Abstract
In this paper we adopt the approach presented in Agostiniani and Mazzieri (J Math Pures Appl 104:561–586, 2015; Commun Math Phys 355:261–301, 2017) to study nonsingular vacuum static spacetimes with nonzero cosmological constant. We introduce new integral quantities, and under suitable assumptions we prove their monotonicity along the level set flow of the static potential. We then show how to use these properties to derive a number of sharp geometric and analytic inequalities, whose equality case can be used to characterize the rotational symmetry of the underlying static solutions. As a consequence, we are able to prove some new uniqueness statements for the de Sitter and the antide Sitter metrics. In particular, we show that the de Sitter solution has the least possible surface gravity among threedimensional static metrics with connected boundary and positive cosmological constant.
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References
V. Agostiniani, L. Mazzieri, Riemannian aspects of potential theory. J. Math. Pures Appl. 104(3), 561–586 (2015)
V. Agostiniani, L. Mazzieri, Monotonicity formulas in potential theory (2016). https://arxiv.org/abs/1606.02489.
V. Agostiniani, L. Mazzieri, Comparing monotonicity formulas for electrostatic potentials and static metrics. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28(1), 7–20 (2017)
V. Agostiniani, L. Mazzieri, On the geometry of the level sets of bounded static potentials. Commun. Math. Phys. 355(1), 261–301 (2017)
V. Agostiniani, S. Borghini, L. Mazzieri, On the torsion problem for domains with multiple boundary components (in preparation)
V. Agostiniani, M. Fogagnolo, L. Mazzieri, Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. arXiv preprint arXiv:1812.05022 (2018)
L. Ambrosio, G. Da Prato, A. Mennucci, Introduction to measure theory and integration, in Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 10 (Edizioni della Normale, Pisa 2011)
L. Ambrozio, On static threemanifolds with positive scalar curvature. J. Differ. Geom. 107(1), 1–45 (2017)
S. Borghini, L. Mazzieri, On the mass of static metrics with positive cosmological constantII. 2017. ArXiv Preprint Server https://arxiv.org/abs/1711.07024
S. Borghini, L. Mazzieri, On the mass of static metrics with positive cosmological constant: I. Classical and Quantum Gravity 35(12), 125001 (2018)
S. Borghini, G. Mascellani, L. Mazzieri, Some sphere theorems in linear potential theory. Trans. Am. Math. Soc. (2019). https://doi.org/10.1030/tran/7637
W. Boucher, G.W. Gibbons, G.T. Horowitz, Uniqueness theorem for antide Sitter spacetime. Phys. Rev. D (3) 30(12), 2447–2451 (1984)
P.T. Chruściel, On analyticity of static vacuum metrics at nondegenerate horizons. Acta Phys. Polon. B 36(1), 17–26 (2005)
P.T. Chruściel, Remarks on rigidity of the de sitter metric. http://homepage.univie.ac.at/piotr.chrusciel/papers/deSitter/deSitter2.pdf
P.T. Chruściel, M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 212(2), 231–264 (2003)
E. De Giorgi, Complementi alla teoria della misura (n − 1)dimensionale in uno spazio ndimensionale, in Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–1961 (Editrice Tecnico Scientifica, Pisa, 1961)
E. De Giorgi, Frontiere Orientate di Misura Minima (Editr. Tecnico scientifica, 1961)
W. De Sitter, On the curvature of space. Proc. Kon. Ned. Akad. Wet 20, 229–243 (1917)
H. Federer, The Gauss–Green theorem. Trans. Am. Math. Soc. 58, 44–76 (1945)
H. Federer, A note on the GaussGreen theorem. Proc. Am. Math. Soc. 9, 447–451 (1958)
H. Federer, Geometric measure theory, in Die Grundlehren der mathematischen Wissenschaften, Band 153 (Springer, New York, 1969)
M. Fogagnolo, L. Mazzieri, A. Pinamonti, Geometric aspects of pcapacitary potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire. https://doi.org/10.1016/j.anihpc.2018.11.005
G.W. Gibbons, S.A. Hartnoll, C.N. Pope, Bohm and EinsteinSasaki metrics, black holes, and cosmological event horizons. Phys. Rev. D (3) 67(8), 084024 (2003)
O. Hijazi, S. Montiel, Uniqueness of the AdS spacetime among static vacua with prescribed null infinity. Adv. Theor. Math. Phys. 18(1), 177–203 (2014)
O. Hijazi, S. Montiel, S. Raulot, Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant. Ann. Glob. Anal. Geom. 47(2), 167–178 (2015)
F. Kottler, Über die physikalischen grundlagen der Einsteinschen gravitationstheorie. Ann. Phys. (Berlin) 361(14), 401–462 (1918)
S.G. Krantz, H.R. Parks, A primer of real analytic functions, in Birkhäuser Advanced Texts: Basler Lehrbücher, 2nd edn. [Birkhäuser Advanced Texts: Basel Textbooks] (Birkhäuser, Boston, 2002)
J. Lafontaine, Sur la géométrie d’une généralisation de l’équation différentielle d’Obata. J. Math. Pures Appl. (9) 62(1), 63–72 (1983)
L. Lindblom, Static uniformdensity stars must be spherical in general relativity. J. Math. Phys. 29(2), 436–439 (1988)
S. Łojasiewicz, Introduction to Complex Analytic Geometry (Birkhäuser, Basel, 1991). Translated from the Polish by Maciej Klimek
H. Nariai, On a new cosmological solution of Einstein’s fieldequations of gravitation. Sci. Rep. Tohoku Univ. Ser. I 35(1), 62–67 (1951)
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)
R. Penrose, Asymptotic properties of fields and spacetimes. Phys. Rev. Lett. 10, 66–68 (1963)
J. Qing, On the uniqueness of AdS spacetime in higher dimensions. Ann. Henri Poincaré 5(2), 245–260 (2004)
K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory. Gen. Relativ. Gravit. 35(5), 951–959 (2003). Translated from the original German article [Sitzungsber. Königl. Preussich. Akad. Wiss. Berlin Phys. Math. Kl. 1916, 189–196] by S. Antoci and A. Loinger
J. Serrin, Isolated singularities of solutions of quasilinear equations. Acta Math. 113, 219–240 (1965)
J. Souček, V. Souček, MorseSard theorem for realanalytic functions. Comment. Math. Univ. Carol. 13, 45–51 (1972)
X. Wang, The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 57(2), 273–299 (2001)
X. Wang, On the uniqueness of the AdS spacetime. Acta Math. Sin. (Engl. Ser.) 21(4), 917–922 (2005)
H.F. Weinberger, Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal. 43, 319–320 (1971)
H.M. Zum Hagen, On the analyticity of static vacuum solutions of Einstein’s equations. Proc. Camb. Philos. Soc. 67, 415–421 (1970)
Acknowledgements
The authors would like to thank P. T. Chruściel for his interest in our work and for stimulating discussions during the preparation of the manuscript. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and are partially founded by the GNAMPA Project “Principi di fattorizzazione, formule di monotonia e disuguaglianze geometriche”.
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Appendix A: Technical Results
Appendix A: Technical Results
This appendix will be dedicated to the proof of the technical results that we have used in our work. Specifically, we will give a complete proof of Theorems 1.2, 1.5 (for the ease of reference, we have restated them here as Theorem A.1 and Theorem A.7), we will prove an estimate on the static solution near the conformal boundary in the case Λ < 0, and we will state the version of the divergence theorem that we have used in the proofs of Propositions 5.1, 5.2.
Theorem A.1
Let (M, g _{0}, u) be a solution of (1.4) satisfying Assumption 1 . The set MAX(u) is discrete (and finite) and
for every 0 ≤ p ≤ n − 1.
In the proof of this theorem, we will need the following result, that will be proven later.
Proposition A.2
Let (M, g _{0}, u) be a solution of (1.4) and let y _{0} ∈MAX(u). Then for every d > 0 it holds
We first show how to use this result to prove Theorem A.1.
Proof of Theorem A.1
First we notice that the functions U _{p}(t) can be written as follows
From formula (1.10) in Theorem 1.1, we have that the term in square bracket is less or equal to 1. Thus, for every p ≤ n − 1, we have
hence U _{p}(t) ≥ U _{n−1}(t) and, in particular
so it is enough to prove the inequality (A.1) for p = n − 1.
Now we pass to analyze the set MAX(u). Suppose that it contains an infinite number of points. Then for each \(k\in \mathbb {N}\) we can consider k points in MAX(u). Let 2d be the minimum of the distances between our points. Applying Proposition A.2 in a neighborhood of radius d of each of these points, we obtain
Since this is true for every \(k\in \mathbb {N}\), we conclude that \(\lim _{t\to 1^} U_{n1}(t)=+\infty \) and, using (A.3), we find that \(\lim _{t\to 1^} U_1(t)=+\infty \). But this is impossible, since from the monotonicity of U _{1}(t) (stated in Theorem 1.1(ii)) we know that
Therefore MAX(u) contains only a finite number of points. Repeating the argument above with k = MAX(u) we obtain the inequality in the thesis. □
Now we turn to the proof of Proposition A.2, that will be done in various steps. Our strategy consists in choosing a suitable neighborhood of the point y _{0} where we are able to control the quantities in our limit, and then proceed to estimate them.
Notation 1
Here and throughout the paper, we agree that for \(f \in {\mathcal C}^{\infty }(M)\) , \(\tau \in \mathbb {R}\) and \(k \in \mathbb {N}\) it holds
where the J’s are multiindexes.
Consider a normal set of coordinates (x ^{1}, …, x ^{n}) in B _{d}(y _{0}), that diagonalize the hessian in y _{0}. Note that, since y _{0} is a maximum of u, the derivatives \(\partial ^2_{\alpha }u_{{ }_{y_0}}\) are non positive numbers for all α = 1, …, n, hence it makes sense to introduce the quantities \(\lambda _\alpha ^2=\partial ^2_\alpha u_{{ }_{y_0}}\) for α = 1, …, n. Since Δu = −nu, we have \(\sum _{\alpha =1}^n \lambda _\alpha ^2=n\). In particular, at least one of the λ _{α}’s is different from zero. We have the following Taylor expansion of u in a neighborhood of y _{0}
From (A.4) we easily compute
Now we consider polar coordinates (r, θ ^{1}, …, θ ^{n−1}), where \(\theta =(\theta ^1,\dots ,\theta ^{n1})\in \mathbb {R}^{n1}\) are stereographic coordinates on \(\mathbb {S}^{n1}\setminus \{north\ pole\}\).
Lemma A.3
With respect to the coordinates (r, θ) = (r, θ ^{1}, …, θ ^{n−1}), the metric g _{0} writes as
where σ = o _{2}(r), σ _{i} = o _{2}(r ^{2}), σ _{ij} = o _{2}(r ^{3}), as r → 0^{+}.
Proof
To ease the notation, in this proof we use the Einstein summation convention. It is known that, with respect to the normal coordinates (x ^{1}, …, x ^{n}) the metric g _{0} writes as
where δ _{αβ} is the Kronecker delta and η _{αβ} = o _{2}(r) (actually, the term η _{αβ} can be estimated better, but this is enough for our purposes).
Moreover, it is easy to check that the quantities ϕ ^{α} = x ^{α}∕r are smooth functions of the coordinates (θ ^{1}, …, θ ^{n−1}) only, and that
From r ^{2} = δ _{αβ} x ^{α} x ^{β} one also finds the equality δ _{αβ} ϕ ^{α} ϕ ^{β} = 1. Deriving it with respect to θ ^{i} we get
We are now ready to compute
where σ, σ _{i}, σ _{ij} are infinitesimals of the wished order. □
We can rewrite formulæ (A.4), (A.5) in terms of (r, θ) as
where , . Moreover, since we know from (A.4) that ∂w∕∂x ^{α} = o(r) for any α, we have the following estimates on the order of the derivatives of w with respect to (r, θ)
To estimate the limit in (A.2), we need to rewrite the set {u = t}∩ B _{d}(y _{0}) and the density
as functions of our coordinates. In order to do so, it will prove useful to restrict our neighborhood B _{d}(y _{0}) to a smaller domain where we have a better characterization of the level set {u = t}. In this regard, it is convenient, for any ε > 0, to define the set
The following result shows that, for t small enough, the level set {u = t}, is a graph over C _{ε}.
Lemma A.4
For any 0 < ε < 1, there exists η = η(ε) > 0 such that

(i)
the estimates \(w(r,\theta )<\frac {\varepsilon ^2}{4}\,r^2\) , \(\partial w/\partial r(r,\theta )<\frac {\varepsilon }{2}\,r\) , h(r, θ) < ε ^{2} r ^{2} holds on the whole B _{η}(y _{0}).

(ii)
it holds \(\frac {\partial u}{\partial r}(r,\theta )<0\) in (0, η) × C _{ε}.

(iii)
for every 0 < δ < η, there exists τ = τ(δ, ε) such that for any τ < t < 1, there exists a smooth function r _{t} : C _{ε} → (0, δ) such that
$$\displaystyle \begin{aligned}\{u=t\}\cap B_\delta(y_0)\cap C_\varepsilon\,=\,\{(r_t(\theta),\theta)\,:\,\theta\in C_\varepsilon\}\,. \end{aligned}$$
Proof
Since the functions w, h in (A.7), (A.8) are o(r ^{2}), while ∂w∕∂r is o(r) thanks to (A.9), it is clear that statement (i) is true for some η small enough. Moreover, from expansion (A.7) we compute
This proves point (ii). To prove (iii), fix t ∈ (0, 1) and consider the function u(r, θ) − t. Since u(r, θ) → 1^{−} as r → 0^{+}, we have u(r, θ) − t > 0 for small values of r.
On the other hand, from expansion (A.7) we find
and the quantity on the right is negative for any \(t>\tau =1\frac {\varepsilon }{4}\delta ^2\).
Therefore, fixed a θ ∈ C _{ε} the function r↦u(r, θ) − t is positive for small values of r and negative for r = δ. Moreover from point (ii) we have that \(\frac {\partial u}{\partial r}(r,\theta )<0\) for any (r, θ) ∈ (0, δ) × C _{ε}, hence for any θ ∈ C _{ε}, there exists one and only one value 0 < r _{t}(θ) < δ such that (r _{t}(θ), θ) ∈{u = t}. The smoothness of the function r _{t}(θ) is a consequence of the Implicit Function Theorem applied to the function u(r, θ). □
As anticipated, Lemma A.4 will now be used to estimate the density of the restriction of the metric g _{0} on {u = t}∩ ((0, δ) × C _{ε}).
Lemma A.5
There exists 0 < δ < η(ε) such that it holds
for every θ ∈ C _{ε} , τ(δ, ε) < t < 1.
Proof
Let r _{t} be the function introduced in Lemma A.4. Taking the total derivative of u(r _{t}(θ), θ) = t, we find, for any θ ∈ C _{ε}
where
To ease the notation, in the rest of the proof we avoid to explicitate the dependence of the functions on the variables r _{t}(θ), θ. In order to compute the restriction of the metric on {u = t}∩ ((0, δ) × C _{ε}), we substitute the term dr in formula (A.6) with the formula for dr _{t} computed above. We obtain
Set \(\xi =\sum _{j=1}^{n1}\xi _j d\theta ^j\). We have the following
where in the second equality we have used the Matrix Determinant Lemma.
On the other hand, since σ _{i} = o(r ^{2}) and σ _{ij} = o(r ^{3}), we deduce that
hence
with ω = o(1) as r → 0^{+}. In particular, we can choose δ small enough so that ω < ε on (0, δ) × C _{ε}. Combining Eqs. (A.11) and (A.12) we have the thesis. □
We also need an estimate of the integrand in (A.2), which is provided by the following lemma.
Lemma A.6
We can choose 0 < δ < η(ε) such that
for every (r, θ) ∈ (0, δ) × C _{ε}.
Proof
To ease the notation, in this proof we avoid to explicitate the dependence of the functions on the coordinates r, θ. From expansions (A.7) and (A.8) we deduce
Using the CauchySchwarz Inequality we have
hence, recalling that \(\sum _{\alpha =1}^n (\lambda _\alpha ^2 \phi ^\alpha { }^2)>\varepsilon \) on C _{ε}, we have also \(\sum _{\alpha =1}^n (\lambda _\alpha ^4 \phi ^\alpha { }^2)>\varepsilon ^2\). Therefore, from Lemma A.4(i) we easily compute
In particular, we can choose δ small enough so that the right hand side of the inequality above is greater than 1 − ε. Hence, we get
where in the first inequality we have used that u ≤ 1 on M and in the latter inequality we have used (A.13). □
Now we are finally able to prove our proposition.
Proof of Proposition A.2
For every ε > 0 and 0 < δ ≤ d, we have the following estimate of the left hand side of condition (A.2)
Since u ≤ 1, we have 2∕(1 + u) ≥ 1. Moreover, from (A.7) and Lemma A.4(i), we obtain
Now fix a δ small enough so that Lemmas A.5, A.6 are in charge. Taking the limit of integrand (A.14) as t → 1^{−} we obtain the estimate
It is clear that the functions converge to as ε → 0^{+}, where
Therefore, taking the limit of (A.15) as ε → 0^{+} and using the Monotone Convergence Theorem, we find
To end the proof, it is enough to show that the set \(\mathbb {S}^{n1}\setminus C_0\) is negligible. But this is clear. In fact, since \(\sum _{\alpha =1}^n\lambda _\alpha ^2=n\), there exists at least one integer β such that λ _{β} ≠ 0. Thus \(\mathbb {S}^{n1}\setminus C_0\) is contained in the hypersurface {ϕ ^{β} = 0}, hence its nmeasure is zero. This proves inequality (A.2) and the thesis. □
This concludes the proof for the de Sitter case. In the antide Sitter case we can prove the following analogue of Theorem A.1.
Theorem A.7
Let (M, g _{0}, u) be a conformally compact static solution of problem 1.17 satisfying Assumption 2 . Then the set MIN(u) is discrete (and finite) and
for every p ≤ n − 1.
The proof follows the exact same scheme as the de Sitter case, the only small modifications being in the proof of Lemma A.5 and in the computation (A.15), where we have used the fact that u ≤ 1. This is not true anymore, however, since we are working around a minimum point, we can suppose u < 1 + κ, where κ is an infinitesimal quantity that can be chosen to be as small as necessary. Aside from this little expedient, the proof is virtually the same as the de Sitter case, thus we omit it.
We pass now to the proof of some other results that we have used in our work. The next lemma is useful in order to study the behavior of the static solutions of problem (1.17) near the conformal boundary.
Lemma A.8
Let (M, g _{0}, u) be a conformally compact static solution to problem (1.17). Suppose that \(1/\sqrt {u^21}\) is a defining function, so that the metric g = g _{0}∕(u ^{2} − 1) extends to the conformal boundary ∂M. Then

(i)
\(\lim _{x\to \bar {x}}(u^21{\mathrm {D}} u{ }^2)\) is welldefinite and finite for every \(\bar x\in \partial M\) ,

(ii)
∂M is a totally geodesic hypersurface in \((\overline {M},g)\).
Proof
For the proof of this result, it is convenient to use the notations introduced in Sect. 3. Let φ be the function defined by (3.18). By hypothesis, M is the interior of a compact manifold \(\overline {M}\) and the metric g is well defined on the whole \(\overline {M}\). In particular, the scalar curvature R_{g} is a smooth finite function at ∂M. Therefore, from Eq. (3.24) we easily deduce that \(\lim _{x\to \bar x}u^2(1\nabla \varphi { }_g^2)\) is welldefinite and finite for every \(\bar x\in \partial M\). Since
this proves point (i).
To prove statement (ii), we first observe that, since ∇φ_{g} = 1 at ∂M (as it follows immediately from point (i)), there exists δ > 0 such that ∇φ_{g} ≠ 0 on the whole collar \(\mathcal {U}_\delta =\{\varphi <\delta \}\). Therefore, proceeding as in Sect. 3.4, we find a set of coordinates {φ, 𝜗 ^{1}, …, 𝜗 ^{n−1}} on \(\mathcal {U}_{\delta }\), such that the metric g writes as
With respect to these coordinates, the second fundamental form of the boundary ∂M = {φ = 0} is
On the other hand, from the first equation of problem (3.23), we easily deduce that ∇^{2} φ = 0 on ∂M. This concludes the proof of point (ii). □
Finally, in order to prove the integral identities in Sect. 5, we need an extension of the classical Divergence Theorem to the case of open domains whose boundary has a (not too big) nonsmooth portion. Note that [4, Theorem A.1] is not enough for our purposes, because hypothesis (ii) is not necessarily fulfilled. To avoid problems, we state the following generalization, due to De Giorgi and Federer.
Theorem A.9 ([16, 17, 19, 20])
Let (M, g) be a ndimensional Riemannian manifold, with n ≥ 2, let E ⊂ M be a bounded open subset of M with compact boundary ∂E of finite (n − 1)dimensional Hausdorff measure, and suppose that ∂E = Γ ⊔ Σ, where the subsets Γ and Σ have the following properties:

(i)
For every x ∈ Γ, there exists an open neighborhood U _{x} of x in M such that Γ ∩ U _{x} is a smooth regular hypersurface.

(ii)
The subset Σ is compact and \(\mathcal {H}^{n1}(\Sigma ) = 0\).
If X is a Lipschitz vector field defined in a neighborhood of \(\overline {E}\) then the following identity holds true
where n denotes the exterior unit normal vector field.
Appendix B: BoucherGibbonsHorowitz Method
In this section we discuss an alternative approach to the study of the rigidity of the de Sitter and antide Sitter spacetime, which does not require the machinery of Sect. 3. Without the need of any assumptions, this method will allow to derive results that are comparable to Theorems 2.2, 2.6 (case Λ > 0) and Theorems 2.14, 2.18 (case Λ < 0). In the case Λ > 0, the computations that we are going to show are quite classical (see [12, 14]). However, to the author’s knowledge, the analogous calculations in the case Λ < 0 are new.
As usual, we start with the case Λ > 0. Recalling the Bochner formula and the equations in (1.4) we compute
Now, if we consider the field
we can compute its divergence using (B.1).
More generally, for every nonzero \(\mathcal {C}^1\) function α = α(u):
where \(\dot \alpha \) is the derivative of α with respect to u. The computation above suggests us to choose
With this choice of α, we have
Proposition B.1
Let (M, g _{0}, u) be a static solution to problem (1.4). Then, for every t ∈ [0, 1) it holds
Moreover, if there exists t _{0} ∈ (0, 1) such that
then the triple (M, g _{0}, u) is isometric to the de Sitter solution.
Remark B.1
Recalling Remark 1, it is easy to realize that the integral on the left hand side of (B.3) is well defined also when t is a singular value of u.
Remark B.2
Note that the right hand side of inequality (B.3) is always nonnegative, as opposed to formula (5.8), where we needed to suppose Assumption 3 to achieve the same result. This is one of the reasons why this approach works without the need to suppose any assumption.
Proof of Proposition B.1
Suppose for the moment that {u = t} is a regular level set. Integrating by parts identity (B.2), we obtain
where n = −Du∕Du is the outer gunit normal of the set {u ≥ t} at its boundary. On the other hand, from the first formula in (3.27) it is easy to deduce that
Substituting in (B.5) proves formula (B.3) in the case where {u = t} is a regular level set.
In the case where t > 0 is a singular value of u, we need to apply a slightly refined version of the Divergence Theorem, namely Theorem A.9 in the Appendix A, in order to perform the integration by parts which leads to identity (B.5). The rest of the proof is identical to what we have done for the regular case. We set
so that ∂E = {u = t}.
As usual, we denote by Crit(u) = {x ∈ M  Du(x) = 0} the set of the critical points of u, From [30] (see also [27, Theorem 6.3.3]), we know that there exists an open (n − 1)dimensional submanifold N ⊆Crit(u) such that \(\mathcal {H}^{n1}({\mathrm {Crit}}(u)\setminus N)=0\). Set Σ = ∂E ∩ (Crit(u) ∖ N) and Γ = ∂E ∖ Σ, so that ∂E can be written as the disjoint union of Σ and Γ. We have \(\mathcal {H}^{n1}(\Sigma )=0\) by definition, while Γ is the union of the regular part of ∂E and of N, which are open (n − 1)submanifolds. Therefore the hypotheses of Theorem A.9 are met, and we can apply it to conclude that Eq. (B.5) holds true also when t is a singular value of u.
To prove the second part, we observe that from (B.3) and (B.4) one immediately gets D^{2} u = ( Δu∕n) g _{0} in {u ≥ t _{0}}. Since u is analytic, the same equality holds on the whole manifold M. Now we can use the results in [28] to conclude that (M, g _{0}, u) is the de Sitter solution. □
The proposition above is particularly interesting when applied at the boundary ∂M = {u = 0}.
Corollary B.2
Let (M, g _{0}, u) be a static solution to problem (1.4). Then it holds
Moreover, if the equality holds then the triple (M, g _{0}, u) is isometric to the de Sitter solution.
Proof
First we compute from the equations in (1.4) and formula (3.27), that
where ν = Du∕Du as usual. In particular, we have H = 0 on ∂M. Hence we can use the Gauss–Codazzi identity to find
Substituting t = 0 in formula (B.3) and applying Proposition B.1, we have the thesis. □
Now we turn our attention to the case Λ < 0. Mimicking the computations done in the case Λ > 0, but using the equations in (1.17) instead of the ones in (1.4) we obtain
Incidentally, we notice that this equation coincides with the analogous formula (B.2) in the case Λ > 0. We are now ready to state the analogous of Proposition B.1.
Proposition B.3
Let (M, g _{0}, u) be a static solution to problem (1.17). Then, for every t ∈ (1, +∞) it holds
Moreover, if there exists t _{0} ∈ (1, +∞) such that
then the triple (M, g _{0}, u) is isometric to the antide Sitter solution.
Remark B.3
Recalling Remark 1, it is easy to realize that the integral on the left hand side of (B.8) is well defined also when t is a singular value of u.
Proof of Proposition B.3
The proof is almost identical to the proof of Proposition B.3. The only change is that, when we apply the divergence theorem, we need the outer gunit normal of the set {u ≤ t}, that is n = Du∕Du instead of −Du∕Du. This is the reason of the different signs in formulæ (B.3), (B.8). □
Now suppose that the manifold M is conformally compact. We would like to use Proposition B.3 to study the behavior of a static solution at the conformal boundary ∂M. In order to simplify the computations and to emphasize the analogy with the case Λ > 0, it will prove useful to suppose that Assumption 2bis holds. Therefore, from now on we suppose that \(1/\sqrt {u^21}\) is a defining function, and that \(\lim _{x\to \bar x}(u^21{\mathrm {D}} u{ }^2)=0\) for every \(\bar x\in \partial M\). We are now ready to prove the analogous of Corollary B.2 in the case of a negative cosmological constant.
Corollary B.4
Let (M, g _{0}, u) be a conformally compact static solution to problem (1.17) satisfying assumption 2bis , and let g = g _{0}∕(u ^{2} − 1). Then it holds
Moreover, if
where ν _{g} = Du∕Du_{g} , then the triple (M, g _{0}, u) is isometric to the antide Sitter solution.
Proof
First we compute from the equations in (1.17) and formula (3.27), that
where ν = Du∕Du as usual. Therefore, we can rewrite formula (B.8) as
Now we use Eq. (3.16) in order to rewrite the term in the square brackets in the following way
Now it is easy to obtain from inequality (B.12) the following formula
Since \(\lim _{x\to \bar x} (u^21{\mathrm {D}} u{ }^2)=0\), in particular \({\mathrm {D}} u/\sqrt {u^21}\) goes to zero as t → +∞. Therefore, taking the limit as t → +∞ of the formula above, we obtain
where ν _{g} = Du∕Du_{g}. Since ∂M is a totally geodesic hypersurface by Lemma A.8(ii), from the Gauss–Codazzi equation and formula (3.24) we obtain
Substituting in Eq. (B.13) we obtain formula (B.10).
To prove the rigidity statement, we observe that we can rewrite formula (B.11) as
Now we recall that u [Ric(ν, ν) + (n − 1)] = −HDu − (n − 1) Δu∕n and we conclude using Proposition B.3. □
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Borghini, S., Mazzieri, L. (2019). Monotonicity Formulas for Static Metrics with Nonzero Cosmological Constant. In: Dipierro, S. (eds) Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Series, vol 33. Springer, Cham. https://doi.org/10.1007/9783030189211_3
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DOI: https://doi.org/10.1007/9783030189211_3
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