An Obata-type theorem on compact Einstein manifolds with boundary

We show a kind of Obata-type theorem on a compact Einstein n-manifold (W,g¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(W, \bar{g})$$\end{document} with smooth boundary ∂W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial W$$\end{document}. Assume that the boundary ∂W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial W$$\end{document} is minimal in (W,g¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(W, \bar{g})$$\end{document}. If (∂W,g¯|∂W)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\partial W, \bar{g}|_{\partial W})$$\end{document} is not conformally diffeomorphic to (Sn-1,gS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S^{n-1}, g_S)$$\end{document}, then for any Einstein metric gˇ∈[g¯]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\check{g} \in [\bar{g}]$$\end{document} with the minimal boundary condition, we have that, up to rescaling, gˇ=g¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\check{g} = \bar{g}$$\end{document}. Here, gS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_S$$\end{document} and [g¯]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\bar{g}]$$\end{document} denote respectively the standard round metric on the (n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-1)$$\end{document}-sphere Sn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{n-1}$$\end{document} and the conformal class of g¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{g}$$\end{document}. Moreover, if we assume that ∂W⊂(W,g¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial W \subset (W, \bar{g})$$\end{document} is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of (W,∂W,[g¯])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(W, \partial W, [\bar{g}])$$\end{document}.


Introduction and main results
In [14,15], Obata has proved the following uniqueness theorem for constant scalar curvature metrics (csc metrics for brevity) on a closed Einstein manifold.
Here, [g] and g S = g S n denote respectively the conformal class of g and the round metric of constant curvature one on the n-sphere S n .
We will first review briefly a uniqueness problem on compact manifolds with boundary, which is related to Obata Theorem. Let W be a compact n-manifold (n ≥ 2) with non-empty smooth boundary ∂ W . Letḡ be a metric on W such that ∂ W is minimal in (W ,ḡ). We shall call suchḡ a relative metric on W (cf. [1]). For the conformal class [ḡ], its subset is called the relative conformal class ofḡ, where H g denotes the mean curvature ∂ W in (W , g), and ν = νḡ denotes the inner unit normal vector field along ∂ W with respect toḡ. Set S n + := {x = (x 1 , · · · , x n , x n+1 ) ∈ S n ⊂ R n+1 | x n+1 ≥ 0}, S n−1 := {x ∈ S n | x n+1 = 0}, Conf(S n + , [g S ]) := {ϕ ∈ Conf(S n , [g S ]) | ϕ(S n + ) = S n + }, where g S denotes both the round metric on S n and its restriction to S n + . Then, we consider the following naive question: Problem Letḡ be a relative Einstein metric on a compact n-manifold W with boundary ∂ W . Then, for any relative csc metricǧ ∈ [ḡ] 0 , the question is whether the following uniqueness holds or not.
Like the case of non-positive csc metrics on a closed manifold, ifḡ has non-positive scalar curvature Rḡ ≤ 0, then the uniqueness for relative csc metrics in [ḡ] 0 holds, up to rescaling (see the argument of [11, Proof of Theorem 3.6] for proof). Hence, in the above problem, it is enough to consider only the case of positive scalar curvature Rḡ > 0. Unfortunately, there exists a counterexample to the second assertion (2) (see Escobar [7,p. 875] for instance). However, if we assume that ∂ W is totally geodesic in (W ,ḡ), then the following holds, which is a relative version of Obata Theorem.
The proof of Obata Theorem consists of two steps. The first step is to prove thatǧ ∈ [g] is also an Einstein metric on M. So ignoring the first step, we may regard Obata Theorem as the uniqueness result for Einstein metrics in a given conformal class.
Our main result of this paper is the following, which is released from the assumption that ∂ W is totally geodesic in (W ,ḡ).
The equality in (2) also holds if and only if (W ,ḡ) is homothetic to (S n + , g S n ). This paper is organized as follows. In Sect. 2, we recall some background materials, particularly a variational characterization of relative Einstein metrics and the Yamabe problem on compact manifolds with boundary. We also give another counterexample to the second assertion (2) of Problem, which is different from the one in Escobar [7, p. 875]. In Sect. 3, we prove Theorem 1.2. In Sect. 4, we finally prove Theorem 1.1.

Preliminaries
We first recall to the variational characterization of Einstein metrics on a closed manifold. Let M be a closed n-manifold (n ≥ 3). It is well known that a Riemannian metric on a closed manifold M is Einstein if and only if it is a critical point of the normalized Einstein-Hilbert functional E on the space M(M) of all Riemannian metrics on M where R g , dμ g and Vol(M, g) denote respectively the scalar curvature and volume measure of g, and the volume of (M, g). However, if we consider the analogue of the case of the functional E on a compact n-manifold W with non-empty boundary ∂ W , then the set of critical points of E on the space M(W ) is empty (see Proposition 2.1 below), where M(W ) denotes the space of all Riemannian metrics on W . In this case, we need to fix a kind of boundary condition for all metrics, and then E must be restricted to a subspace of M(W ).
For a fixed conformal class C ∈ C(∂ W ) on ∂ W , set several subspaces of M(W ) for a as below: By Proposition 2.1 below, it is reasonable to restrict the functional E to the subspace either M 0 (W , C) or M 0 (W ).

and only ifḡ is an Einstein metric such that ∂ W is totally geodesic in
Here, for instance, Crit(E) and Crit(E| M 0 (W ,C) ) denote respectively the set of all critical metrics of E and the set of those of its restriction to M 0 (W , C).

)]):
From this, note that [ḡ] 0 = ∅ for anyḡ ∈ M(W ). From now on, we throughout assume that g ∈ M 0 (W ), namely a relative metric on W . The restriction of E to any relative conformal class [ḡ] 0 is always bounded from below similarly to the case of closed manifolds. Hence, we can consider the following conformal invariant of (W , which is called the relative Yamabe constant of (W , ∂W , [ḡ]), or (W , [ḡ]) simply (cf. [1]). Then, the following Aubin-type inequality for relative Yamabe constants, the so-called relative Aubin's inequality holds (cf. [8, (4)]): for any compact Riemannian n-manifold (W ,ḡ) with minimal boundary ∂ W .
The relative Yamabe constant is related to the Yamabe problem on a compact n-manifold W with boundary below If such aǧ exists, it is called a relative Yamabe metric in [ḡ] 0 , which is a csc metric with On the relative Yamabe problem, although the formulation (3) in Escobar [8] is slightly different from the above, but these are same each other. The relative Yamabe problem was solved by Cherrier [5] and Escobar [8] under some restrictions. Indeed, Cherrier proved the existence of a minimizer for the Yamabe functional Escobar also solved the relative Yamabe problem under one of the restrictions we list below.
Here is the list of conditions given in [8, Theorem 6.1] (on further developments, see [4] for instance): (i) n = 3, 4, or 5, (ii) W has a non totally geodesic point on ∂ W , (iii) ∂ W is totally geodesic and (W ,ḡ) is locally conformally flat, (iv) ∂ W is totally geodesic, n ≥ 6 and the Weyl tensor does not vanish identically on ∂ W . Note that, in the case of either (ii) or (iv), the above strict inequality (3) holds. By the above results, it is easy to see that the relative Yamabe problem is generically solvable. Next, we will give some examples for Escobar Theorem.
Example 2.2 (1) Letḡ be the standard product Einstein metric on W := S 2 (1) × S 2 + (1), where S 2 (1) denotes the round 2-sphere of radius 1 in R 3 . It is obvious that ∂ W is totally geodesic. By Escobar Theorem, this metricḡ is a unique relative csc metric in [ḡ] 0 , up to rescaling.
(2) Set W := CP 2 − B 4 . Let g P be the Page metric on CP 2 #(−CP 2 ) = W ∂ W (−W ), which is an Einstein metric and not a metric of constant curvature. Then, there exists a natural isometric involution ι on CP 2 #(−CP 2 ) such that the fixed point set of ι is equal to ∂ W (cf. [3, Proof of Theorem 9.125]). Hence, ∂ W is totally geodesic. By Escobar Theorem again, the restriction g P | W of g P to W is a unique relative csc metric in [g P | W ] 0 , up to rescaling.
We now give another counterexample to the second assertion (2) in Problem of Sect. 1, which is different from the one in [7, p. 875].
Counterexample Consider the Clifford torus (T 2 ): It is a minimal flat torus. Set It is well known that Vol(V 1 ) = Vol(V 2 ), each of which V i (i = 1, 2) is a solid torus with minimal boundary ∂ V i = (T 2 ). Hence,ḡ := g S | V 1 is a metric of constant curvature one on V 1 , and thus it is a relative Einstein metric. However, since (T 2 ) is not totally geodesic in V 1 , it is follows from Escobar's result [8,Theorem 4 . Hence, from Cherrier's result, there exists a relative Yamabe metricǧ ∈ [ḡ] 0 such that , thenǧ =ḡ, and thus the uniqueness assertion for relative csc metrics does not hold. Note also that, from Theorem 1.1, (1.2),ǧ is not an Einstein metric. More generally, any separating embedded minimal hypersurface N in S n (1) (n ≥ 3) which is not totally geodesic produces a counterexample as below. Let ). By the same reason as the above, we get the counterexample (W 1 , g S n | W 1 ) to the second assertion (2) in Problem.
We finally prepare a necessary result for our proof of Theorem 1.1. The following result follows from Frankel-Petersen-Wilhelm theorem [16, Theorem 3] (cf. [9]) for minimal hypersurfaces in a Riemannian manifold with positive Ricci curvature.

Proofs of Theorem 1.2
In order to prove Theorem 1.2, we first give another proof of Escobar Theorem, which is slightly different from the one given in Escobar [7, Proof of Theorem 4.1].
Step 2 The metricḡ on W can be extended naturally to a metric g on D(W ). In particular, g can be expressed on the ε-tubular For a general relative metricḡ on W , g is only a C 0,1 metric on D(W ) since Aḡ does not vanish generally. But, by the assumption that ∂ W is totally geodesic (i.e., Aḡ = 0 on ∂ W ), g is a C 2,1 metric on D(W ). Note also that there exists a natural isometric involution of D(W ) such that ι(t, Step 3 Cover U ε (∂ W ) by a family of harmonic coordinates. Then, applying DeTurck-Kazdan's regularity theorem [6] to the relative Einstein C 2,1 metric g on D(W ), we conclude that (D(W ), g) is a real analytic Einstein manifold. Setǧ = u 4/(n−2) ·ḡ on W for u ∈ C ∞ >0 (W ). Then u satisfies the following: u can be naturally extended to a positive function u on D(W ), similarly to g. In particular, Note that ∂u ∂t = 0 on ∂ W sinceǧ is also relative metric. Then, u ∈ C 1,1 >0 (D(W )), and hence, by the elliptic regularity, u ∈ C ∞ >0 (D(W )). Setting g := u 4/(n−2) · g on D(W ), we get that g is a csc C ∞ metric on D(W ).
Step 4 Consider the two conformal metrics g and g on D(W ). By the first step of the proof of Obata Theorem (cf. [17, Proof of Proposition 1.4]), g is also an Einstein metric, and that it satisfies the following Hessian equation: where ∇ denotes the Levi-Civita connection of g. If φ is not constant on D(W ), it is known that (D(W ), g) is homothetic to (S n , g S ) [14], [18] (cf. [13,Theorem 24]). By the existence of the isometric involution ι of (D(W ), g), the following assertions are equivalent: . By the fact that ι is also an isometric involution of (D(W ), g), Escobar Theorem now follows from Obata Theorem.
Proof of Theorem 1.2 Set g :=ḡ| ∂ W . Sinceḡ is Einstein, then the Gauss equation for the hypersurface ∂ W implies that Moreover, by Escobar Theorem,ḡ is a relative Yamabe metric on W . Then, and hence we get the desired inequality (1). It is obvious that the equality in the above inequality holds if and only if g is a Yamabe metric on ∂ W . Now, we assume that the equality in (1) holds, equivalently that g is a Yamabe metric on ∂ W . Without loss of generality, we may also assume that Ricḡ = (n − 1)ḡ. By Gromov's isoperimetric inequality [10] for (W ,ḡ) in (D(W ), g), we then have that .
Moreover, the equality in (5) holds if and only if (W ,ḡ) is isometric to (S n + , g S n ). Combining the inequality (5) with the equality in (1), we obtain the reverse inequality (2) as below:

Proof of Theorem 1.1
Proof of Theorem 1.1 Step 1 Consider the two conformal relative metricsḡ andǧ = φ −2 · g ( φ ∈ C ∞ >0 (W ) ) given in Theorem 1.1. Then the following formula holds [15] (cf. [17]): where Eǧ := Ricǧ − Rǧ n ·ǧ. By the assumption that bothḡ andǧ are Einstein, we get By the assumption that bothḡ andǧ are relative metrics, we then have that ∂φ ∂t = 0 along ∂ W in terms of Fermi coordinates (t, x) = (t, x 1 , · · · , x n−1 ) around ∂ W . From this, we also note that where ∂ t = ∂ ∂t , ∂ x i = ∂ ∂ x i and ∇ denotes the Levi-Civita connection of g :=ḡ| ∂ W . Combining the above Hessian equation on W with these equations, we then get Step  [13,Lemmas 13 and 18,Corollary 19] is still valid for polar coordinates on (B,ḡ) centered at p ∈ ∂ W ∩ B. Then, we have also thatḡ is a metric of positive constant curvature on W , and thusǧ is so too.
In any case, φ ≡ const > 0 on W unlessḡ is a metric of positive constant curvature on W . Hence, this completes the proof of the second assertion (2) in Theorem 1.1.
Step 3 We finally assume thatḡ is a metric of positive constant curvature on W . This implies thatǧ is so too. Without loss of generality, we may also assume that bothḡ andǧ are of constant curvature one. Let π : ( W , g) → (W ,ḡ) be the universal Riemannian covering. We claim here that ∂ W is connected even if W may not be compact. Suppose that ∂ W is not connected. Since the deck transformation group of π acts transitively on each fiber π −1 (q) (q ∈ W ) and that the base space W is compact, one can check that the same argument as the proof of Frankel-Petersen-Wilhelm theorem [16,Theorem 3] is still applicable to our case. Namely, there exist two connected components V 1 , V 2 of ∂ W and a minimizing geodesic γ joining V 1 and V 2 . Then, combining the Synge's second variational formula for the length of γ with zero mean curvature of V 1 , V 2 and the positivity of Ricci curvature of g, we can get a contradiction. Hence, ∂ W is connected.