(0,\delta)$, it holds

$|\mathfrak{q}(z)-\mathfrak{q}(0)|\geq
C\left(\inf\\{|\varphi-\overline{\varphi}|;\;\overline{\varphi}\in
B_{\delta}(0),\nabla\mathfrak{q}(\overline{\varphi})=0\\}\right)^{2+\gamma}.$

We can prove now the main local stability estimate.

######  Proposition 3.2 (Local quantitative stability).

Let $(M,g)$ be a closed $d$-dimensional Riemannian manifold, $d>2$. Suppose
that (2.5) holds with $A,B>0$ and $q\in(2,2^{*}]$. Fix $v\in\mathcal{M}^{1}$.
Then, there exist positive constants $C,\gamma,\delta$ depending on $v$ such
that

$\mathcal{Q}_{q}(u)-1\geq
Cd_{\delta}(u,\mathcal{M}^{1})^{2+\gamma},\qquad\forall
u\in\mathcal{B}_{\delta}(v).$

Moreover, if $v$ is integrable or non-degenerate, we may take $\gamma=0$.

###### Proof.

In this proof, $C$ is a general positive constant depending on $v$ (in
particular on $(M,g)$ and the value of $q$) whose value might change from line
to line.

Taking $\delta>0$ in Proposition 2.6 smaller if necessary, we can ensure that
for any $\epsilon>0$, we have $\|u_{\mathcal{L}}-u\|_{W^{1,2}}\leq\epsilon$,
as well as $\|u_{\mathcal{L}}-v\|_{W^{1,2}}\leq\epsilon$. In particular, we
can write

$\mathcal{Q}_{q}(u)-1=(\mathcal{Q}_{q}(u)-\mathcal{Q}_{q}(u_{\mathcal{L}}))+(\mathcal{Q}_{q}(u_{\mathcal{L}})-1)=:{\rm
I}+{\rm II}.$

We analyse the two term separately. For the first term, write
$u=u_{\mathcal{L}}+u^{\perp}$, and Taylor expand the functional to get

(3.2) $\displaystyle\begin{split}{\rm
I}&=\nabla_{\mathcal{B}}\mathcal{Q}_{q}(u_{\mathcal{L}})[u^{\perp}]+\frac{1}{2}\nabla_{\mathcal{B}}^{2}\mathcal{Q}_{q}(\zeta)[u^{\perp},u^{\perp}]\\\
&=\frac{1}{2}\nabla_{\mathcal{B}}^{2}\mathcal{Q}_{q}(v)[u^{\perp},u^{\perp}]+o(1)\|u^{\perp}\|_{W^{1,2}}^{2}\geq
C\lambda_{1}\|u^{\perp}\|_{W^{1,2}}^{2}=C\|u-u_{\mathcal{L}}\|_{W^{1,2}}^{2},\end{split}$

where $\lambda_{1}$ is the smallest non-zero eigenvalue of the Hessian of
$\mathcal{Q}_{q}$. Here $\zeta$ is a geodesic point in $\mathcal{B}$ between
$u$ and $u_{\mathcal{L}}$ and $o(1)$ is a quantity that goes to zero as
$\|u-v\|_{W^{1,2}}$ goes to zero. For the second equality, we used the fact
that $\nabla_{\mathcal{B}}\mathcal{Q}_{q}(u_{\mathcal{L}})[u^{\perp}]=0$, as
well as continuity of the Hessian, cf. Lemma 2.4 property (ii). The last
inequality holds by taking $\delta>0$ small enough.

We can now turn to the second term $\mathrm{II}$ specializing the discussion
depending on $v$ being non-degenerate, integrable or non-integrable. First, if
$v$ is non-degenerate, we have $u_{\mathcal{L}}=v$, so II $=0$ and (3.2)
immediatly yields the conclusion. When $v$ is integrable, we infer
$\mathcal{Q}_{q}(u_{\mathcal{L}})=\mathfrak{q}(0)=Q_{q}(v)$, where the first
equality follows from $\mathfrak{q}$ being constant in a neighborhood of the
origin (cf. above (3.1)). In particular, also in this situation II $=0$ and
since $u_{\mathcal{L}}\in\mathcal{M}^{1}\cap B_{\delta}(v)$ the estimate (3.2)
combined with the simple observation

$\|u-u_{\mathcal{L}}\|_{W^{1,2}}^{2}\geq\left(\inf\big{\\{}\|u-\tilde{v}\|_{W^{1,2}}\colon\tilde{v}\in\mathcal{M}^{1}\cap
B_{\delta}(v)\big{\\}}\right)^{2},$

immediately give the conclusion. It remains to handle the case when $v$ is
non-integrable. Denoting $\varphi=\pi_{K}(u-v)$, we deduce

$\displaystyle{\rm II}=\mathfrak{q}(\varphi)-\mathfrak{q}(0)$
$\displaystyle\geq
C\left(\inf\\{|\varphi-\overline{\varphi}|;\;\overline{\varphi}\in K\cap
B_{\delta}(0),\nabla\mathfrak{q}(\overline{\varphi})=0\\}\right)^{2+\gamma}$
$\displaystyle\geq
C\inf\\{\|u_{\mathcal{L}}-\tilde{v}\|_{W^{1,2}};\;\tilde{v}\in\mathcal{M}^{1}\cap\mathcal{B}_{\delta}(v)\\}^{2+\gamma},$

thanks to Proposition 3.1, and where we used the second and third properties
of Proposition 2.6, while for the last inequality we argued as in [ENS22, page
405]. Thus, the conclusion of the proof follows by noticing

$\displaystyle\mathcal{Q}_{q}(u)-1$ $\displaystyle\geq
C\|u-u_{\mathcal{L}}\|_{W^{1,2}}^{2}+C\inf\\{\|u_{\mathcal{L}}-\hat{v}\|_{W^{1,2}};\;\hat{v}\in\mathcal{M}^{1}\cap\mathcal{B}_{\delta}(v)\\}^{2+\gamma}$
$\displaystyle\geq
C\left(\|u-u_{\mathcal{L}}\|_{W^{1,2}}+\inf\\{\|u_{\mathcal{L}}-\hat{v}\|_{W^{1,2}};\;\hat{v}\in\mathcal{M}^{1}\cap\mathcal{B}_{\delta}(v)\\}\right)^{2+\gamma}$
$\displaystyle\geq
C\left(\inf\big{\\{}\|u-\tilde{v}\|_{W^{1,2}}\colon\tilde{v}\in\mathcal{M}^{1}\cap
B_{\delta}(v)\big{\\}}\right)^{2+\gamma}.$

∎

###  3.2. Proof of Theorem 1.1 and Theorem 1.2

We start with the proof of Theorem 1.1 and then explain how to modify it to
obtain Theorem 1.2.

Fix any $q\in(2,2^{*}]$ and set here $A=A^{\rm opt}_{q}(M)$ and $B={\rm
Vol}_{g}(M)^{2/q-1}$. We claim that it is enough to prove Theorem 1.1 for
every $u\in\mathcal{B}=\\{u\in W^{1,2}(M)\colon\|u\|_{L^{q}}=1\\}$. Indeed,
the main estimate is zero-homogeneous in $u$ and

$\inf\\{\|u-v\|_{W^{1,2}}\colon
v\in\mathcal{M}_{q}(A)\cap\mathcal{B}\\}\geq\inf\left\\{\frac{\|u-v\|_{W^{1,2}}}{\|u\|_{W^{1,2}}}\colon
v\in\mathcal{M}_{q}(A)\right\\}=:d(u,\mathcal{M}_{q}(A)).$

Thus, fix $v\in\mathcal{M}_{q}(A)$ with $\|v\|_{L^{q}(M)}=1$. Let us invoke
Proposition 3.2 with $\mathcal{M}^{1}=\mathcal{M}_{q}(A)\cap\mathcal{B}$ to
get the existence of constants $\gamma(v),\delta(v)$, and $C(v)$. In
particular, by compactness of $\mathcal{M}_{q}(A)\cap\mathcal{B}$, c.f.
Proposition 2.1, we can take a finite subcover (indexed by $I$) of the open
cover of balls $\mathcal{B}(v,\delta(v)/2)$. Notice that, in the case
$q=2^{*}$, our extra assumption on $A^{\rm opt}_{2^{*}}(M)$ plays a crucial
role. Set now

$\delta_{0}=\min_{I}\delta(v_{i})/2,\quad\gamma_{0}=\max_{I}\gamma(v_{i})/2,\quad\text{and}\quad
C_{0}=\min_{I}C(v_{i}).$

We now consider two cases: let $u\in\mathcal{B}$ satisfy either
$d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})\leq\delta_{0}/4$, or
$d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})>\delta_{0}/4$. Assuming the former,
there exists $i\in I$ such that $\|v_{i}-u\|_{W^{1,2}}\leq\delta_{i}/2$. In
particular, the triangle inequality implies that if $\overline{v}$ is the
closest element of $\mathcal{M}_{q}(A)\cap\mathcal{B}$ to $u$, then
$\|\overline{v}-u\|_{W^{1,2}}\leq\delta_{i}$. Thus, local quantitative
stability implies, cf. Proposition 3.2,

$\mathcal{Q}_{q}(u)-1\geq
C(v_{i})d_{\delta_{i}}(u,\mathcal{M}_{q}(A)\cap\mathcal{B})^{2+\gamma_{i}}\geq
C_{0}d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})^{2+\gamma_{0}}.$

On the other hand, in the case
$d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})\geq\delta_{0}/4$, we argue as follows.
By (the contrapositive of) Proposition 2.1, we get the existence of
$\epsilon>0$ such that

$u\in\mathcal{B},\quad
d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})>\frac{\delta_{0}}{4}\quad\Rightarrow\quad\frac{1-{\rm
Vol}_{g}(M)^{2/q-1}\|u\|_{L^{2}}^{2}}{\|\nabla u\|_{L^{2}}^{2}}\leq A^{\rm
opt}_{q}(M)-\epsilon,$

(notice that, as we require
$d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})>\delta_{0}/4$, it follows
automatically that $u$ cannot be a constant, hence the ratio is well defined).
Rearranging, the last conclusion becomes

$\mathcal{Q}_{q}(u)-1>\epsilon\|\nabla u\|_{L^{2}}^{2}.$

We claim that, there exist $l>0$ so that

$u\in\mathcal{B},\quad
d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})>\frac{\delta_{0}}{4}\quad\Rightarrow\quad\|\nabla
u\|_{L^{2}}>l.$

If not, we could find a sequence $(u_{n})\subseteq W^{1,2}(M)$ with
$\|u_{n}\|_{L^{q}}=1$ for all $n\in\mathbb{N}$ so that
$d(u,\mathcal{M}_{q}(A)\cap\mathcal{B})>\delta_{0}/4$ holds for all
$n\in\mathbb{N}$ but $\|\nabla u_{n}\|_{L^{2}}\to 0$ as $n\uparrow\infty$. Up
to a not relabelled subsequence, we thus infer that there is $u\in W^{1,2}(M)$
so that $u_{n}$ converges strong in $L^{2}$ and weak in $W^{1,2}(M)$ to $u$.
However, by lower semicontinuity, it follows that $\|\nabla u\|_{L^{2}}=0$. In
particular, $u$ must be constant by the Poincaré inequality, say $u\equiv c$
for some $c\in\mathbb{R}\setminus\\{0\\}$ and therefore we get directly
$\|u_{n}\|_{W^{1,2}}\to\|u\|_{W^{1,2}}$ as $n\uparrow\infty$. Since $W^{1,2}$
is Hilbert and $u_{n}$ is already $W^{1,2}$-weak convergent, there is
therefore strong $W^{1,2}$-convergence to the constant $c$. In particular, the
Sobolev inequality implies that $u_{n}\rightarrow c$ in $L^{q}$ (note that
this is not trivial when $q=2^{*}$). In particular, $\|c\|_{L^{q}}=1$ and
therefore $c$ is a competitor in $\mathcal{M}_{q}(A)\cap\mathcal{B}$. However,
this would violate the uniform condition

$\frac{\delta_{0}}{4}<d(u_{n},\mathcal{M}_{q}(A)\cap\mathcal{B})\leq\frac{\|u_{n}-c\|_{W^{1,2}}}{\|u_{n}\|_{W^{1,2}}},$

for $n$ large enough. Finally, setting $C=\min\\{C_{0},l^{2}\epsilon\\}$ we
conclude the proof of Theorem 1.1 (note that $d(u,\mathcal{M}_{q}(A))\leq 1$
always hold).

Let us now discuss the straightforward modifications of the above arguments to
obtain the proof of Theorem 1.2. In this case, we simply need to consider
$q=2^{*},\,A={\sf S}_{d}^{2},\,B=B_{2^{*}}^{\rm opt}(M)$ and
$\mathcal{M}^{1}=\mathcal{M}_{2^{*}}(B)\cap\mathcal{B}$. When
$d(u,\mathcal{M}_{2^{*}}(B)\cap\mathcal{B})\leq\delta_{0}/4$ we repeat
_verbatim_ the above arguments, the only difference being the use of Theorem
2.3 and (2.3) replacing the role of Proposition 2.1. Finally, the case
$d(u,\mathcal{M}_{2^{*}}(B)\cap\mathcal{B})\geq\delta_{0}/4$ is now excluded
by the last assumption in Theorem 1.2 choosing $\delta$ accordingly. ∎

######  Remark 3.3.

As already discussed in the Introduction, the concentration-compactness result
in Proposition 2.1 made it possible to deduce in Theorem 1.1 that, when $u$ is
far away from the set of normalized optimizers, $\mathcal{Q}_{q}(u)$ is far
away from its minimum value. This principle does not appear in [DD01] (recall
Theorem 2.3 and (2.3)). Consequently, the only information we get when
applying our strategy in Theorem 1.2 is a stability for functions that are
_assumed_ sufficiently close to the optimizers.

##  4\. Degenerate stability

In this part, we prove degenerate stability results for $A^{\rm opt}_{q}(M)$.

###  4.1. Proof of Theorem 1.3

Denote by $\lambda(M)$ the first non-zero eigenvalue of the Laplace-Beltrami
operator on $M$, and let $\varphi\in W^{1,2}(M)$ be the corresponding
eigenfunction satisfying

(4.1) $-\Delta\varphi=\lambda(M)\varphi,\qquad\int\varphi\,d{\rm Vol}_{g}=0.$

Recall that $\lambda(M)$ admits the following variational characterization

$\lambda(M)=\inf_{\begin{subarray}{c}f\in W^{1,2}(M)\\\ f\neq
cst.\end{subarray}}\frac{\|\nabla
f\|_{L^{2}}^{2}}{\|f-\bar{f}\|_{L^{2}}^{2}},$

where $\bar{f}=\int f\,d{\rm Vol}_{g}$. Under the standing hypotheses, we
claim that

(4.2) $\frac{q-2}{\lambda(M)}{\rm Vol}_{g}(M)^{\frac{2}{q}-1}=A^{\rm
opt}_{q}(M).$

That the left-hand side is bounded by the right-hand is follows from the well-
known principle that the Sobolev inequality implies the Poincaré inequality on
$M$, see e.g. the argument in [BGL14, Proposition 6.2.2]. For the converse
inequality, we argue as follows. By definition of $A^{\rm opt}_{q}(M)$, let us
consider a sequence $(u_{n})\subseteq W^{1,2}(M)$ of non-constant functions so
that

$\frac{\|u_{n}\|_{L^{q}}-{\rm Vol}_{g}(M)^{2/q-1}\|u_{n}\|_{L^{2}}}{\|\nabla
u_{n}\|_{L^{2}}}\to A^{\rm opt}_{q}(M),\qquad\text{as }n\uparrow\infty.$

Without loss of generality, we can suppose that $\|u_{n}\|_{L^{q}}=1$ for all
$n\in\mathbb{N}$. We are in a position to invoke Proposition 2.1 to deduce
that, up to a non-relabelled subsequence, $u_{n}\to u$ in $L^{q}$ and in
$W^{1,2}$, for some non-zero $u\in\mathcal{M}_{q}(A)$. By assumption, we must
have that $u\equiv c$ for $c={\rm Vol}_{g}(M)^{-1/q}$ (as constant functions
are the only extremal functions and such $c$ has unitary $L^{q}$-norm). By the
quantitative linearization principle for Sobolev inequalities [NV22, Lemma
6.7] (and by suitably scaling the volume in the norms, as there it is assumed
a probability reference measure), setting $f_{n}:=u_{n}-c$, we deduce

$\displaystyle A^{\rm opt}_{q}(M)$
$\displaystyle=\lim_{n\uparrow\infty}\frac{\|u_{n}\|_{L^{q}}-{\rm
Vol}_{g}(M)^{2/q-1}\|u_{n}\|_{L^{2}}}{\|\nabla u_{n}\|_{L^{2}}}$
$\displaystyle=(q-2){\rm
Vol}_{g}(M)^{\frac{2}{q}-1}\lim_{n\uparrow\infty}\frac{\int|f_{n}-\bar{f}_{n}|^{2}\,d{\rm
Vol}_{g}}{\|\nabla f_{n}\|_{L^{2}}}\leq\frac{q-2}{\lambda(M)}{\rm
Vol}_{g}(M)^{\frac{2}{q}-1}.$

This completes the proof of (4.2).

Let us now consider the constant extremal $u=c={\rm Vol}_{g}(M)^{-1/q}$. By
scaling, we can assume that the eigenfunction $\varphi$ in (4.1) satisfies
$\|\varphi\|_{L^{q}}=1$. Then, the condition $\int\varphi\,d{\rm Vol}_{g}=0$
implies that $\varphi\in T_{u}\mathcal{B}$. Hence, we can Taylor expand the
functional $\mathcal{Q}_{q}$ around $u=c$, in the direction $\varphi$ to infer

$\displaystyle\mathcal{Q}_{q}(c+\epsilon\varphi)-1$
$\displaystyle=\mathcal{Q}_{q}(c+\epsilon\varphi)-\mathcal{Q}_{q}(c)=\epsilon\nabla_{\mathcal{B}}\mathcal{Q}_{q}(c)[\varphi]+\frac{\epsilon^{2}}{2}\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(\xi)[\varphi,\varphi]$
$\displaystyle=\epsilon^{2}\frac{1}{2}\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(c)[\varphi,\varphi]+o(1)\|\epsilon\varphi\|_{W^{1,2}}^{2},$

where $\xi\in\mathcal{B}$ is a geodesic point between $c$ and
$c+\epsilon\varphi$, and $o(1)$ goes to zero as $\epsilon\downarrow 0$. By the
computation of Lemma 2.4, we know that

$\displaystyle\frac{1}{2}\nabla^{2}_{\mathcal{B}}\mathcal{Q}_{q}(c)[\varphi,\varphi]$
$\displaystyle=A^{\rm opt}_{q}(M)\int\lambda(M)\varphi^{2}\,d{\rm
Vol}_{g}+{\rm Vol}_{g}(M)^{\frac{2}{q}-1}\int\varphi^{2}\,d{\rm Vol}_{g}$
$\displaystyle\qquad-(q-1)\mathcal{Q}_{q}(c)\int c^{q-2}\varphi^{2}\,d{\rm
Vol}_{g}$ $\displaystyle=(q-1){\rm
Vol}_{g}(M)^{\frac{2}{q}-1}\int\varphi^{2}\,d{\rm Vol}_{g}-{\rm
Vol}_{g}(M)^{\frac{2}{q}-1}(q-1)\int\varphi^{2}\,d{\rm Vol}_{g}=0,$

having used (4.2), and the choice $c={\rm Vol}_{g}(M)^{-1/q}$. Finally, by
contradiction, if $\gamma=0$ were possible, we would reach

$C\inf_{v\in\mathcal{M}_{q}(A)}\|c+\epsilon\varphi-v\|_{W^{1,2}}^{2}\leq\mathcal{Q}_{q}(c+\epsilon\varphi)-1\approx\|\epsilon\varphi\|^{2}_{W^{1,2}}\cdot
o(1)$

which is false for $\epsilon\ll 1$. This concludes the proof.∎

###  4.2. Proof of Corollary 1.4

We prove first i). Let $d>2$ and consider any $q\in(2,2^{*})$. Recall [BE85,
BVV91, Bec93] (also rearranging in [Fra22, Eq. (5)]) that

$A^{\rm opt}_{q}(\mathbb{S}^{d})=\frac{q-2}{d}{\rm
Vol}(\mathbb{S}^{d})^{2/q-1},$

and that the only extremal functions are constant functions. Thus, Theorem 1.3
guarantees that the stability is degenerate. To check that $\gamma=2$ is
optimal, we can rearrange the quantitative stability in [Fra22, Theorem 2] and
we have for some $c_{d,q}>0$

$\frac{A^{\rm opt}_{q}(\mathbb{S}^{d})\|\nabla u\|_{L^{2}}^{2}-{\rm
Vol}(\mathbb{S}^{d})^{2/q-1}\|u\|_{L^{2}}^{2}}{\|u\|_{L^{q}}^{2}}-1\geq
c_{d,q}\left(\inf_{c\in\mathbb{R}}\frac{\|u-c\|_{W^{1,2}}}{\|u\|_{W^{1,2}}}\right)^{4},$

for every $u\in W^{1,2}(\mathbb{S}^{d})\setminus\\{0\\}$. Since in [Fra22,
Theorem 2] the exponent $4$ is proven to be sharp, we directly deduce that in
$\mathbb{S}^{d}$, and in the range $q\in(2,2^{*})$, Theorem 1.1 holds sharp
with $\gamma=2$.

We conclude by proving ii). For $d>2$, consider the closed Riemannian manifold

(4.3)
$M_{*}:=\mathbb{S}^{1}\left(\tfrac{1}{\sqrt{d-2}}\right)\times\mathbb{S}^{d-1},$

equipped with the product metric denoted by $g$. In [Fra22], the author
consider the following conformally invariant Sobolev inequality:

(4.4) $Y(M_{*})\|u\|_{L^{2^{*}}}^{2}\leq\mathcal{E}_{g}(u),\qquad\forall u\in
W^{1,2}(M_{*})$

where $\mathcal{E}_{g}(u):=\int|\nabla u|^{2}+\frac{d-2}{4(d-1)}{\rm
R}_{g}u^{2}\,d{\rm Vol}_{g},$ and $Y(M_{*})$ is the _Yamabe constant_ of
$M_{*}$. Notice

${\rm R}_{g}=(d-2)(d-1),\qquad{\rm Vol}_{g}(M_{*})=\frac{2\pi}{\sqrt{d-2}}{\rm
Vol}(\mathbb{S}^{d-1}).$

Moreover, the Yamabe constant can be also computed explicitly (see [Fra22] and
[Sch89])

$Y(M_{*})=\frac{(d-2)^{2}}{4}{\rm Vol}_{g}(M_{*})^{2/d}.$

Hence, rearranging in (4.4), we have for all $u\in W^{1,2}(M_{*})$

(4.5) $\|u\|_{L^{2^{*}}}^{2}\leq\frac{4}{(d-2)^{2}}{\rm
Vol}_{g}(M_{*})^{-2/d}\|\nabla u\|_{L^{2}}^{2}+{\rm
Vol}_{g}(M_{*})^{-2/d}\|u\|_{L^{2}}^{2}.$

Since (4.4) is sharp by definition of the Yamabe constant $Y(M_{*})$, we
deduce that

$A^{\rm opt}_{2^{*}}(M_{*})=\frac{4}{(d-2)^{2}}{\rm Vol}_{g}(M_{*})^{-2/d}.$

Now, constant functions are the only extremal functions in (4.4) and hence
also in (4.5). Thus, to apply Theorem 1.3 we need to check that

(4.6) ${\sf S}_{d}^{2}<A^{\rm opt}_{2^{*}}(M_{*}),$

Recalling that ${\sf S}_{d}^{2}=\frac{4}{d(d-2)}{\rm
Vol}(\mathbb{S}^{d})^{-2/d}$ and using the above computations, (4.6) becomes

$\frac{4}{d(d-2)}<\frac{4}{(d-2)^{2}}(d-2)^{1/d}.$

which is always true for $d>2$. Thus, Theorem 1.3 guarantees that the
stability is degenerate. We now check that $\gamma=2$ is optimal and conclude.
By [Fra22, Theorem 3], there is a constant $c_{d}>0$ so that for every $u\in
W^{1,2}(M_{*})\setminus\\{0\\}$ it holds sharp

(4.7) $\mathcal{E}_{g}(u)-Y(M_{*})\|u\|_{L^{2^{*}}}^{2}\geq
c_{d}\inf_{c\in\mathbb{R}}\frac{\mathcal{E}_{g}(u-c)^{2}}{\mathcal{E}_{g}(u)}.$

Notice that $\mathcal{E}_{g}(u)$ is comparable up to dimensional constants to
$\|u\|_{W^{1,2}}^{2}$ (thanks to the fact that ${\rm R}_{g}$ is constant).
Hence, we can rearrange in (4.7) using (4.5) to deduce

$\frac{A^{\rm opt}_{2^{*}}(M_{*})\|\nabla u\|_{L^{2}}^{2}+{\rm
Vol}(M_{*})^{-2/d}\|u\|_{L^{2}}^{2}}{\|u\|_{L^{2^{*}}}}-1\geq
c_{d}\left(\inf_{c\in\mathbb{R}}\frac{\|u-c\|_{W^{1,2}}}{\|u\|_{W^{1,2}}}\right)^{4},$

for every $u\in W^{1,2}(M_{*})\setminus\\{0\\}$, and for a possibly different
constant $c_{d}$. Finally, this concludes the proof of ii) since the
optimality in (4.7) implies that $\gamma=2$ is sharp.∎

Acknowledgments. F.N. is a member of INDAM-GNAMPA and acknowledges partial
support of the Academy of Finland project _Approximate incidence geometry_ ,
Grant No. 355453 and of the MIUR Excellence Department Project awarded to the
Department of Mathematics, University of Pisa, CUP I57G22000700001. F.N. would
like to thank I. Y. Violo for numerous conversations around the topics of this
note. Both authors would like to thank I.Y. Violo, and L. Spolaor for useful
discussions and interesting comments.

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