# A note on the critical Laplace equation and Ricci curvature

Mattia Fogagnolo  M. Fogagnolo, Centro di Ricerca Matematica Ennio De Giorgi,
Scuola Normale Superiore, Piazza dei Cavalieri 3, 56126 Pisa (PI), Italy
<EMAIL_ADDRESS>,  Andrea Malchiodi  A. Malchiodi, Scuola Normale
Superiore, Piazza dei Cavalieri 7, 56126 Pisa (PI), Italy
<EMAIL_ADDRESS>and  Lorenzo Mazzieri  L. Mazzieri, Università degli
Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
<EMAIL_ADDRESS>
###### Abstract.

We study strictly positive solutions to the critical Laplace equation

$-\Delta u=n(n-2)u^{\frac{n+2}{n-2}},$

decaying at most like $d(o,x)^{-(n-2)/2}$, on complete noncompact manifolds
$(M,g)$ with nonnegative Ricci curvature, of dimension $n\geq 3$. We prove
that, under an additional mild assumption on the volume growth, such a
solution does not exist, unless $(M,g)$ is isometric to $\mathbb{R}^{n}$ and
$u$ is a Talenti function. The method employs an elementary analysis of a
suitable function defined along the level sets of $u$.

MSC (2020): 35R01, 35B33, 53C21, 40E10 .

Keywords: critical equations, classification results, manifolds with
nonnegative Ricci curvature, level sets

##  1\. Introduction and statement of the main results

The critical Laplace equation

$-\Delta u=n(n-2)u^{\frac{n+2}{n-2}},$ (1.1)

on $\mathbb{R}^{n}$, with $n\geq 3$, together its possible generalizations,
has been widely studied in the mathematical literature for many reasons. Let
us just mention that entire solutions to (1.1) provide critical functions for
the $L^{2}$ Sobolev inequality and that $u^{4/(n-2)}g_{\mathbb{R}^{n}}$ is a
metric of positive, constant scalar curvature. Since the works of Obata
[Oba71], Talenti [Tal76], Aubin [Aub76], Gidas-Ni-Nirenberg [GNN81], it became
clear that under additional decay or variational assumptions the only entire
solutions to (1.1) are radially symmetric. They are often called _Talentian_.
The grounbreaking contribution by Caffarelli-Gidas-Spruck [CGS89] actually
established that no additional assumptions are required, except for
positivity.

In this note, we face the problem of classifying entire solutions to (1.1) on
complete noncompact manifolds $(M,g)$ with nonnegative Ricci curvature of
dimension $n\geq 3$. We prove that, under a suitable decay assumption on $u$
and an additional mild assumption on the volume growth, such a solution does
not exist, unless $(M,g)$ is isometric to $\mathbb{R}^{n}$ and $u$ is a
Talenti function.

######  Theorem 1.1.

Let $(M,g)$ be a complete noncompact Riemannian manifold with nonnegative
Ricci curvature, of dimension $n\geq 3$, satisfying

$\frac{\lvert B(o,t\rvert}{\lvert B(o,s)\rvert}\geq
C\left(\frac{t}{s}\right)^{b}$ (1.2)

for some $1<b\leq n$ and for any $t\geq s>0$. Assume that there exists a
global, strictly positive solution $u$ to

$-\Delta u=n(n-2)u^{\frac{n+2}{n-2}},$

such that $u(x)d(o,x)^{(n-2)/2}$ is uniformly bounded in $M$ as a function of
$x\in M$, for some $o\in M$. Then, $(M,g)$ is isometric to
$(\mathbb{R}^{n},g_{\mathbb{R}^{n}})$ and

$u(x)=\left(\frac{a}{a^{2}+d(o,x)^{2}}\right)^{\frac{(n-2)}{2}},$ (1.3)

for some $a>0$.

We observe that the volume condition (1.2) is satisfied any time

$C^{-1}r^{b}\leq\lvert B(o,r)\rvert\leq Cr^{b}$

for $1<b\leq n$ and for any $r\geq 1$. In particular, Euclidean volume growth
manifolds fulfill it.

As a consequence of the previous result, apart from the Euclidean case, the
Ricci-flat manifolds $(M,g)$ fulfilling the above assumptions do not enjoy
conformal metrics $\tilde{g}=u^{4/(n-2)}g$ of positive constant scalar
curvature with conformal factor decaying at infinity as above. In particular,
this applies to ALE (such as Eguchi-Hanson), ALF and ALG gravitational
instantons. We refer the interested reader to [Min10], [CC15], [CC15a],
[CC16].

We point out that, to our knowledge, a study of (1.1) on manifolds with
nonnegative Ricci curvature was yet to be addressed in literature. In the case
of subcritical exponent $1\leq\alpha<(n+2)/(n-2)$ in the right hand side of
(1.1), it has been shown in [GS81, Theorem 1.2] that entire solutions are not
allowed. Theorem 1.1 can thus be interpreted as a first extension of such
study to the critical regime. Concerning the case of complete Riemannian
manifolds with other curvature assumptions, we mention [MS21] by Muratori-
Soave, who considered solutions to (1.1) that are radially symmetric or that
minimize the Sobolev quotient in Cartan-Hadamard manifolds.

Our proof combines elementary computations on a vector field with nonnegative
divergence already considered by [Oba71] and [GS81], a Harnack-type argument
triggered by the decay assumption coupled with a Yau-type gradient estimate,
and a novel auxiliary function $V$ defined along the level sets of $u$. In
proving Theorem 1.1, we first show an intermediate result that adds to the
decay assumption on $u$ a _finite energy condition_ , that is to say

$\int_{M}\lvert\nabla u\rvert^{2}{\,\rm d}\mu<+\infty,$ (1.4)

but does not require any volume restriction. It reads as follows.

######  Theorem 1.2.

Let $(M,g)$ be a complete noncompact Riemannian manifold with nonnegative
Ricci curvature and of dimension $n\geq 3$. Assume that there exists a global,
strictly positive _finite energy_ solution $u$ to

$-\Delta u=n(n-2)u^{\frac{n+2}{n-2}},$

such that $u(x)d(o,x)^{(n-2)/2}$ is uniformly bounded in $M$ as a function of
$x\in M$. Then, $(M,g)$ is isometric to $(\mathbb{R}^{n},g_{\mathbb{R}^{n}})$
and

$u(x)=\left(\frac{a}{a^{2}+d(o,x)^{2}}\right)^{\frac{(n-2)}{2}},$ (1.5)

for some $a>0$.

It is worth pointing out that, thanks to the decay estimate on the gradient
that we observe in Proposition 2.2, if the decay on $u$ is reinforced to
$u\leq Cd(o,x)^{(2-n)/2-\varepsilon}$, for $\varepsilon>0$, then (1.4) is just
a consequence of this. In particular, Theorem 1.2 fully recovers the main
result in [GNN81], and extends it to a rigidity statement in the geometry of
nonnegative Ricci curvature.

Summary. In Section 2 we gather some basic estimates about $u$ and its
derivatives. In Section 3 we introduce the fundamental vector field with
nonnegative divergence and set its connection with the auxiliary function $V$,
and finally in Section 4 we prove Theorems 1.2 and 1.1.

_Added note_

During the checking process of the presentation of the manuscript, the
preprint [CM22] appeared on the arXiv. It refines and generalizes our results,
with different methods.

##  2\. Preliminary a priori estimates

In this section we work out some preliminary estimates in force for solutions
to (1.1), that we later particularize for those such that
$u(x)d(o,x)^{(n-2)/2}$ is uniformly bounded.

First of all, we crucially observe that the technique Yau [Yau75] pioneered in
order to obtain gradient estimates for harmonic functions under lower Ricci
curvature bounds yields for solutions to the critical Laplace equation the
following inequality.

######  Proposition 2.1 (Yau-type estimate for solutions to the critical
Laplace equation).

Let $(M,g)$ be a complete, noncompact Riemannian manifold with nonnegative
Ricci curvature of dimension $n\geq 3$. Then, a solution to (1.1) satisfies

$\sup_{B(x,R)}\frac{\lvert\nabla u\rvert^{2}}{u^{2}}\leq
C\left[\frac{1}{R^{2}}+\sup_{B(x,2R)}u^{\frac{4}{(n-2)}}\right]$ (2.1)

for any $x\in M$, for any $R\geq 1$ and for some dimensional constant $C>0$.

The proof of (2.1) is obtained by following very closely the one for
eigenfunctions proposed for [Li, Theorem 1.1] and, as such, we omit the proof.

As a consequence of the above result, we have the following decay estimate for
$\lvert\nabla u\rvert$ when $u$ satisfies the same bound as in Theorem 1.1.
Here and in the sequel of the paper, we consider $o\in M$ to be any fixed
point, and use the notation $r(x)=d(o,x)$ for $x\in M$.

######  Corollary 2.2 (Decay of the gradient).

Let $(M,g)$ be a complete, noncompact Riemannian manifold with nonnegative
Ricci curvature of dimension $n\geq 3$, and let $u$ be a solution to (1.1)
that satisfies $u\leq Cr^{-(n-2)/2}$. Then,

$\frac{\lvert\nabla u\rvert^{2}}{u^{2}}(x)\leq C\frac{1}{r(x)^{2}}$ (2.2)

for any $x$ such that $r(x)=d(o,x)\geq 4$.

###### Proof.

It suffices to apply Proposition 2.1 with $2R=r(x)/2$. Coupled with the decay
assumption on $u$, it yields

$\frac{\lvert\nabla u\rvert^{2}}{u^{2}}\leq
C\left[\frac{1}{r(x)^{2}}+u^{\frac{4}{n-2}}(y_{x})\right]\leq
C\left[\frac{1}{r(x)^{2}}+\frac{1}{r(y_{x})^{2}}\right],$ (2.3)

for some $y_{x}\in B(x,r(x)/2)$. On the other hand, we have $r(x)/3\leq
r(y_{x})\leq 2r(x)$, and plugging this information into (2.3) leaves us with
(2.2). ∎

Through integrating by parts the Bochner formula, we do now work out an
integral estimate on the Hessian of $u$. The computations, that in our case is
particularly simple, got some inspiration from analogous ones in the
celebrated work [CC96]. Here and in the remainder of this paper we consider
$o\in M$ to be fixed and denote with $A_{R_{1},R_{2}}$ the open annulus
$B(o,R_{2})\setminus\overline{B}(o,R_{1})$.

######  Proposition 2.3.

Let $(M,g)$ be a complete, noncompact Riemannian manifold with nonnegative
Ricci curvature of dimension $n\geq 3$. Then, for any $k>1$ there exist
numbers $1<\alpha<\beta<k$ such that a solution to (1.1) satisfies, for any
$R\geq 1$,

$\int\limits_{A_{\alpha R,\beta R}}\lvert\nabla\nabla u\rvert^{2}{\,\rm
d}\mu\leq C\left(R^{-2}\int\limits_{A_{R,kR}}\lvert\nabla u\rvert^{2}{\,\rm
d}\mu+\int\limits_{A_{R,kR}}u^{\frac{4}{(n-2)}}\lvert\nabla u\rvert^{2}{\,\rm
d}\mu\right),$ (2.4)

for some constant $C$ not depending on $R$.

###### Proof.

By the Bochner formula and by the nonnegativity of the Ricci tensor, we have

$\lvert\nabla\nabla u\rvert^{2}\leq\frac{1}{2}\Delta\lvert\nabla
u\rvert^{2}-\langle\nabla\Delta u,\nabla u\rangle.$ (2.5)

We multiply both sides by an annular Euclidean-like cut-off function $\varphi$
supported in $B(o,kR)\setminus\overline{B(o,R)}$ such that

$\displaystyle\varphi\equiv 1\,\,$ $\displaystyle\text{on}\,\,B(o,\beta
R)\setminus\overline{B(o,\alpha R)},$ (2.6)
$\displaystyle\lvert\nabla\varphi\rvert$ $\displaystyle\leq
C\,\frac{\varphi^{1/2}}{R},$ (2.7) $\displaystyle\lvert\Delta\varphi\rvert$
$\displaystyle\leq C\,\frac{1}{R^{2}}.$ (2.8)

for $1<\alpha<\beta<k$. Such function is well known to exist since [CC96], and
can be more precisely built by slightly re-adapting the proof of [BS18,
Corollary 2.3]. Integrating by parts twice the first term on the right-hand
side, we get

$\begin{split}\int\limits_{A_{\alpha R,\beta R}}\lvert\nabla\nabla
u\rvert^{2}{\,\rm d}\mu&\leq\int\limits_{A{r,kR}}\lvert\nabla\nabla
u\rvert^{2}\varphi{\,\rm d}\mu\\\
&\leq\frac{1}{2}\int\limits_{A{r,kR}}\lvert\nabla
u^{2}\rvert\Delta\varphi+\int\limits_{A_{R,kR}}u^{\frac{4}{(n-2)}}\lvert\nabla
u\rvert^{2}\varphi{\,\rm d}\mu\\\ &\leq
C\left(R^{-2}\int\limits_{A_{R,kR}}\lvert\nabla u\rvert^{2}{\,\rm
d}\mu+\int\limits_{A_{R,kR}}u^{\frac{4}{(n-2)}}\lvert\nabla u\rvert^{2}{\,\rm
d}\mu\right),\end{split}$

as claimed. ∎

When $u$ is assumed to satisfy $u\leq Cr^{-(n-2)/2}$ we get the following
integral decay estimate for its Hessian.

######  Corollary 2.4 (Integral decay of the Hessian).

Let $(M,g)$ be a complete, noncompact Riemannian manifold with nonnegative
Ricci curvature of dimension $n\geq 3$, and let $u$ be a solution to (1.1)
that satisfies $u\leq Cr^{-(n-2)/2}$. Then, for any $k>1$ there exist numbers
$1<\alpha<\beta<k$ such that a solution to (1.1) satisfies, for any $R\geq 1$,

$\int\limits_{A_{\alpha R,\beta R}}\lvert\nabla\nabla u\rvert^{2}{\,\rm
d}\mu\leq\frac{C}{R^{2}}\int\limits_{A_{R,kR}}\lvert\nabla u\rvert^{2}{\,\rm
d}\mu\leq\frac{C}{R^{2}},$ (2.9)

for some positive constant $C$.

###### Proof.

The first inequality just follows by plugging the assumption on $u$ into
(2.4), the second one by coupling it with (2.2), and observing that, by
Bishop-Gromov monotonicity, we have

$\lvert A_{r,kr}\rvert\leq\lvert B(o,kR)\rvert\leq\omega_{n}(kR)^{n},$

where $\omega_{n}$ is the Lebesgue measure of the unit ball in
$\mathbb{R}^{n}$. ∎

We close this preliminary section by recording a basic and fundamental control
on $u$ from below.

######  Proposition 2.5 (Control from below for $u$).

Let $(M,g)$ be a complete, noncompact Riemannian manifold with nonnegative
Ricci curvature of dimension $n\geq 3$, and let $u$ be a solution to (1.1)
that goes to zero at infinity. Then, there exists a strictly positive constant
$C$ such that

$u(x)\geq Cr(x)^{2-n}$ (2.10)

for any $x\in M\setminus B(o,1)$.

###### Proof.

Let $C=\min_{\partial B(0,1)}u$, and recall that $\Delta r^{2-n}$ is sub-
harmonic in the sense of distributions in $M\setminus\\{o\\}$ (see e.g. the
proof of [AFM20, Lemma 2.2]). Then, the function $u/C-r^{2-n}$ is super-
harmonic in $B(o,R)\setminus\overline{B(o,1)}$ and in particular it satisfies

$(u/C-r^{2-n})(x)\geq\sup_{\partial B(o,1)}(u/C-r^{2-n})+\sup_{\partial
B(o,R)}(u/C-r^{2-n})\geq\sup_{\partial B(o,R)}(u/C-r^{2-n})$

for any $x\in B(o,R)\setminus\overline{B(o,1)}$, where the last inequality is
due to the specific choice of $C$. Since both $u$ and $r^{2-n}$ vanish at
infinity, we get that the claim by letting $R$ at infinity and by the
arbitrariness of $x$. ∎

##  3\. A key vector field and a useful auxiliary function

In this section we introduce a vector field with nonnegative divergence,
vanishing exactly in the case in the flat model situation and set the main
properties of the integral auxiliary function $V$.

We introduce the vector field with nonnegative divergence that rules our
problem. Its intuition dates back to the early work of Obata [Oba71], and it
constitutes a key ingredient also in the study of subcritical elliptic
equations performed in [GS81]. As in the latter, the vector field is better
understood in terms of the function

$v=u^{-\frac{2}{n-2}},$ (3.1)

with $u$ a strictly positive solution to (1.1). This is due to the fact that,
in the model situation of the flat $\mathbb{R}^{n}$ with $u$ given by a
Talentian (1.3), $v$ becomes an affine function of $d(o,x)^{2}$, that in
particular solves

$\nabla\nabla v=\frac{\Delta v}{n}g.$

In fact, the squared norm of the trace-free Hessian of $v$ will constitute,
together with a Ricci curvature term, the nonnegative divergence of the vector
field.

We define

$X=\frac{1}{v^{n-1}}\left[\frac{1}{2}\nabla\lvert\nabla
v\rvert^{2}-\frac{1}{2}\frac{\lvert\nabla v\rvert^{2}\nabla
v}{v}-2\frac{{\nabla v}}{v}\right].$ (3.2)

We actually give, for the reader’s sake, a self-contained computation of its
divergence. In order to do it, we first point out that the critical Laplace
equation translates as

$\Delta v=\frac{n}{2}\frac{\lvert\nabla v\rvert^{2}}{v}+\frac{2n}{v}$ (3.3)

in terms of $v$.

######  Proposition 3.1.

Let $(M,g)$ be a Riemannian manifold, $\Omega\subset M$ an open set and
$u:\Omega\to\mathbb{R}$ a strictly positive function solving (1.1). Then,
letting $v$ be as in (3.1), the vector field defined by (3.2) satisfies

${\mathrm{div}}X=\frac{1}{v^{n-1}}\left[\left|\nabla\nabla v-\frac{\Delta
v}{n}g\right|^{2}+\mathop{\rm Ric}\nolimits\left(\nabla v,\nabla
v\right)\right].$ (3.4)

###### Proof.

We have

$\begin{split}{\mathrm{div}}X\,=\,&\frac{1}{v^{n-1}}\left[\frac{1}{2}{\mathrm{div}}(\nabla\lvert\nabla
v\rvert^{2})-\frac{1}{2}{\mathrm{div}}\left(\lvert\nabla
v\rvert^{2}\frac{\nabla v}{v}\right)-2{\mathrm{div}}\left(\frac{\nabla
v}{v}\right)\right]\\\
&-\frac{(n-1)}{v^{n-1}}\left[\frac{1}{2}\left\langle\nabla\lvert\nabla
v\rvert^{2},\frac{\nabla v}{v}\right\rangle-\frac{\lvert\nabla
v\rvert^{4}}{2v^{2}}-2\frac{\lvert\nabla
v\rvert^{2}}{v^{2}}\right].\end{split}$ (3.5)

We do now compute the three divergence terms. We have, using Bochner identity
and (3.3),

$\begin{split}{\mathrm{div}}(\nabla\lvert\nabla
v\rvert^{2})&=2\left[\left|\nabla\nabla v-\frac{\Delta
v}{n}\right|^{2}+\frac{(\Delta v)^{2}}{n}+\left\langle\nabla\Delta v,\nabla
v\right\rangle+\mathop{\rm Ric}\nolimits(\nabla v,\nabla v)\right]\\\
&=2\left[\left|\nabla\nabla v-\frac{\Delta
v}{n}\right|^{2}-\frac{n}{4}\frac{\lvert\nabla
v\rvert^{4}}{v^{2}}+\frac{4n}{v^{2}}+\frac{n}{2}\left\langle\nabla\lvert\nabla
v\rvert^{2},\frac{\nabla v}{v}\right\rangle+\mathop{\rm Ric}\nolimits(\nabla
v,\nabla v)\right].\end{split}$ (3.6)

Moreover,

${\mathrm{div}}\left(\lvert\nabla v\rvert^{2}\frac{\nabla
v}{v}\right)=\left\langle\nabla\lvert\nabla v\rvert^{2},\frac{\nabla
v}{v}\right\rangle+\frac{1}{2}(n-2)\frac{\lvert\nabla
v\rvert^{4}}{v^{2}}+2n\frac{\lvert\nabla v\rvert^{2}}{v^{2}}.$ (3.7)

Finally,

${\mathrm{div}}\left(\frac{\nabla
v}{v}\right)=\frac{(n-2)}{2}\frac{\lvert\nabla
v\rvert^{2}}{v^{2}}+\frac{2n}{v^{2}}.$ (3.8)

Plugging (3.6), (3.7) and (3.8) into (3.5) leaves us with identity (3.4). ∎

We now define, a priori only for regular level sets of $v$, the function

$V:(\inf_{M}v,+\infty)\setminus v(\mathrm{Crit}(v))\to(0+\infty),$

where $\mathrm{Crit}(v)=\\{\nabla v=0\\}$, given by

$V(s)=\int\limits_{\\{v=s\\}}\lvert\nabla v\rvert^{3}{\,\rm d}\sigma.$ (3.9)

In particular, by Sard’s Theorem, this function is defined for almost every
$s\in(\inf_{M}v,+\infty)$. In the next result, we show that it is actually
equivalent to an absolutely continuous function, and compute its derivative.
In order to justify the formal computation, we argue as in the recent paper
[CM21], but we provide, for completeness, all details.

######  Proposition 3.2.

Let $(M,g)$ be a Riemannian manifold, and let $u$ be a positive solution to
(1.1) that tends to zero at infinity. Then, the function $V$ defined above is
equivalent to an absolutely continuous function
$V:(\inf_{M}v,+\infty)\to\mathbb{R}$, and moreover

$V^{\prime}(s)=\int\limits_{\\{v=s\\}}\left\langle\nabla\lvert\nabla
v\rvert^{2},\frac{\nabla v}{\lvert\nabla v\rvert}\right\rangle{\,\rm
d}\sigma+\frac{n}{2}s^{-1}\int\limits_{\\{v=s\\}}\lvert\nabla
v\rvert^{3}{\,\rm d}\sigma+2ns^{-1}\int_{\\{v=s\\}}\lvert\nabla v\rvert{\,\rm
d}\sigma$ (3.10)

holds true for almost any value $s\in(\inf_{M}v,+\infty)$.

###### Proof.

Let $s$ be a regular value for $v$, so that

$V(s)=\int_{\\{v=s\\}}\lvert\nabla v\rvert^{2}\left\langle\nabla
v,\frac{\nabla v}{\lvert\nabla v\rvert}\right\rangle{\,\rm
d}\sigma=\int\limits_{\\{v<s\\}}{\mathrm{div}}(\lvert\nabla v\rvert^{2}\nabla
v){\,\rm d}\mu,$

where we used the divergence theorem and that, since $v$ tends to infinity, it
is a proper function and the outer unit normal to $\\{v<s\\}$ is given by
$\nabla v/\lvert\nabla v\rvert$.

Observe now that, arguing exactly as in the basic proof of [CM21, Lemma 1.1],
the critical set of $u$ is locally contained in smooth $(n-1)$-dimensional
manifolds, in particular it is of null $n$-dimensional Hausdorff measure and
moreover all the level sets of $u$ (and hence of $v$) have zero
$n$-dimensional Hausdorff measure. Indeed, this argument just relies on the
fact that $\Delta u<0$. In particular, the function
$\tilde{V}:(\inf_{M}v,+\infty)\to\mathbb{R}$ defined by

$\tilde{V}(s)=\int_{\\{v=s\\}}\lvert\nabla v\rvert^{2}\left\langle\nabla
v,\frac{\nabla v}{\lvert\nabla v\rvert}\right\rangle{\,\rm
d}\sigma=\int\limits_{\\{v<s\\}}{\mathrm{div}}(\lvert\nabla v\rvert^{2}\nabla
v){\,\rm d}\mu,$

is immediately seen to be continuous. Indeed, for $\varepsilon>0$, we have

$\tilde{V}(s+\varepsilon)-\tilde{V}(s)=\int\limits_{\\{s\leq
v<s+\varepsilon\\}}{\mathrm{div}}(\lvert\nabla v\rvert^{2}\nabla v){\,\rm
d}\mu=\int\limits_{\\{s<v<s+\varepsilon\\}}{\mathrm{div}}(\lvert\nabla
v\rvert^{2}\nabla v){\,\rm d}\mu$

where we have used $\mu\\{v=s\\}=0$, and, in particular, the right-hand side
above approaches $0$ as $\varepsilon\to 0^{+}$ by the Dominated Convergence
Theorem.

We proved so far that $V$ is equivalent to the continuous function
$\tilde{V}$, and in particular we can safely identify one with the other. In
order to show the absolute continuity, let, as in the proof of [CM21, Lemma
1.3],

$V_{\delta}(s)=\int\limits_{\\{v<s\\}}\frac{{\mathrm{div}}(\lvert\nabla
v\rvert^{2}{\nabla v})\lvert\nabla v\rvert}{\lvert\nabla v\rvert+\delta}{\,\rm
d}\sigma,$

for $\delta>0$. Observe that, since as already pointed out $\mu\\{\nabla
v=0\\}=0$, the above expression converges by dominated convergence theorem as
$\delta\to 0^{+}$ to $V(s)$. Moreover, by coarea formula, we have

$V_{\delta}(s)=\int\limits_{v_{0}}^{s}\int_{\\{v=t\\}}\frac{{\mathrm{div}}(\lvert\nabla
v\rvert^{2}\nabla v)}{\lvert\nabla v\rvert+\delta}{\,\rm d}\sigma{\,\rm d}t,$

where we let $v_{0}=\inf_{M}v$. By the Monotone Convergence Theorem, we can
pass to the limit as $m\to\\+\infty$ also in the right-hand side of the
identity above, and deduce that

$V(s)=\int\limits_{v_{0}}^{s}\int_{\\{v=t\\}}\frac{{\mathrm{div}}(\lvert\nabla
v\rvert^{2}\nabla v)}{\lvert\nabla v\rvert}{\,\rm d}\sigma{\,\rm d}t,$ (3.11)

for any $s\in(\inf_{M}v,+\infty)$, that is the absolute continuity of $V$.
Finally, computing, with the aid of (3.3),

${\mathrm{div}}(\lvert\nabla v\rvert^{2}\nabla
v)=\left\langle\nabla\lvert\nabla v\rvert^{2},{\nabla
v}\right\rangle+\frac{n}{2}\frac{\lvert\nabla
v\rvert^{4}}{v}+2n\frac{\lvert\nabla v\rvert^{2}}{v},$

and plugging it into (3.11), we also showed (3.10). ∎

We finally link the function $V$ and its derivative to the vector field $X$.
We have, by the Divergence Theorem,

$\int\limits_{\\{v<s\\}}{\mathrm{div}}X{\,\rm
d}\mu=\frac{\int_{\\{v=s\\}}\left\langle\nabla\lvert\nabla v\rvert^{2},{\nabla
v}{\lvert\nabla v\rvert}^{-1}\right\rangle{\,\rm
d}\sigma}{s^{n-1}}-\frac{1}{2}\frac{\int_{\\{v=s\\}}\lvert\nabla
v\rvert^{3}{\,\rm d}\sigma}{s^{n}}-2\frac{\int_{\\{v=s\\}}\lvert\nabla
v\rvert{\,\rm d}\sigma}{s^{n}}$ (3.12)

for any regular value $s$. Coupling this with (3.4) and (3.1), we deduce the
following.

######  Corollary 3.3.

Let $(M,g)$ be a complete Riemannian manifold. Let $u$ be a strictly positive
solution to (1.1) that vanishes at infinity. Then, we have

$\begin{split}\int\limits_{\\{v<s\\}}\frac{1}{v^{n-1}}\left[\left|\nabla\nabla
v-\frac{\Delta v}{n}g\right|^{2}+\mathop{\rm Ric}\nolimits\left(\nabla
v,\nabla v\right)\right]{\,\rm
d}\mu=\frac{1}{2}\frac{V^{\prime}(s)}{s^{n-1}}&-\frac{1}{4}(n+2)\frac{V(s)}{s^{n}}\\\
&-(n+2)\frac{\int_{\\{v=s\\}}\lvert\nabla v\rvert{\,\rm
d}\sigma}{s^{n}}\end{split}$ (3.13)

for almost any $s\in v(M)$.

The main aim, for proving Theorems 1.2 and 1.1, will be showing that in their
assumptions, the left-hand side of (3.13) vanishes for any $s$.

##  4\. Proof of the main results

In this section we prove Theorems 1.1 and 1.2. We first prove the latter, i.e.
the rigidity result for finite energy solutions, and then prove that in the
setting of Theorem 1.1 the function $u$ actually enjoys finite energy. We
start with the following simple splitting principle, consequence of the basic
result according to which the existence of a nontrivial function with
vanishing trace-less Hessian implies a warped product splitting of the metric
[Pet16, Theorem 5.7.4], [CMM12], [CC96, Section 1].

######  Lemma 4.1.

Let $(M,g)$ be a complete manifold, and let $u$ be a solution to (1.1) that
vanishes at infinity. If

$\left|\nabla\nabla v-\frac{\Delta v}{n}g\right|^{2}+\mathop{\rm
Ric}\nolimits\left(\nabla v,\nabla v\right)\equiv 0$ (4.1)

on $M$, then $(M,g)$ is isometric to flat $\mathbb{R}^{n}$ and $u$ is of the
form (1.3).

###### Proof.

It is classically deduced from the vanishing of the trace-less Hessian of $v$,
e.g. appealing to [Pet16, Theorem 5.7.4], that $(M,g)$ must split a warped
product $(I\times N,{\,\rm d}\rho\otimes{\,\rm
d}\rho+\varphi^{2}(\rho)g_{N})$, for some hypersurface $N$ in $M$ and with $I$
coinciding with $\mathbb{R}$ or $[0,+\infty)$. By the proof presented in the
contribution above (see also the maybe more transparent computations carried
out in the proof of [CMM12, Theorem 1.1], one also realizes that $N$ can be
identified with a smooth level set of $v$, and that in particular it is a
closed hypersurface. Again by construction of the splitting, one also has that
in these coordinates $v^{\prime}(\rho)=2\alpha\varphi(\rho)$ for some constant
$\alpha$. Consequently, since the Ricci curvature of $g$ can be computed as

$\mathop{\rm Ric}\nolimits=-(n-1)\frac{\varphi^{\prime\prime}}{\varphi}{\,\rm
d}\rho\otimes{\,\rm d}\rho+\mathop{\rm
Ric}\nolimits_{N}-\big{(}(n-2)(\varphi^{\prime})^{2}+\varphi\varphi^{\prime\prime}\big{)}^{2}g_{N},$
(4.2)

plugging in $\mathop{\rm Ric}\nolimits(\nabla v,\nabla v)=\lvert
v^{\prime}(\rho)\rvert^{2}\mathop{\rm Ric}\nolimits(\nabla\rho,\nabla\rho)=0$
we deduce that $\varphi=A+B\rho$, for some constants $A,B$. If $B=0$, then
$(M,g)$ would be a cylinder with cross-section $N$, and $v=A\rho+C$, for some
other constant $C$. Consequently, $\Delta v=0$, and this is in contradiction
with the equation (3.3).

We can thus suppose, by possibly translating the coordinate $\rho$, that $A=0$
and $\varphi=B\rho$ for some constant $B\neq 0$. In particular, since $g$ must
be smooth also as $\rho\to 0^{+}$, we have $I=[0,+\infty)$, $(N,g_{N})$ the
sphere $\mathbb{S}^{n-1}$ with its standard round metric and $b=1$. In other
words, $M$ is the Euclidean space and $g$ is its flat norm.

Moreover, $v=\alpha\rho^{2}+\beta$, with $\rho$ constituting the Euclidean
distance from some origin $o$. Plugging this function into (3.3), one
immediately gets $\alpha\beta=1$, and thus, by (3.9), $u$ has the form (1.3).
∎

Exploiting crucially (3.13) and the above splitting principle, we prove
Theorem 1.2.

###### Proof of Theorem 1.2.

Due to the finite energy condition (1.4) and the coarea formula, we have

$\int\limits_{0}^{t_{0}}\int_{\\{u=t\\}}\lvert\nabla u\rvert{\,\rm
d}\sigma{\,\rm d}t<+\infty.$

for any $t_{0}\in u(M)$. Performing a change of variables, such condition
translates in terms of $v$ as

$\int\limits_{s_{0}}^{+\infty}\int_{\\{v=s\\}}\frac{\lvert\nabla
v\rvert}{s^{n}}{\,\rm d}\sigma{\,\rm d}s<+\infty.$ (4.3)

We want to prove that

$\int\limits_{\\{v<s_{j}\\}}\frac{1}{v^{n-1}}\left[\left|\nabla\nabla
v-\frac{\Delta v}{n}g\right|^{2}+\mathop{\rm Ric}\nolimits\left(\nabla
v,\nabla v\right)\right]{\,\rm d}\mu\to 0^{+}$ (4.4)

for some $s_{j}\to+\infty$. Indeed, since the Ricci curvature, and thus the
integrand, are nonnegative, this would complete the proof by Lemma 4.1. Assume
now by contradiction that (4.4) holds for no diverging sequences
$\\{s_{j}\\}_{j\in\mathbb{N}}$. Then, by (3.13), we deduce that for almost any
$s\in v(M)$

$V^{\prime}(s)\geq\delta s^{n-1}$

for some $\delta>0$. Integrating the above inequality in $(s_{0},s)$ for some
$s_{0}\in v(M)$, we deduce the following growth for the absolutely continuous
function $V$

$V(s)\geq\frac{\delta}{n}s^{n}+F(s_{0}),$ (4.5)

with $F(s_{0})=V(s_{0})-\frac{\delta}{n}{s_{0}}^{n}$. Moreover, by choosing
$s_{0}$ big enough, by coupling (2.2) with (2.10), one gets

$\lvert\nabla v\rvert^{2}(x)\leq C\frac{v^{2}}{r^{2}}(x)\leq Cv(x)$ (4.6)

for any $x\in\\{v>s_{0}\\}$. But then, by (4.5) and (4.6), the left-hand side
of (4.3) must also satisfy

$\int\limits_{s_{0}}^{+\infty}\int_{\\{v=s\\}}\frac{\lvert\nabla
v\rvert}{s^{n}}{\,\rm d}\sigma{\,\rm
d}s=\int\limits_{s_{0}}^{+\infty}\int_{\\{v=s\\}}\frac{\lvert\nabla
v\rvert^{3}}{\lvert\nabla v\rvert^{2}s^{n}}{\,\rm d}\sigma{\,\rm
d}s\geq\frac{1}{C}\int\limits_{s_{0}}^{+\infty}\frac{V(s)}{s^{n+1}}{\,\rm
d}s\geq\frac{1}{C}\int\limits_{s_{o}}^{+\infty}\frac{(\delta/n)s^{n}+F(s_{0})}{s^{n+1}}{\,\rm
d}s,$ (4.7)

which contradicts (4.3). ∎

The proof of Theorem 1.1 fully builds on the following improved decay estimate
on $u$. It crucially uses again the vector field $X$, this time integrating
its divergence on geodesic balls in place of sub-level sets of $v$, and a
Harnack-type argument.

######  Proposition 4.2.

Let $(M,g)$ and $u$ satisfy the assumptions of Theorem 1.1. Then, for any
positive $\varepsilon<(n-2)/6$, we have

$u(x)r^{(n-2)/2+\varepsilon}(x)\to 0$ (4.8)

as $r(x)=d(o,x)\to+\infty$.

###### Proof.

We apply the Divergence Theorem to the vector field $X$ of (3.2) in a geodesic
ball $B(o,R)$. We get, by (3.4), that

$\begin{split}\int\limits_{B(o,R)}\frac{1}{v^{n-1}}\left[\left|\nabla\nabla
v-\frac{\Delta v}{n}g\right|^{2}+\mathop{\rm Ric}\nolimits\left(\nabla
v,\nabla v\right)\right]{\,\rm d}\mu=&\frac{1}{2}\int\limits_{\partial
B(o,R)}\frac{1}{v^{n-1}}\langle\nabla\lvert\nabla v\rvert^{2},\nu\rangle{\,\rm
d}\sigma\\\ &-\frac{1}{2}\int\limits_{\partial
B(o,R)}\frac{1}{v^{n}}\langle\lvert\nabla v\rvert^{2}\nabla v,\nu\rangle{\,\rm
d}\sigma\\\ &-2\int\limits_{\partial B(o,R)}\frac{1}{v^{n}}\langle{\nabla
v},\nu\rangle{\,\rm d}\sigma,\end{split}$ (4.9)

where $\nu$ is the outward unit normal to $\partial B(o,R)$. We claim that, if
(4.8) does not hold for some positive $\varepsilon<(n-2)/6$ , then there
exists a sequence of radii $R_{j}\to+\infty$, as $j\to+\infty$, such that the
right-hand side of (4.9) for $R=R_{j}$ vanishes in the limit as $j\to+\infty$.
Indeed, in this case, by the nonnegativity of the left hand side, we deduce
that its whole integrand is null and deduce by Lemma 4.1 that $u$ is (an
Euclidean) Talentian function of the form (1.3). But this actually satisfies
(4.8), giving a contradiction.

Assume then that there exists a sequence of points $x_{j}$, with
$r(x_{j})=d(o,x_{j})\to+\infty$ as $j\to+\infty$, such that

$u(x_{j})r^{\frac{(n-2)}{2}+\varepsilon}(x_{j})\geq C>0$ (4.10)

for any $x_{j}$. By [Min09, Proposition 0.4], in force by condition (1.2),
there exists $k>1$ such that for any $R>1$, for any two points $x,y$ in
$\overline{B(o,R)}\setminus B(o,k^{-1}R)$ there exists a geodesic
$\gamma_{x,y}$ connecting them _fully contained_ in such an annulus. In
particular, setting $R_{j}=r(x_{j})$, for any point
$y\in\overline{B(o,R_{j})}\setminus B(o,k^{-1}R_{j})$ we have

$\log u(y)-\log
u(x_{j})\leq\int\limits_{0}^{\lvert\gamma_{x_{j},y}\rvert}\lvert\nabla\log
u\rvert(\gamma_{x_{j},y}(t)){\,\rm d}t\leq
C\frac{\lvert\gamma_{x_{j},y}\rvert}{R_{j}},$ (4.11)

where we denoted by $\lvert\gamma_{x_{j},y}\rvert$ the length of the geodesic,
and where we employed in the last step the decay estimate for the gradient
(2.2). Moreover, as an application of the Bishop-Gromov monotonicity the ratio
on the rightmost hand side of (4.11) is uniformly bounded, see the annular
diameter estimate [AG90, Lemma 1.4]. Consequently, any point
$y\in\overline{B(o,R_{j})}\setminus B(o,k^{-1}R_{j})$ in fact satisfies
(4.10). Coupling this information with (2.10) and translating in terms of $v$,
we get

$\frac{1}{C}r\leq v\leq Cr^{\frac{2\varepsilon}{n-2)}+1}$ (4.12)

on any annulus $\overline{B(o,R_{j})}\setminus B(o,k^{-1}R_{j})$, for some
constant $C>0$ not depending on $R_{j}$. The gradient decay (2.2), applied to
$v$, improves to

$\lvert\nabla v\rvert(x)\leq Cr(x)^{\frac{2\varepsilon}{n-2}}$ (4.13)

for any $x\in\overline{B(o,R_{j})}\setminus B(o,k^{-1}R_{j})$, for some $C>0$
not depending $R_{j}$. We can thus estimate, for almost any
$\hat{R}_{j}\in(k^{-1}R_{j},R_{j})$,

$\int_{\partial B(o,\hat{R}_{j})}\frac{\lvert\nabla
v\rvert^{b}}{v^{n}}\leq\frac{C}{\hat{R}_{j}^{1-2b\varepsilon/(n-2)}},$ (4.14)

where we used also the lower bound in (4.13) and that $\lvert\partial
B(o,R)\rvert\leq\lvert\mathbb{S}^{n-1}\rvert R^{n-1}$ for almost any $R>0$,
again by the Bishop-Gromov monotonicity (of perimeters). Since
$\varepsilon<(n-2)/6$, the above integral vanishes as $j\to+\infty$, when
$b=3$ and $b=1$, and this in particular will imply that the second and the
third term in (4.9) vanish along $\hat{R_{j}}$. We are left to work out an
inequality suitable for controlling the first one. By direct computation, we
have

$\nabla\nabla v=\frac{n}{2}\frac{\lvert\nabla
v\rvert^{2}}{v}-\frac{2}{(n-2)}v^{\frac{n}{2}}\nabla\nabla u.$

Integrating on an annulus $A_{R_{1},R_{2}}$, with generic radii
$0<R_{1}<R_{2}$, we thus get

$\int\limits_{A_{R_{1},R_{2}}}\frac{\lvert\nabla\nabla v\rvert{\lvert\nabla
v\rvert}}{v^{n-1}}{\,\rm d}\mu\leq
C\left[\int\limits_{A_{R_{1},R_{2}}}\frac{\lvert\nabla
v\rvert^{3}}{v^{n}}{\,\rm
d}\mu+\int\limits_{A_{R_{1},R_{2}}}\frac{\lvert\nabla\nabla
u\rvert\lvert\nabla v\rvert}{v^{\frac{n}{2}-1}}\right]{\,\rm d}\mu.$ (4.15)

The first term in the right-hand side is estimated, choosing
$R_{1}=k^{-1}R_{j}$ and $R_{2}=R_{j}$ and arguing as above with $b=3$, with
$R_{j}^{6\varepsilon/(n-2)}$. We have used in particular that the volume of
the annulus is controlled by $R_{j}^{n}$, by Bishop-Gromov’s inequality.
Restrict now to a smaller annulus $A_{\alpha R_{j},\beta R_{j}}$, with
$k^{-1}<\alpha<\beta<1$ such that

$\int\limits_{A_{\alpha R_{j},\beta R_{j}}}\lvert\nabla\nabla
u\rvert^{2}{\,\rm d}\mu\leq\frac{C}{R_{j}}^{2},$

that exists by Corollary 2.4. In such an annulus, the second term in the right
hand side of (4.15) can be estimated as

$\int\limits_{A_{\alpha R_{j},\beta R_{j}}}\frac{\lvert\nabla\nabla
u\rvert\lvert\nabla v\rvert}{v^{\frac{n}{2}-1}}{\,\rm
d}\mu\leq\frac{\left(\int_{A_{\alpha R_{j},\beta R_{j}}}\lvert\nabla\nabla
u\rvert^{2}{\,\rm d}\mu\int_{A_{\alpha R_{j},\beta R_{j}}}\lvert\nabla
v\rvert^{2}{\,\rm d}\mu\right)^{1/2}}{R_{j}^{\frac{n}{2}-1}}\leq
CR_{j}^{4\varepsilon/(n-2)},$ (4.16)

where we have employed the Hölder inequality together with the estimates on
$v$ and $\lvert\nabla v\rvert$ recalled above. We can thus deduce that

$\int\limits_{A_{\alpha R_{j},\beta R_{j}}}\frac{\lvert\nabla\nabla
v\rvert{\lvert\nabla v\rvert}}{v^{n-1}}{\,\rm d}\mu\leq
C\left(R_{j}^{6\varepsilon/(n-2)}+R_{j}^{4\varepsilon/(n-2)}\right).$ (4.17)

Through the coarea formula, we can in particular infer the existence, for any
$j$, of a set of positive measure $S_{j}\subset(\alpha R_{j},\beta R_{j})$
such that

$\int\limits_{\partial B(0,\hat{R}_{j})}\frac{\lvert\nabla\nabla
v\rvert{\lvert\nabla v\rvert}}{v^{n-1}}{\,\rm d}\sigma\leq
C\left(\frac{1}{\hat{R}_{j}^{1-6\varepsilon/(n-2)}}+\frac{1}{\hat{R}_{j}^{1-4\varepsilon/(n-2)}}\right)$
(4.18)

for any $\hat{R}_{j}\in S_{j}$. Coupling this with (4.14) with $b=1$ and
$b=3$, that holds true in particular for almost any $\hat{R}_{j}\in S_{j}$, we
can thus bound, up to a multiplicative constant, the absolute value of the
right hand side of (4.9) on spheres of such radii $\hat{R}_{j}$ with

$\begin{split}\int\limits_{\partial B(0,\hat{R}_{j})}\frac{\lvert\nabla\nabla
v\rvert{\lvert\nabla v\rvert}}{v^{n-1}}{\,\rm d}\sigma+\int\limits_{\partial
B(o,\hat{R}_{j})}\frac{\lvert\nabla v\rvert^{3}}{v^{n}}{\,\rm
d}\sigma+\int\limits_{\partial B(o,\hat{R}_{j})}\frac{\lvert\nabla
v\rvert}{v^{n}}{\,\rm d}\sigma\leq
C\Bigg{(}\frac{1}{\hat{R}_{j}^{1-6\varepsilon/(n-2)}}&+\frac{1}{\hat{R}_{j}^{1-4\varepsilon/(n-2)}}\\\
&+\frac{1}{\hat{R}_{j}^{1-2\varepsilon/(n-2)}}\Bigg{)},\end{split}$ (4.19)

that vanishes as $j\to+\infty$ since $\varepsilon<(n-2)/6$, completing the
proof. ∎

Theorem 1.1 now follows as an immediate corollary of Theorem 1.2 and
Proposition 4.2.

###### Proof of Theorem 1.1.

By combining (4.8) for some $0<\varepsilon<(n-2)/6$ with (2.2), we deduce that

$\lvert\nabla u\rvert^{2}\leq Cr(x)^{-n-2\varepsilon}$

for any $x\in M\setminus B(o,R)$, with $R$ big enough. Consequently, we have

$\int\limits_{M\setminus B(o,R)}\lvert\nabla u\rvert^{2}{\,\rm d}\mu\leq
C\int\limits_{R}^{+\infty}\int_{B(o,r)}r^{-n-2\varepsilon}{\,\rm
d}\sigma{\,\rm d}r\leq
C\int\limits_{R}^{+\infty}\frac{1}{r^{1+2\varepsilon}}<+\infty,$

where we have used the coarea formula and $\lvert
B(o,r)\rvert\leq\lvert\mathbb{S}^{n-1}\rvert r^{n-1}$ for almost any $r>0$. We
conclude by Theorem 1.2. ∎

## References

  * [AFM20] V. Agostiniani, M. Fogagnolo and L. Mazzieri  “Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature”  In _Inventiones mathematicae_ , 2020  DOI: 10.1007/s00222-020-00985-4
  * [AG90] U. Abresch and D. Gromoll  “On complete manifolds with nonnegative Ricci curvature”  In _J. Amer. Math. Soc._ 3.2, 1990, pp. 355–374  DOI: 10.2307/1990957
  * [Aub76] T. Aubin  “Problèmes isopérimétriques et espaces de Sobolev”  In _J. Differential Geometry_ 11.4, 1976, pp. 573–598  URL: http://projecteuclid.org/euclid.jdg/1214433725
  * [BS18] D. Bianchi and A. G. Setti  “Laplacian cut-offs, porous and fast diffusion on manifolds and other applications”  In _Calc. Var. Partial Differential Equations_ 57.1, 2018, pp. Paper No. 4, 33  DOI: 10.1007/s00526-017-1267-9
  * [CC15] G. Chen and X. Chen  “Gravitational instantons with faster than quadratic curvature decay (I)”, 2015  arXiv:1505.01790 [math.DG]
  * [CC15a] G. Chen and X. Chen  “Gravitational instantons with faster than quadratic curvature decay (II)”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ , 2015  DOI: 10.1515/crelle-2017-0026
  * [CC16] G. Chen and X. Chen  “Gravitational instantons with faster than quadratic curvature decay (III)”  In _Mathematische Annalen_ , 2016  DOI: 10.1007/s00208-020-01984-9
  * [CC96] J. Cheeger and T. H. Colding  “Lower bounds on Ricci curvature and the almost rigidity of warped products”  In _Ann. of Math. (2)_ 144.1, 1996, pp. 189–237  DOI: 10.2307/2118589
  * [CGS89] L. A. Caffarelli, B. Gidas and J. Spruck  “Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth”  In _Comm. Pure Appl. Math._ 42.3, 1989, pp. 271–297  DOI: 10.1002/cpa.3160420304
  * [CM21] T. H. Colding and W. P. Minicozzi  “Optimal growth bounds for eigenfunctions”, 2021  arXiv:2109.04998 [math.DG]
  * [CM22] G. Catino and D. D. Monticelli  “Semilinear elliptic equations on manifolds with nonnegative Ricci curvature”, 2022  arXiv:2203.03345 [math.DG]
  * [CMM12] G. Catino, C. Mantegazza and L. Mazzieri  “On the global structure of conformal gradient solitons with nonnegative Ricci tensor”  In _Commun. Contemp. Math._ 14.6, 2012, pp. 1250045, 12  DOI: 10.1142/S0219199712500459
  * [GNN81] B. Gidas, W. M. Ni and L. Nirenberg  “Symmetry of positive solutions of nonlinear elliptic equations in ${\bf R}^{n}$”  In _Mathematical analysis and applications, Part A_ 7, Adv. in Math. Suppl. Stud.  Academic Press, New York-London, 1981, pp. 369–402
  * [GS81] B. Gidas and J. Spruck  “Global and local behavior of positive solutions of nonlinear elliptic equations”  In _Comm. Pure Appl. Math._ 34.4, 1981, pp. 525–598  DOI: 10.1002/cpa.3160340406
  * [Li] P. Li  “Lectures on Harmonic Function” Lectures at UCI
  * [Min09] V. Minerbe  “Weighted Sobolev inequalities and Ricci flat manifolds”  In _Geom. Funct. Anal._ 18.5, 2009, pp. 1696–1749  DOI: 10.1007/s00039-009-0701-3
  * [Min10] V. Minerbe  “On the asymptotic geometry of gravitational instantons”  In _Ann. Sci. Éc. Norm. Supér. (4)_ 43.6, 2010, pp. 883–924  DOI: 10.24033/asens.2135
  * [MS21] M. Muratori and N. Soave  “Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds”  In _Annali Scuola Normale Superiore - Classe di Scienze_ Scuola Normale Superiore - Edizioni della Normale, 2021, pp. 30  DOI: 10.2422/2036-2145.202105˙071
  * [Oba71] M. Obata  “The conjectures on conformal transformations of Riemannian manifolds”  In _J. Differential Geometry_ 6, 1971/72, pp. 247–258  URL: http://projecteuclid.org/euclid.jdg/1214430407
  * [Pet16] P. Petersen  “Riemannian geometry” 171, Graduate Texts in Mathematics  Springer, Cham, 2016, pp. xviii+499  DOI: 10.1007/978-3-319-26654-1
  * [Tal76] G. Talenti  “Best constant in Sobolev inequality”  In _Ann. Mat. Pura Appl. (4)_ 110, 1976, pp. 353–372  DOI: 10.1007/BF02418013
  * [Yau75] S.-T. Yau  “Harmonic functions on complete Riemannian manifolds”  In _Comm. Pure Appl. Math._ 28, 1975, pp. 201–228  DOI: 10.1002/cpa.3160280203