Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below

We show that the hyperbolic manifold ℍⁿ/ℤⁿ⁻² is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound −𝑛(𝑛 − 1), and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related spl...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2023
Hauptverfasser:	Hao, Tianze, Hu, Yuhao, Liu, Peng, Shi, Yuguang
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Veröffentlicht:	Інститут математики НАН України 2023
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author	Hao, Tianze Hu, Yuhao Liu, Peng Shi, Yuguang
author_facet	Hao, Tianze Hu, Yuhao Liu, Peng Shi, Yuguang
citation_txt	Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below. Tianze Hao, Yuhao Hu, Peng Liu and Yuguang Shi. SIGMA 19 (2023), 083, 28 pages
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description	We show that the hyperbolic manifold ℍⁿ/ℤⁿ⁻² is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound −𝑛(𝑛 − 1), and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related splitting results.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 083, 28 pages Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below Tianze HAO a, Yuhao HU ab, Peng LIU a and Yuguang SHI a a) Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China E-mail: haotz@pku.edu.cn, 1801110011@pku.edu.cn, ygshi@math.pku.edu.cn b) School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China E-mail: yuhao.hu@sjtu.edu.cn Received April 08, 2023, in final form October 20, 2023; Published online November 01, 2023 https://doi.org/10.3842/SIGMA.2023.083 Abstract. We show that the hyperbolic manifold Hn/Zn−2 is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound −n(n− 1), and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related splitting results. Key words: scalar curvature; rigidity; ALH manifolds; µ-bubbles 2020 Mathematics Subject Classification: 53C21; 53C24 Dedicated to Jean-Pierre Bourguignon on the occasion of his 75th birthday 1 Introduction In [21, Section 3] and [22, p. 240], Gromov stated the following generalization of Min-Oo’s hyperbolic rigidity theorem [30]. Statement 1.1 (“generalised Min-Oo rigidity theorem”). Parabolic quotients Z = Hn/Γ of the hyperbolic n-space admit no non-trivial, compactly supported ‘deformation’ with scalar curvature R ≥ −n(n− 1). According to [21], a deformation can change not only the metric, but also the topology of a compact region in Z. If the deformation is topologically a connected sum with a closed n- manifold, Statement 1.1 is known to be true for (at least) Z = Hn/Zn−1, with idea of proof already outlined by [19, Section 55 6 ] (for a detailed treatment, see also [2, Theorem 1.1]). The situation turns out to be more subtle if broader types of deformations are considered, allowing, for example, surgeries along an embedded, non-contractible loop. In this latter case we construct a counterexample to Statement 1.1, which, more precisely, demonstrates the following. Theorem 1.2. For n ≥ 3, let Hn/Zn−2 be equipped with the standard hyperbolic metric. There exists a complete Riemannian manifold (Mn, g), not (globally) hyperbolic, and compact subsets K ⊂ M and K ′ ⊂ Hn/Zn−2, such that (1) Rg ≥ −n(n − 1) and (2) M \ K is isometric to ( Hn/Zn−2 ) \K ′. This paper is a contribution to the Special Issue on Differential Geometry Inspired by Mathemati- cal Physics in honor of Jean-Pierre Bourguignon for his 75th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Bourguignon.html mailto:haotz@pku.edu.cn mailto:1801110011@pku.edu.cn mailto:ygshi@math.pku.edu.cn mailto:yuhao.hu@sjtu.edu.cn https://doi.org/10.3842/SIGMA.2023.083 https://www.emis.de/journals/SIGMA/Bourguignon.html 2 T. Hao, Y. Hu, P. Liu and Y. Shi Remark 1.3. (1) While the theorem above concerns non-rigidity of Hn/Zn−2, it is also interesting to ask whether its statement still holds if Hn/Zn−2 is replaced by Hn/Zn−1; this will be answered in the affirmative in Section 2.4.1. Thus, we obtain counterexamples to the “weak rigidity of Hn/Zn−1” mentioned in [20, p. 678]. (2) En route to proving Theorem 1.2, we obtain counterexamples (see Proposition 2.6) to the following statement in [21, p. 12]: Represent Hn/Zn−2 as a warped product ( H2×Tn−2, gH ) (see formula (2.1)), and, for a geodesic 2-disk D2 ⊂ H2, let X = D2 × Tn−2 ⊂ H2 × Tn−2 with the restricted metric gH \|X ; then no Riemannian manifold ( Mn, g ) with boundary isometric to ∂X can have scalar curvature Rg ≥ −n(n − 1) and mean curvature1 of ∂M greater than that of ∂X. (3) Our proof of Theorem 1.2 is constructive, which, a little to our surprise, shows thatM can be chosen to be homeomorphic to Hn/Zn−2 (see Section 2.4.2); moreover, Rg > −n(n− 1) for some points in K. From the perspective of our construction, the non-rigidity of Hn/Zn−2 seems closely related to the fact: A deformation supported in a compact subset K can ‘break’ the incompressibility 2 of some submanifold that is disjoint from K. On the other hand, rigidity does hold if one only considers deformations that preserve such incompressibility, as the next theorem shows (cf. [11, Theorem 1.8]). Theorem 1.4. For 3 ≤ n ≤ 7, let (Mn, g) be a complete Riemannian manifold 3 with scalar curvature Rg ≥ −n(n−1). Suppose that there exist compact subsets K ⊂M , K ′ ⊂ Hn/Zn−2, and an isometry f : M \K → ( Hn/Zn−2 ) \K ′. Representing Hn/Zn−2 topologically as R2 +×Tn−2, let p ∈ R2 + be such that T = {p}×Tn−2 is disjoint from K ′, and suppose that the map f−1\|T : T →M is incompressible. Then (M, g) is isometric to Hn/Zn−2. Technically, we will derive Theorem 1.4 as a consequence of Theorem 1.5 below. The latter can be regarded as a kind of positive mass type theorem for manifolds with an ALH end; its statement relies on a gluing construction, which we now describe. Gluing construction: Let Nn be a smooth manifold, and suppose that ϕ : Tk → N (1 ≤ k ≤ n− 2) is an embedding with trivial normal bundle. Moreover, write Hn/Zn−1 (topologically) as the product R× Tn−k−1 × Tk, and define ψ : Tk → R× Tn−k−1 × Tk ∼= Hn/Zn−1 by ψ(p) = (t, q, p) for some fixed t ∈ R and q ∈ Tn−k−1. By removing tubular neighborhoods of ϕ ( Tk ) ⊂ N and ψ ( Tk ) ⊂ Hn/Zn−1 and then identifying the respective boundaries in the obvious way, we obtain a manifold M . For brevity, M will be referred to as obtained by gluing N and Hn/Zn−1 along Tk via (ϕ, ψ). In particular, for c sufficiently large, (c,∞)× Tn−1 ⊂ Hn/Zn−1 remains an ‘end’ of M , and this end is denoted by E . Theorem 1.5. For 3 ≤ n ≤ 7, let Nn be a smooth manifold that is either closed or non-compact without boundary, and let Mn be obtained by gluing N with Hn/Zn−1 along Tk via (ϕ, ψ) (see description above). Suppose that (a) the map ϕ : Tk → N is incompressible; 1Unless specified otherwise, in this article the mean curvature along a boundary will always be computed with respect to the outward unit normal. 2A continuous map f : X → Y between topological spaces is said to be incompressible if the induced map f∗ : π1(X) → π1(Y ) is injective; when f is an inclusion, we say ‘X is incompressible in Y ’. 3In this article, all manifolds are assumed to be orientable, and all hypersurfaces 2-sided. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 3 (b) g is a complete Riemannian metric on M with Rg ≥ −n(n− 1); (c) (E , g) is asymptotically locally hyperbolic (ALH ) (see Definition 3.1). Then m̄E,g ≥ 0 (see Definition 3.2). In addition, suppose that (d) the curvature tensor of (M, g) and its first covariant derivatives are bounded; (e) there exists some α > 0 such that Rg ≤ −α outside a compact set. Then κ = 0 (see (3.1) for the definition of κ) only if (M, g) is Einstein. Readers familiar with positive mass theorems may have noticed that the second half of Theo- rem 1.5 is not in an ideal form; in other words, one wants to know whether the vanishing of m̄E,g, and not just κ, implies that (M, g) is isometric to Hn/Zn−1, even without the assumptions (d) and (e). In our proof, these assumptions play a role in making sure that the normalized Ricci flow (NRF) starting at g has desired properties (see Lemma 3.4); on the other hand, it seems subtle to prove hyperbolicity from (M, g) being Einstein and the assumed ALH decay rate. Thus we decide to leave the stronger statement for future investigation. Theorem 1.5 has the following corollary. Corollary 1.6. For 3 ≤ n ≤ 7, let Nn be a closed manifold, and suppose that Mn is obtained by gluing N with Hn/Zn−1 along Tk via (ϕ, ψ). Suppose that g is a complete metric on M such that (M, g) is isometric to the hyperbolic manifold Hn/Zn−1 outside a compact set,4 and suppose that (a) the map ϕ : Tk → N is incompressible; (b) Rg ≥ −n(n− 1). Then (M, g) is isometric to Hn/Zn−1. In fact, Corollary 1.6 remains true if N is allowed to be non-compact, which can be deduced as a corollary of Theorem 1.10 below (see Remark 4.5). Besides rigidity problems modeled on complete manifolds, it is often natural to consider similar problems for manifolds with boundary and scalar/mean curvature bounds. In this regard, we present a splitting result of ‘cuspidal-boundary’ type (see [21, Section 4, last paragraph]). Our proof relies on an approximation scheme developed in [38] involving µ-bubbles. Theorem 1.7. Let ( M4, g ) be a complete, non-compact Riemannian 4-manifold with compact, connected boundary ∂M . Suppose that π2(M) = π3(M) = 0 and that the scalar curvature Rg ≥ −12. Then inf ∂M H ≤ 3, where H is the mean curvature of ∂M . Moreover, if inf ∂M H = 3, then (M, g) is isometric to ( (−∞, 0]×Σ, dt2 + e2tg0 ) , where t is the coordinate on (−∞, 0] and (Σ, g0) is a closed 3-manifold with a flat metric. 4That is, there exists an isometry f : M \ K → (Hn/Zn−1) \ K′ for some compact sets K ⊂ M and K′ ⊂ Hn/Zn−1. 4 T. Hao, Y. Hu, P. Liu and Y. Shi Remark 1.8. Theorem 1.7 would fail if one allows M to be compact. Indeed, take M = S1 × B3, g = cosh2 ρdθ2 + dρ2 + sinh2 ρgS2 , ρ ≤ ρ0, where θ ∈ S1, ρ is the radial coordinate on B3, and gS2 is the standard round metric on S2. In this example, M has contractible universal cover, so both its π2 and π3 vanish. Moreover, since g is hyperbolic, Rg = −12, but the mean curvature H∂M = 2 coth ρ0 + tanh ρ0 > 3. Counterexamples also exist if one drops the assumption on π2(M) and π3(M). In fact, let us take the manifold (M ′, g′) in Section 2.4.1 and then, for sufficiently small z0 > 0, remove the subset {0 < z < z0} from M ′; the result is a manifold M ′′ with π2(M ′′) ̸= 0, H∂M ′′ = 3 and R ≥ −12. Clearly, M ′′ ̸∼= [c,∞)× ∂M ′′ ∼= [c,∞)× T3. Finally, we present an analogue of Theorem 1.7 in more general dimensions. Definition 1.9 (cf. [11]). We say that a closed, connected manifold Σ belongs to the class Cdeg, if � Σ is aspherical,5 and � any compact manifold Σ′ that admits a map to Σ of nonzero degree cannot be endowed with a PSC metric (i.e., metric with positive scalar curvature). It is well known that Tn ∈ Cdeg for n ≤ 7; also note that the second item in Definition 1.9 is redundant when dimΣ ≤ 5, according to [13]. Theorem 1.10. For 3 ≤ n ≤ 7, let (Mn, g) be a complete and non-compact Riemannian manifold with compact, connected boundary ∂M . Suppose that (a) ∂M is incompressible in M, (b) ∂M ∈ Cdeg, (c) Rg ≥ −n(n− 1), then inf ∂M H ≤ n− 1, where H is the the mean curvature of ∂M . Moreover, if inf ∂M H = n− 1, then (M, g) is isometric to ((−∞, 0]× Σ,dt2 + e2tg0) where t is the coordinate on (−∞, 0] and (Σ, g0) is a closed (n− 1)-manifold with a flat metric. Additional notes on the literature. (a) All our main theorems are fundamentally related to Gromov’s fill-in problems (e.g., [21, Problems A and B]; [22, p. 234, Question (c)]). (b) The- orem 1.10 can be viewed as a generalization of [36, Theorem 3.2]. (c) It is a classical theme to relate incompressibility conditions with scalar curvature (see [23, Section 11]). (d) To adapt to modern language, our Theorem 1.5 considers manifolds with a prescribed end and some ‘arbitrary ends’; the study of positive-mass type theorems on such manifolds has generated con- siderable interest recently (see, for example, [10, 12, 29, 40]). (e) While in this paper we focus 5A closed, connected manifold is said to be aspherical if it has contractible universal cover. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 5 on rigidity results for complete, non-compact manifolds with boundary and scalar curvature lower bounds, similar results in the compact case (with boundary) are obtained by Gromov in [21, Section 4]. In both cases, the proofs rely on the µ-bubble technique. (f) It would be interesting to compare Theorem 1.5 with some recent progress in proving positive mass and rigidity results for ALH manifolds (see [1, 16, 17, 26]); in this latter development, manifolds are often assumed to have nonempty inner boundary with the mean curvature bound H ≤ n − 1 (now H is computed with respect to the inner unit normal); such mean curvature bounds serve as barrier conditions in the method of ‘marginally outer trapped surfaces’ (MOTS), which can be viewed as a generalization of the µ-bubble technique. Organization of this article. The proof of Theorem 1.2 is technically independent from the rest of the work and is included in Section 2. Section 3 serves as a preliminary to proving Theorem 1.5, presenting results concerning NRF and conformal deformations. In Section 4, we prove Theorem 1.5, followed by proofs of Corollary 1.6 and Theorem 1.4. In Section 5, we prove Theorem 1.7 and Theorem 1.10. Several of the proofs rely on the so-called ‘µ-bubble’ technique, a brief discussion of which is included in Appendix A. Appendix B includes two topological lemmas. 2 Non-rigidity of Hn/Zn−2 Let the hyperbolic n-space Hn be represented by the upper half-space model Rn + = {(x,y, z): x ∈ R,y ∈ Rn−2, z > 0}, and let Zn−2 act by translating along the orthogonal lattice 2πZn−2 ⊂ Rn−2 while keeping the x, z-coordinates fixed. The quotient space is denoted by Hn/Zn−2 and has the hyperbolic metric gH = z−2 ( dz2 + dx2 ) + z−2gTn−2 , (2.1) where the subscript ‘H’ stands for ‘hyperbolic’, and gTn−2 is the associated flat metric on Tn−2. Henceforth, we will regard (x, z) as coordinates on the hyperbolic plane H2; manifestly that( Hn/Zn−2, gH ) is a warped product of H2 and ( Tn−2, gTn−2 ) . The following lemma is easily verified by standard computation, so we omit its proof. Lemma 2.1. Let ∇, ∇2 denote the gradient and Hessian with respect to gH (same below). We have (a) ∇z = z2∂/∂z, (b) ∇2z(∂/∂x, ∂/∂x) = −∇2z(∂/∂z, ∂/∂z) = −1/z, (c) ∇2z(∂/∂z, ∂/∂x) = 0. Next, we proceed to prove Theorem 1.2 by constructing an example that satisfies all its conditions. The idea is to remove a suitable compact subset, Xp,r, from Hn/Zn−2 and then ‘glue’ the result with a compact manifold, X̄r, along their boundaries; Xp,r and X̄r will be defined in Sections 2.1 and 2.2 respectively, and then we handle the gluing step in Section 2.3. 2.1 First preliminary construction Let p ∈ H2, and define Xp,r := Dr(p)× Tn−2 ⊂ Hn/Zn−2 and Yp,r := ∂Xp,r, (2.2) where Dr(p) ⊂ H2 is the geodesic disc, centered at p, of radius r > 0; the inclusion in (2.2) makes sense since Hn/Zn−2 is a warped product of H2 and Tn−2, as we already noted. 6 T. Hao, Y. Hu, P. Liu and Y. Shi Now we have two sets of coordinates for H2: (x, z) and the polar coordinates (ϱ, θ) centered at p. In terms of the polar coordinates, the metric on H2 reads gH2 = dϱ2 + sinh2 ϱdθ2. Lemma 2.2. The boundary Yp,r ⊂ (Xp,r, gH) has the mean curvature Hp,r = coth r − (n− 2)z−1∂z ∂ϱ . (2.3) Moreover, (a) \|Hp,r − coth r\| ≤ n− 2; (b) There exists a constant r0 > 0 such that Hp,r > 0 for all r ≤ r0. Proof. The formula (2.3) is straightforward to check by using the representation gH = dϱ2 + sinh2 ϱ dθ2 + z−2gTn−2 . Moreover, since both z−1∇z and ∇ϱ have unit norm with respect to gH ,∣∣∣∣∂z∂ϱ ∣∣∣∣ = ∣∣〈∇z,∇ϱ〉∣∣ = ∣∣z〈z−1∇z,∇ϱ 〉∣∣ ≤ z. (2.4) This implies (a), and (b) follows since coth r → ∞ as r → 0. ■ Lemma 2.3. There exists a constant Cr > 0, depending only on r, such that on ∂Dr(p) we have \|∂θz(r, θ)\| ≤ z sinh r and ∣∣∂2θz(r, θ)∣∣ ≤ Crz. Proof. Since both z−1∇z and (sinh r)−1(∂/∂θ) have unit norm with respect to gH , we have \|∂θz(r, θ)\| = ∣∣〈∇z, ∂/∂θ〉∣∣ ≤ z sinh r. Moreover, a calculation shows that ∇2z(∂/∂θ, ∂/∂θ) = ∂2θz + (∂ϱz) sinh ϱ cosh ϱ. (2.5) By Lemma 2.1 (b), (c), the left-hand side of (2.5) has its magnitude bounded by (sinh2 ρ)z; thus, using (2.4) and evaluating (2.5) at ϱ = r, we get∣∣∂2θz(r, θ)∣∣ ≤ sinh r(sinh r + cosh r)z. Taking Cr = sinh r(sinh r + cosh r) finishes the proof. ■ 2.2 Second preliminary construction Let D be a 2-disc with polar coordinates ( ϱ̄, θ̄ ) , where 0 ≤ ϱ̄ ≤ π/3 and 0 ≤ θ̄ < 2π. Equip D with the metric gD = dϱ̄2 + 4 sin2(ϱ̄/2)dθ̄2. Thus, (D, gD) is isometric to a ‘cap’ in the round sphere of radius 2. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 7 Now let r > 0 and z(ϱ, θ) be as in Section 2.1 above. Consider X̄r := S1 ×D × Tn−3 equipped with the metric ḡ = sinh2 r dθ2 + ( z(r, θ) )−2 gD + ( z(r, θ) )−2 gTn−3 , (2.6) and let Ȳr := ∂X̄r. By construction, the boundaries (Yp,r, gH \|Yp,r) and ( Ȳr, ḡ\|Ȳr ) are isometric under the obvious identification. Lemma 2.4. The boundary Ȳr ⊂ ( X̄r, ḡ ) has the mean curvature H̄r = √ 3 2 z(r, θ). (2.7) Proof. Standard computation by using (2.6). ■ Regarding the scalar curvature of a warped-product metric, the following is well-known. Lemma 2.5 (cf. [23, Proposition 7.33]). Let ( Nn−1, h ) be an (n− 1)-dimensional Riemannian manifold with scalar curvature Rh. Given any smooth function ϕ(θ) defined on an interval I and a constant a > 0, the warped product metric g = a2dθ2 + ϕ(θ)2h defined on I ×N has the scalar curvature Rg = n− 1 a2 [ −2 ( ϕ′ ϕ )′ − n ( ϕ′ ϕ )2 ] + ϕ−2Rh. (2.8) In our case, to compute the scalar curvature of ḡ, it suffices to substitute h = gD + gTn−3 , ϕ(θ) = 1/z(r, θ) and a = sinh r into (2.8). Noting that Rh = 1/2, we have Rḡ = (n− 1)(sinh r)−2 { −2∂θ[z∂θ(1/z)]− n[z∂θ(1/z)] 2 } + z2/2 = (n− 1)(sinh r)−2 { 2 ( ∂2θz ) /z − (n+ 2)[(∂θz)/z] 2 } + z2/2, (2.9) where z, ∂θz and ∂2θz are evaluated at (r, θ). Now we are ready to observe the following. Proposition 2.6. For fixed r > 0, the manifold (X̄r, ḡ) satisfies: (a) The scalar curvature Rḡ ≥ 1 2 [z(r, θ)]2 − Cn,r for a constant Cn,r > 0 depending only on n and r. In particular, we have Rḡ > −n(n−1) provided that p ∈ H2 is chosen to have a large enough z-coordinate; (b) Under the obvious identification (isometry) between Yp,r and Ȳr, we have H̄r > Hp,r pro- vided that the z-coordinate of p is large enough. Proof. (a) follows from (2.9) and Lemma 2.3; (b) follows from Lemma 2.2 (a) and (2.7). ■ 8 T. Hao, Y. Hu, P. Liu and Y. Shi 2.3 The gluing step Lemma 2.7 ([8, Theorem 5]). Let Ω be a compact n-manifold with boundary ∂Ω, and let g and g̃ be two smooth Riemannian metrics on Ω such that (a) g − g̃ = 0 at each point on ∂Ω; (b) the mean curvatures satisfy Hg̃ −Hg > 0 at each point on ∂Ω. Then, given any ϵ > 0 and any neighborhood U of ∂Ω, there exists a smooth metric ĝ on Ω with the following properties: (1) Rĝ ≥ min{Rg, Rg̃} − ϵ in Ω; (2) ĝ = g̃ in Ω \ U ; (3) ĝ = g in a neighborhood of ∂Ω. Remark 2.8. By an arbitrary extension, in Lemma 2.7 it suffices to assume that g is defined only in a neighborhood of ∂Ω. To prove Theorem 1.2, a basic idea is to apply Lemma 2.7 to obtain a metric ĝ on X̄r which agrees with gH in a neighborhood of ∂X̄r ∼= ∂Xp,r, so ĝ extends smoothly into ( Hn/Zn−2 ) \Xp,r by gH . A compromise is the ϵ-cost to the scalar curvature estimate. Thus, one would like to have a bit more scalar curvature to begin with, so that the cost can be absorbed, maintaining the desired lower bound Rĝ ≥ −n(n− 1). This can be achieved by a suitable deformation of gH in a neighborhood of Yp,r ⊂ Hn/Zn−2, as the following lemma shows. Lemma 2.9. Let u(ϱ) = { 1− e 1 ϱ−r0 , ϱ < r0, 1, ϱ ≥ r0, (2.10) and define g′H := [ u(ϱ) ]2 dϱ2 + sinh2 ϱdθ2 + [ z(ϱ, θ) ]−2 gTn−2 . As long as r0 > 0 is small enough, we can find δ > 0 such that Rg′H + n(n− 1) > 0 for ϱ ∈ [r0 − 2δ, r0). Proof. By [34, Claim 2.1], we have Rg′H = RgH + ( 1− u−2 ) (Rγ(ϱ) −RgH ) + 2u−3u′(ϱ)Hp,ϱ, (2.11) where γ(ϱ) = sinh2 ϱdθ2 + [ z(ϱ, θ) ]−2 gTn−2 and RgH = −n(n− 1). We want to estimate the right-hand side of formula (2.11). To start with, by Lemma 2.5, Rγ(ϱ) = (n− 2)(sinh ϱ)−2 { 2 ( ∂2θz ) /z − (n+ 1)[(∂θz)/z] 2 } . Thus, by the proof of Lemma 2.3, there exists a constant Cn,r0 , depending on n, r0 only, such that \|Rγ(ϱ)\| ≤ Cn,r0 for ϱ ∈ [r0/2, r0]. (2.12) Next, by the definition of u, we have, for ϱ ≤ r0, 0 ≥ 1− u−2 = u−2 ( −2e 1 ϱ−r0 + e 2 ϱ−r0 ) ≥ −2u−2e 1 ϱ−r0 ≥ −2u−3e 1 ϱ−r0 . (2.13) Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 9 Moreover, for sufficiently small r0, we have Hp,ϱ ≥ 1 for any ϱ ≤ r0 (Lemma 2.2 (b)), and so 2u−3u′(ϱ)Hp,ϱ ≥ 2u−3e 1 ϱ−r0 (ϱ− r0) −2. (2.14) On combining (2.11), (2.12), (2.13) and (2.14), we obtain that Rg′H −RgH ≥ 2u−3e 1 ϱ−r0 [ (r0 − ϱ)−2 − Cn,r0 − n(n− 1) ] for ϱ ∈ [r0/2, r0]. Clearly, we can choose a small δ > 0 such that Rg′H −RgH > 0 for ϱ ∈ [r0 − 2δ, r0). This completes the proof. ■ Proof of Theorem 1.2. Let r0 be small enough, and let u(ϱ), g′H and δ be as in Lemma 2.9. Define c := min ϱ∈[r0−2δ,r0−δ] Rg′H + n(n− 1) > 0. Take r := r0 − δ, and note that we still have the freedom of choosing p ∈ H2. Suppose that the isometry between Yp,r and Ȳr maps q ∈ Yp,r to q̄ ∈ Ȳr. Furthermore, by using Fermi coordinates, any point in a small neighborhood of Yp,r ⊂ Xp,r is uniquely represented by a pair (q, d′), where d′ is the g′H -distance to Yp,r. Similarly, (q̄, d̄) coordinatizes a neighborhood of Ȳr ⊂ X̄r. By identifying (q, d) with (q̄, d), we have arranged that g′H = ḡ along Ȳr. To apply Lemma 2.7, assign Ω = X̄r, g = g′H (defined in a neighborhood U of Ȳr ⊂ X̄r, via the identification above) and g̃ = ḡ (defined on X̄r). As noted above, Lemma 2.7 (a) is satisfied. Furthermore, the mean curvature of Yp,r ⊂ Xp,r with respect to g′H is H ′ p,r := Hp,r/u(r) ≥ Hp,r, but by choosing p to have large z-coordinate, we can still arrange that H̄r > H ′ p,r (see the proof of Proposition 2.6 (b)). Next, by shrinking U if needed, we can assume that Rg′H ≥ c−n(n−1) on U , and we can assume the same lower bound for Rḡ by choosing p suitably (Proposition 2.6 (a)). Finally, take ϵ = c/2. With the above setting, Lemma 2.7 applies and yields a metric ĝ defined on X̄r, satisfying � Rĝ ≥ −n(n− 1) + c/2; � ĝ = ḡ on X̄r \ U ; � ĝ = g′H in a neighborhood of Ȳr ⊂ X̄r. Thus, ĝ and g′H piece together to become a smooth metric g defined on M := [( Hn/Zn−2 ) \Xp,r ] ∪ X̄r/∼, where ∼ indicates boundary identification, with (non-constant) scalar curvature Rg ≥ −n(n−1). (For the reader’s convenience, Figure 1 includes a schematic, 1-dimensional illustration of the construction.) In the statement of Theorem 1.2, take (M, g) = (M , g), K = X̄r ∪ (Xp,r0\Xp,r) ⊂ M and K ′ = Xp,r0 , and the proof is complete. ■ 10 T. Hao, Y. Hu, P. Liu and Y. Shi Figure 1. A schematic picture of (M , g). Figure 2. An illustration of ( S1 × R2 ) \ ( D2 ϵ × S1 ) , where D2 ϵ × S1 is shaded. 2.4 Further remarks 2.4.1 Surgery applied to Hn/Zn−1 The construction above only modifies a portion of Hn/Zn−2 that is contained in between x0 < x < x1 for some x0, x1 ∈ R. By a translation, we can always arrange that x0 = 0. Now, T : (x,y, z) 7→ (x + x1,y, z) maps a neighborhood of {x = 0} isometrically to a neighborhood of {x = x1}. Thus, by removing the subsets {x < 0} and {x > x1} from M and then identifying {x = 0} and {x = x1} via T , we obtain a smooth Riemannian manifold (M ′, g′) that satisfies Rg′ ≥ −n(n − 1). In fact, (M ′, g′) can be viewed as a compactly supported ‘deformation’ of a hyperbolic cusp Hn/Zn−1, where Zn−1 acts on (x,y) ∈ Rn−1 by translating along the lattice x1Z× 2πZn−2. This serves as yet another counterexample to Gromov’s Statement 1.1. 2.4.2 A note on topology It is interesting to determine the topology of both M and M ′ above. Topologically, M is obtained by a surgery along {p}×S1 ⊂ R2×S1 and then taking product with Tn−3. The result of that surgery is homeomorphic to S1 × R2. To see this, view S3 as the union of D2×S1 and S1×D2 with the boundaries identified. Then R2×S1 is simply S3 with the core circle C = S1 ×{q} removed. Surgery of S3 along {p}× S1 yields S1 × S2. Then removing C from S1 × S2 gives S1 × R2. In conclusion, M ∼= S1 × R2 × Tn−3, which is homeomorphic to Hn/Zn−2 ∼= R2 × S1 × Tn−3 via a map that switches the first two factors. Regarding M ′, note that by identifying x = 0 and x = x1 in {0 ≤ x ≤ x1} ⊂ H2, one obtains an open annulus, or equivalently R2\{0}. Thus, M ′ is obtained by a surgery along {p} × S1 ⊂ ( R2\{0} ) × S1 and then taking product with Tn−3. In this case, a similar argument as the above applies, and the result of the surgery is homeomorphic to ( S1 × R2 ) \ ( D2 ϵ × S1 ) , i.e., the result of removing a solid torus that is contained in a 3-ball B3 ⊂ S1×R2 (see Figure 2). Thus M ′ ∼= [( S1 ×R2 ) \ ( D2 ϵ × S1 )] ×Tn−3. In particular, the two ends of M ′ are separated by a hypersurface with the topology S2 × Tn−3; the same is not true for Hn/Zn−1. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 11 3 ALH manifolds, mass and deformations This section includes basic notions and results concerning ALH manifolds, possibly with arbi- trary ends, and their NRF and conformal deformations. These results will be used in proving Theorem 1.5. 3.1 ALH manifolds and mass Definition 3.1. Let (Mn, g) be a complete Riemannian manifold without boundary. Suppose that (1) for some (sufficiently large) compact set K ⊂ M , M \ K has a connected component E that is diffeomorphic to (0, 1)× Tn−1, and (2) restricted to E , the metric g admits an asymptotic expansion of the form g = 1 τ2 [ dτ2 + h+ τn n κ+O ( τn+1 )] , (3.1) where τ is the coordinate on the interval (0, 1); h denotes a flat metric on Tn−1, which represents metric at the conformal infinity E0 (i.e., when τ = 0); κ = κAB dyAdyB is a symmetric tensor defined on Tn−1, where ( yA ) are flat coordinates on Tn−1; finally, O ( τn+1 ) stands for a remainder Q = QAB dyAdyB with the asymptotics∣∣QAB ∣∣+ ∑ \|α\|+k≤2 ∣∣τk∂αy ∂kτ QAB ∣∣ ≤ Cτn+1 as τ → 0, for some constant C, where α = (α1, . . . , αn−1) are multi-indices. Such an (M, g) is called asymptotically locally hyperbolic (ALH ), and E an ALH end. Moreover, if M \ E is non-compact, we say that (M, g) is ALH with arbitrary ends. Definition 3.2 (cf. [28, Definition 1.1]). Given a Riemannian manifold (M, g) with an ALH end E on which g admits the expansion (3.1), we call mE,g := trhκ = hABκAB the mass aspect function associated to the pair (E , g). Furthermore, define m̄E,g := sup Tn−1 mE,g. Throughout, let each τ -level set in E be denoted by Eτ . The following lemma shows how m̄E,g is related to the mean curvature of Eτ ⊂ E . Lemma 3.3. Let ( Mn, g ) be a Riemannian manifold with an ALH end E. If m̄E,g < 0, then there exist constants τ0, C > 0 such that HEτ ≥ (n− 1) + Cτn for τ ≤ τ0, (3.2) where HEτ is the mean curvature of Eτ computed with respect to the ‘outward normal’ −∂/∂τ . Proof. Before making any assumption about m̄E,g, we have HEτ = (n− 1)− n− 2 2n mE,gτ n +O ( τn+1 ) . (3.3) For m̄E,g < 0, let us take C = −m̄E,g/10, and clearly (3.2) holds for some τ0 > 0. ■ 12 T. Hao, Y. Hu, P. Liu and Y. Shi 3.2 NRF deformations Given a Riemannian n-manifold (Mn, g0), the normalized Ricci flow (NRF), with initial met- ric g0, is by definition a smooth family of Riemannian metrics g(t) onM satisfying the evolution equation ∂tg = −2[Ricg + (n− 1)g], g(0) = g0. (3.4) Lemma 3.4. Suppose that (Mn, g0) is a complete Riemannian manifold with an ALH end E that satisfies Rg0 ≥ −n(n − 1) as well as the assumptions (d) and (e) in Theorem 1.5. Then there exists a T > 0 such that, for t ∈ (0, T ], g(t) is complete and satisfies (3.4) along with the following properties: (i) (E , g(t)\|E) remains ALH, and g(t) has the expansion (see equation (3.1)) g(t) = 1 τ2 [ dτ2 + h+ τn n κ(t) +O ( τn+1 )] ; (ii) on M , Rg(t) ≥ −n(n− 1) for all t ∈ (0, T ]; (iii) if g0 is not Einstein, then Rg(t) > −n(n− 1) for all t ∈ (0, T ]; (iv) outside a compact subset in M , Rg(t) ≤ −α/2 for t ∈ (0, T ]; (v) if κ(0) = 0, then κ(t) = 0 for all t ∈ (0, T ]; (vi) if κ(0) = 0, then for any t ∈ (0, T ] we have Rg(t) + n(n− 1) = O(τn+1) as τ → 0. Proof. The existence of g(t), t ∈ (0, T ], satisfying (3.4) follows from the existence of a solu- tion g̃(t), t ∈ (0, T̃ ], of the Ricci flow initiated at g0. They are related by a time-transformation: g(t) := e−2(n−1)tg̃ (Φ(t)) , where Φ(t) = e2(n−1)t − 1 2(n− 1) . Thus, up to constant factors, the curvature tensor Rm(t) of g(t) satisfies the same estimates as R̃m(Φ(t)) of g̃(Φ(t)). In particular, it follows from [33] that, for all t ∈ (0, T ], g(t) is complete, and \|Rm(t)\| is uniformly bounded. Now we turn to proving the properties. (i) follows from [5, Proposition 3.1]. (ii) can be verified by applying the maximum principle (see [15, Theorem 7.42]) to the evolution equation6 satisfied by e2(n−1)t(Rg(t) + n(n − 1)); to prove (iii), invoke the strong maximum principle on the domain Ω × [0, t], where Ω ⊂ M is compact on which g0 is not Einstein, and then let Ω exhaust M . (iv) would follow once we show that the integral∫ t 0 ∂t′R̃mdt′, t ∈ ( 0, T̃ ] (3.5) is uniformly bounded; to see this, note that the first covariant derivatives of R̃m(0) are assumed to be bounded (assumption (d) in Theorem 1.5), by [14, Theorem 14.16], we have∣∣∇2 g̃(t)R̃m(t) ∣∣ ≤ C√ t for some constant C > 0; in addition, the evolution equation of R̃m reads7 ∂tR̃m = ∆g̃(t)R̃m + R̃m ∗ R̃m; of course, 1/ √ t is integrable; combining these, it is easy to see that (3.5) is uniformly bounded for small enough T̃ ; since, by assumption (e) in Theorem 1.5, Rg ≤ −α outside a compact set, (iv) follows. (v) follows from [5, Proposition 4.3]. Finally, (vi) follows from (v) and [5, formulas (3.19)–(3.21)] (note that gij(τ) provides an extra factor of τ2). ■ 6For the evolution equation satisfied by Rg(t), see [5, formula (5.1)]. 7R̃m ∗ R̃m indicates a specific linear combination of the traces of R̃m⊗ R̃m. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 13 3.3 Conformal deformations Throughout this section, cn := 4(n− 1)/(n− 2). Lemma 3.5. Let (M, g) be complete with an ALH end E, and let f ∈ C∞(M) be a non-negative function that satisfies (a) supp f ⊂ K ∪ E for some compact subset K ⊂M ; (b) f ∈ O(τn) as τ → 0 where τ is the function occurring in the expansion (3.1). Then there exist a function v ∈ C∞(M) and a constant δ0 such that 0 < δ0 ≤ v ≤ 1 and −cn∆gv + fv = 0 in M. (3.6) Proof. Let {Ωi}∞i=0 be a sequence of smooth, bounded domains satisfying Ωi ⋐ Ωi+1 and⋃ iΩi =M . For each i, the Dirichlet problem − cn∆gvi + fvi = 0 in Ωi, vi = 1 on ∂Ωi, (3.7) has a positive solution vi. By the maximum principle, 0 < vi ≤ 1. Thus, v := limi→∞ vi is well defined on M , satisfying 0 ≤ v ≤ 1 and (3.6). It remains to show that v has a positive lower bound. Without loss of generality, assume that Σi ⊂ ∂Ωi is the only component of ∂Ωi that is contained in E ; in fact, let us assume that each Σi is a τ -level set. Denote τ0 := τ \|Σ0 . To refine the estimate of vi, we construct an auxiliary function ξ and compare it with vi via the maximum principle. Indeed, let α ∈ (0, n− 1) be any constant, and define ξ = 1− (τ/τ0) α, τ ≤ τ0. Using the fact that − ln τ is, up to adding a constant, the distance function to Σ0, one easily computes that ∆gξ = α(HEτ − α)(τ/τ0) α. (3.8) Thus, by (3.3), for sufficiently small τ0, there exists a constant Cn,α,τ0 > 0 such that ∆gξ ≥ Cn,α,τ0τ α for any τ ≤ τ0. Now, (3.7), the fact that vi ≤ 1, and the assumption that f ∈ O(τn) together imply ∆gvi ≤ C ′ f,nτ n in (Ωi\Ω0) ∩ E , vi > 0 on Σ0, vi = 1 on Σi, where C ′ f,n is a constant depending only on f and n. In comparison, ∆gξ ≥ Cn,α,τ0τ α in E\Ω0, ξ = 0 on Σ0, ξ < 1 on Σi. Thus, for sufficiently small τ0, the maximum principle implies that vi ≥ ξ in (Ωi\Ω0)∩E . Upon taking limit, v ≥ ξ > 0 on E\Ω1. Since v ≥ 0, the strong maximum principle, applied to (3.6), implies that v > 0 on M . 14 T. Hao, Y. Hu, P. Liu and Y. Shi When M \ E is compact (i.e., M having no arbitrary end), the above already implies that v has a positive lower bound. When M \ E is non-compact, since f is supported in K ∪ E , by choosing Ω0 to include K, we have that each vi (i ≥ 1) is harmonic on Ωi\(Ω0 ∪ E); using the maximum principle again, we get min Ωi\(Ω0∪E) vi = min ∂Ω0\E vi i→∞−−−→ min ∂Ω0\E v =: δarb > 0. To finish the proof, it suffices to take δ0 = min{δarb, infΩ1 v, infE\Ω1 ξ}. ■ Proposition 3.6. Let (Mn, g) be complete, with an ALH end E and with Rg ≥ −n(n−1) on M . Let R̄ ∈ C∞(M) be a function that satisfies (a) −n(n− 1) ≤ R̄ ≤ min{Rg, 0}; (b) supp ( Rg − R̄ ) ⊂ E ∪K for some compact subset K ⊂M ; (c) R̄ ≡ −n(n− 1) on E \K ′ for some compact subset K ′ ⊂ E. Then the Yamabe equation − cn∆gu+Rgu− R̄u n+2 n−2 = 0 in M, u→ 1 towards E0 (3.9) has a solution u with 0 < δ0 ≤ u ≤ 1 for some constant δ0. In particular, the metric u4/(n−2)g is complete and has the scalar curvature R̄. Proof. The proof follows a super/sub-solution argument. To start with, define Lg by Lgu = −cn∆gu+Rgu− R̄u n+2 n−2 . Note that Lg1 = Rg − R̄ ≥ 0 by assumption. Thus, 1 is a super-solution of (3.9). To find a sub-solution to (3.9), take f := Rg − R̄ ≥ 0. Note that Rg = −n(n − 1) + O(τn) in E . Thus, Lemma 3.5 applies and yields a solution v to (3.6), satisfying 0 < δ0 ≤ v ≤ 1 for some constant δ0. Now we compute −cn∆gv +Rgv − R̄v n+2 n−2 = −cn∆gv + fv + R̄ ( 1− v 4 n−2 ) v = R̄ ( 1− v 4 n−2 ) v ≤ 0, where the inequality follows from the assumption that R̄ ≤ 0 and the bounds for v. Thus, v is a sub-solution of (3.9). Then one finishes the proof by following the argument of [4, Proposition 2.1]. ■ Next, we will focus on the behavior of u towards the ALH infinity E0. Lemma 3.7. Let u be as in Proposition 3.6. Given any α ∈ (0, n − 1), there exists a constant τ0 > 0 such that 1− (τ/τ0) α ≤ u ≤ 1 for any τ ≤ τ0. Proof. Let ξ := 1− (τ/τ0) α. By (3.8), we have cn∆gξ − ( Rg − R̄ ) ξ = cnα(HEτ − α)(τ/τ0) α − ( Rg − R̄ ) ξ. (3.10) Since Rg − R̄ ∈ O(τn) in E , the right-hand side of (3.10) is positive for τ ≤ τ0, provided that τ0 is sufficiently small. On the other hand, since R̄ ≤ 0 and 0 < u ≤ 1, (3.9) implies that cn∆gu− ( Rg − R̄ ) u ≤ 0. Regarding boundary data, u− ξ ≥ 0 along τ = τ0 and lim τ→0 u = lim τ→0 ξ = 1. Now the maximum principle implies that u ≥ ξ for τ ∈ (0, τ0]. ■ Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 15 Proposition 3.8. Let ( Mn, g ) , R̄ and u be as in Proposition 3.6. Additionally, suppose that Rg + n(n − 1) ∈ O ( τn+1 ) and that, however small τ0 is, Rg > −n(n − 1) at some point in {τ ≤ τ0} ⊂ E. Then u must have the following asymptotic expansion near τ = 0: u = 1 + un0τ n +O ( τn+1−ϵ ) , where un0 < 0 is a smooth function defined on the conformal infinity E0 ∼= Tn−1 and ϵ > 0 is an arbitrary small constant. Proof. Let us take w := u− 1 ≤ 0. By [3, Theorem 1.3], w has the expansion w = ∞∑ i=1 Ni∑ j=0 uijτ i(log τ)j , where uij ∈ C∞(E0). Clearly, the proof would be complete once we verify the conditions: (C1) uij = 0 for i < n; (C2) unj = 0 for j > 0; (C3) un0 < 0. Verification of (C1). By (3.9), w satisfies ∆gw − nw = 1 cn [ Rg(w + 1)− R̄(w + 1) n+2 n−2 ] − nw. Since only a neighborhood of E0 is concerned, we can simply substitute R̄ = −n(n − 1); by rearranging terms, we get ∆gw − nw = 1 cn [Rg + n(n− 1)]u+ n(n− 1) cn [ (w + 1) n+2 n−2 − 1− n+ 2 n− 2 w ] =: A+B. Since limτ→0 u = 1 and 0 ≤ Rg + n(n − 1) ∈ O ( τn+1 ) , we have A ≥ 0 and A ∈ O ( τn+1 ) . On the other hand, B is the remainder of a Taylor expansion truncated at the linear term, so B = O(w2) as τ → 0. By Lemma 3.7, w = O(τα) for any α < n− 1. Of course, we can choose α > (n+1)/2, and thus B = O ( τ2α ) = o ( τn+1 ) . In summary, for sufficiently small τ0, we have 0 ≤ ∆gw − nw ∈ O ( τn+1 ) for τ ≤ τ0. (3.11) Now consider any β ∈ (n− 1, n). Using (3.3), it is easy to verify that ∆gτ β − nτβ = −(β + 1)(n− β)τβ +O ( τn+2 ) . Clearly, there exists τ0 > 0 such that (∆g − n) ( w + λτβ ) ≤ 0 for all τ ≤ τ0 and constants λ ≥ 1. Fix such a τ0, and let us choose λ ≥ 1 such that w\|{τ=τ0} + λτβ0 ≥ 0; moreover, we have limτ→0(w + λτβ) = 0. Thus, by the maximum principle, w ≥ −λτβ for τ ≤ τ0. Since β ∈ (n− 1, n) is arbitrary and w ≤ 0, this verifies (C1). 16 T. Hao, Y. Hu, P. Liu and Y. Shi Verification of (C2). By (C1), we have w = Nn∑ j=0 unjτ n(log τ)j +O ( τn+1−ϵ ) . Further information about unj is obtainable by computing (∆g − n)w using this expansion and then comparing the result with (3.11). In fact, direct computation and (3.3) yield: (∆g − n)τn = O ( τn+2 ) , (∆g − n)[τn(log τ)j ] = [ (n+ 1)j(log τ)j−1 + j(j − 1)(log τ)j−2 ] τn +O ( τn+2 ) with 1 ≤ j ≤ Ni. Now, since unj are all defined on E0, we have ∆gunj ∈ O ( τ2 ) ; and since the remainder O(τn+1−ϵ) does not contribute to the coefficients sj of τn(log τ)j in (∆g − n)w, we have that sj equals to (n+ 1)un1 + 2un2 for j = 0, 2(n+ 1)un2 + 6un3 for j = 1, ... Nn(n+ 1)unNn for j = Nn − 1, 0 for j = Nn. By (3.11), all sj must vanish, which implies that unj ≡ 0 for j = 1, . . . , Nn. This verifies (C2). Verification of (C3). Consider an auxiliary function ζ := −δ ( τn+τn+1 ) where δ > 0 remains to be chosen. Now (∆g − n)ζ = −δ [ (n+ 2)τn+1 +O ( τn+2 )] , so (∆g − n)ζ ≤ 0 provided that τ is small, and let us choose τ0 accordingly (note: this is independent of the choice of δ). By comparison, recall from (3.11) that (∆g−n)w ≥ 0 for τ ≤ τ0. Regarding boundary data, first note that the assumption about Rg implies that w cannot be identically zero for τ ∈ (0, τ0]; thus, the strong maximum principle implies, in particular, that w < 0 along τ = τ0. This allows us to choose δ such that w ≤ ζ along τ = τ0. Moreover, both w, ζ → 0 as τ → 0. Now, by the maximum principle, we get w ≤ ζ = −δ ( τn + τn+1 ) for τ ≤ τ0. This proves that un0 < 0, verifying (C3). ■ Lemma 3.9. Let (Mn, g) be a Riemannian manifold with an ALH end E, on which the asymp- totic expansion (3.1) applies. Suppose that u = 1 + φτn +O(τn+1) is a function defined on E, where φ ∈ C∞(E0). Then, up to a diffeomorphism that restricts to be the identity on E0, the deformed metric ḡ = u 4 n−2 g on E has the expansion ḡ = 1 τ̄2 [ dτ̄2 + h̄+ τ̄n n κ̄+O ( τ̄n+1 )] , where h̄ = h and κ̄ = κ+ 4(n+ 1) n− 2 φh. Proof. A standard argument following the proof of [6, Lemma 6.5]. ■ Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 17 4 Two rigidity results The goal of this section is to prove Theorem 1.5, Corollary 1.6 and Theorem 1.4. The reader may consult Appendix A before proceeding. Proposition 4.1 (cf. [11, Theorem 1.1]). For 3 ≤ n ≤ 7, let Mn be a (connected) non-compact manifold with connected, compact boundary Σ. Let ι : Σ ↪→ M be the inclusion map. Suppose that Σ ∈ Cdeg (see Definition 1.9) and that the map ι is incompressible. Then M admits no complete metric g with Rg ≥ −n(n− 1) and HΣ > n− 1. Proof. To begin with, by the classification of covering spaces, there exists a covering of M , say p : M̂ →M , that satisfies p∗ ( π1 ( M̂ )) = ι∗(π1(Σ)) ⊂ π1(M), (4.1) where base points for the fundamental groups are omitted. Moreover, by the homotopy lifting property, there exists an embedding ι̂ : Σ → M̂ such that ι = p ◦ ι̂. By (4.1) and the incompressibility of ι, the composition J := ( ι∗ −1 ∣∣ ι∗(π1(Σ)) ) ◦ p∗ : π1(M̂) → π1(Σ) is a well-defined group homomorphism. Since Σ is aspherical, by [24, Proposition 1B.9], there exists a map j : M̂ → Σ such that j∗ : π1(M̂) → π1(Σ) is equal to J ; in particular, j∗◦ι̂∗ = idπ1(Σ); then, by applying the uniqueness part of [24, Proposition 1B.9] to Σ, it is easy to see that j ◦ ι̂ is in fact homotopic to idΣ. Since ι is an embedding, each boundary component of M̂ , which is a lifting of Σ, must be diffeomorphic to Σ. In particular, denote Σ̂ = ι̂(Σ). Since j ◦ ι̂ is homotopic to idΣ, we have[ Σ̂ ] = ι̂∗[Σ] ̸= 0 ∈ Hn−1(M̂ ;Z). Now, for the sake of deriving a contradiction, suppose that g is a complete metric on M with Rg ≥ −n(n− 1) and HΣ ≥ (n− 1)(1+ δ) for some constant δ > 0. Let ĝ := p∗g be the pull-back metric on M̂ , and define ρ(x) := distĝ(x, Σ̂) for x ∈ M̂ . For an arbitrarily large T > 0, let DT := { x ∈ M̂ : ρ(x) ≤ T } , and let Σ̂i (0 ≤ i ≤ k) be those components of ∂M̂ that satisfy Σ̂i ∩ DT ̸= ∅, where Σ̂0 = Σ̂. Define (see Figure 3 below) UT = DT ∪ ⋃ 0≤i≤k Σ̂i and UT,ϵ = { x ∈ M̂ : distĝ(x,UT ) < ϵ } . Since M is complete, connected and non-compact, so is M̂ , and we have ŪT ⋐ UT,ϵ. Moreover, for small enough ϵ, UT,ϵ ∩ ( ∂M̂ − ⋃ 0≤i≤k Σ̂i ) = ∅. Thus, by the smooth Urysohn lemma, there exists a function η ∈ C∞(M̂) with η(x) = { 0, x ∈ UT , 1, x ∈ M̂ \ UT,ϵ. 18 T. Hao, Y. Hu, P. Liu and Y. Shi Figure 3. A schematic picture showing UT , UT,ϵ (left figure) and Ba (right figure). The complement of UT,ϵ, which may include more boundary components of M̂ , is not displayed. Let a ∈ (0, 1) be a regular value of η. Automatically, η−1(a) is a smooth, closed hypersurface of M̂ , and η−1(a) ∩ ∂M̂ = ∅. By the above arrangement, Ba := η−1 ( [0, a] ) , equipped with the restriction of the metric ĝ, is a Riemannian band with ∂+ = Σ̂ and ∂− = ∂Ba \ Σ̂ = η−1(a) ∪ ⋃ 1≤i≤k Σ̂i. By letting f = j\|Ba and using Lemma A.8, one easily sees that Ba satisfies the NSep+property (see Definition A.6). Then take ∂⋆ = η−1(a). With these choices, all assumptions of Lemma A.9 are satisfied for (Ba, ĝ\|Ba ; ∂−, ∂+) and ∂⋆. Since ( M̂, ĝ ) is complete and non-compact, the distance distĝ(∂⋆, ∂+) can get arbitrarily large as one chooses large T . This contradicts Lemma A.9. ■ Remark 4.2. Proposition 4.1 still holds ifM is allowed to be compact. In fact, proceeding along the same proof, we still have [ Σ̂ ] ̸= 0 ∈ Hn−1 ( M̂ ;Z ) , so M̂ cannot be compact with a single boundary component. Hence, either (1) M̂ is non-compact, and the previous proof applies verbatim; or (2) M̂ is itself a Riemannian band with ∂+ = Σ̂ that satisfies the NSep+property and the curvature bounds Rĝ ≥ −n(n−1), H∂M̂ ≥ (n−1)(1+δ); however, by Remark A.10 (A), such a band cannot exist, reaching a contradiction. Proposition 4.3. For 3 ≤ n ≤ 7, let ( Mn, g ) be a complete Riemannian manifold without boundary, with an ALH end E ∼= (0, 1) × Tn−1, and satisfying Rg ≥ −n(n − 1). Suppose that Y := M \ E is non-compact and that ∂Y ∼= Tn−1 is incompressible in M . Then m̄E,g ≥ 0. In addition, if the assumptions (d), (e) in Theorem 1.5 hold, then κ = 0 only if (M, g) is Einstein. Proof. Suppose, on the contrary, that m̄E,g < 0. Let τ be a defining function compatible with the ALH structure of E (see (3.1)). Then by Lemma 3.3, there exists a small τ0 > 0 such that the mean curvature of the level set Eτ0 satisfies HEτ0 ≥ (n− 1) + δ0 for some δ0 > 0. Now, remove {0 < τ < τ0}, a subset of E , from M and denote the resulting manifold by M ′. By using the assumptions, it is easy to see that ∂M ′ = Eτ0 ∼= Tn−1 is incompressible in M ′. Clearly, ∂M ′ ∈ Cdeg. By Proposition 4.1, we get a contradiction. This proves the inequality m̄E,g ≥ 0. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 19 Next we turn to the second part of the proposition. Again we argue by contradiction. Assume that κ = 0 without (M, g) being Einstein. Let g(t) be the NRF initiated at g. Then by Lemma 3.4, for some small t0, we have (i) Rg(t0) > −n(n− 1) on M ; (ii) Rg(t0) ≤ −α/2 < 0 outside a compact subset of M ; (iii) Rg(t0) = −n(n− 1) +O ( τn+1 ) on E ; (iv) ( E , g(t0)\|E ) remains ALH with κ(t0) = 0. It is easy to check that a function R̄ as described in Proposition 3.6 exists; thus, there is a positive function u ∈ C∞(M) such that ḡ := u4/(n−2)g(t0) is complete with Rḡ = R̄ ≥ −n(n − 1). Furthermore, thanks to (i) and (iii) above, both Proposition 3.8 and Lemma 3.9 apply. As a consequence, ( E , ḡ\|E ) remains ALH and satisfies κ̄ = 4(n+ 1) (n− 2) un0h, where un0 < 0, and h is a flat metric on Tn−1. Clearly, m̄E,ḡ = trhκ̄ < 0. This contradicts the first part of the proposition. ■ Proof of Theorem 1.5. For convenience, let No (resp., Ho) denote the result of removing a tubular neighborhood of ϕ ( Tk ) from N (resp., ψ ( Tk ) from Hn/Zn−1). Both ∂No and ∂Ho inherit the product structure Sn−k−1 × Tk, which are identified to form M . In symbols, M = Ho ⊔Φ No, where Φ: ∂Ho → ∂No is the identification map. By Proposition 4.3, to prove the theorem it suffices to show that the boundary Σ of M \ E is incompressible in M . To show this, it in turn suffices to show that the Tk-factor of ∂No is incompressible in M , according to Lemma B.2. If this was not the case, let L be a non-contractible loop in {x}×Tk ⊂ ∂No that is contractible in M . Now consider H′ := ( S1 × Tn−k−1 −B ) × Tk, where B is an (n − k)-ball embedded in S1 × Tn−k−1. Topologically, M can be viewed as a subset of M ′ := H′ ⊔Φ No, so L is also contractible in M ′. By [11, Lemma A.3], H′ satisfies the ‘lifting property’ (see [11, Definition A.2]). Thus, [11, Lemma A.4] applies, showing that L is contractible in No and hence in N ; since {x} × Tk and ϕ(Tk) are homotopic in N , ϕ cannot be incompressible, violating the assumption (a). ■ Remark 4.4. The proof above can be made more direct if one assumes that k < n− 2. In this case, both π1(∂No) and π1(∂Ho) are isomorphic to π1 ( Tk ) , and it is easy to see that the maps π1 (∂No) → π1 (No) and π1 (∂Ho) → π1 (Ho) are both injective. By van Kampen’s theorem, we have π1(M) ∼= π1 (Ho) ∗π1(∂No) π1 (No). Thus, a direct application of [31, Theorem 11.67 (i)] shows that ∂No is incompressible in M , and it follows that the Tk-factor of ∂No is also incom- pressible in M . Proof of Corollary 1.6. In this setting, the assumptions (a− e) in Theorem 1.5 are satisfied. Since κ automatically vanishes, we conclude that g is Einstein. Write the metric on Hn/Zn−1 as dt2 + e2tg0 where g0 is a flat metric on Tn−1. Since Hn/Zn−1 is isometric to (M, g) outside a compact set, one can remove the corresponding cusp (i.e., {t < −a} for some a ≫ 0) from M and obtain a complete, non-compact manifold (M ′, g′) with boundary ∂M ′ ∼= Tn−1, satisfying 20 T. Hao, Y. Hu, P. Liu and Y. Shi H∂M ′ ≡ n−1, where the mean curvature is computed with respect to the inward normal. By [18, Theorem 2], (M ′, g′) is isometric to [−a,∞)×Tn−1 with the warped product metric dt′2+e2t ′ g0; by using this fact and the respective distance functions to ∂M ′ ⊂ M and {−a} × Tn−1 ⊂ Hn/Zn−1, it is easy to construct an isometry between (M, g) and Hn/Zn−1. ■ Remark 4.5. The statement of Corollary 1.6 remains true when N is non-compact without boundary. In fact, one only needs to prove the incompressibility of a Tn−1-slice located far into the ALH infinity of M , and this is handled by a corresponding step in the proof of Theorem 1.5. Then the result follows directly from Theorem 1.10. Proof of Theorem 1.4. Let (x, z) be the standard coordinates on R2 +, a topological factor of Hn/Zn−2. Since K is compact, via the isometry f , both x and z can be regarded as coordinate functions on M \K. Thus, for a large enough x0 > 0, we can remove {\|x\| > x0} from M and then identify {x = ±x0} in the same way as we did in Section 2.4.1. The result is a complete Riemannian manifold (M∗, g∗) with an ALH end E , satisfying Rg∗ ≥ −n(n − 1). Moreover, (M∗, g∗) is isometric to Hn/Zn−1 outside a compact set; thus, the assumptions (d), (e) in Theorem 1.5 hold automatically, and κ = 0 for (E , g∗\|E). It is easy to see thatM∗ is of the formM1⊔ΦM2 as described in Lemma B.2 with k = n−2. In particular, M2 can be viewed as a subset of M . By assumption, f−1(T ) is incompressible in M and hence in M2. Using the proof of Theorem 1.5, one can show that f−1(T ) is incompressible in M∗; then by Lemma B.2, ∂(M \ E) ∼= Tn−1 is incompressible in M∗. Thus, all conditions in Proposition 4.3 are verified for (M∗, g∗), and we conclude that g∗ is Einstein. The proof of Corollary 1.6 shows that there is an isometry f̃ : (M∗, g∗) → Hn/Zn−1 that uniquely extends the isometry, induced by f , between the ‘cuspidal ends’ in M∗ and Hn/Zn−1. Let z0 > 0 be sufficiently small; then by using distance functions to the hypersurfaces {z = z0} in both M and Hn/Zn−2, it is easy to construct an isometry between (M, g) and Hn/Zn−2; details are left to the interested reader. ■ 5 Two splitting results of ‘cuspidal-boundary’ type The bulk of this section is dedicated to proving Theorem 1.7. The proof of Theorem 1.10, which largely depends on those of Proposition 4.1 and Theorem 1.7, will be sketched at the end of the section. Now we begin our proof of Theorem 1.7. In addition to its hypothesis, let us assume that H∂M ≥ 3. Under this assumption, the proof would be complete once we show that (M, g) is isometric to ( (−∞, 0] × Σ,dt2 + e2tg0 ) for some closed 3-manifold Σ carrying a flat metric g0. In fact, Σ will occur as a hypersurface in M , obtained by an approximation scheme involving µ-bubbles (Sections 5.1 and 5.2); then we show that Σ must be compact and that (M, g) is isometric to the desired warped product (Section 5.3). The reader is recommended to consult Appendix A before proceeding. 5.1 Specification of µk and Ek Since M is non-compact with compact boundary, there exists a smooth, proper map ρ : M → (−∞, 0] (see [38, Lemma 2.1]) such that ρ−1(0) = ∂M, \|dρ\|g < 1. Fix a smooth function η ∈ C∞((−∞, 0]) satisfying η(t) = 0 for any t ≤ −1 and η(0) = 2; Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 21 define τk by 3 coth(2τk) = 3 + k−1, and then define µ̂k : (−τk, 0] → R by µ̂k(t) = 3 coth ( 2(t+ τk) ) − k−1η(t). Thus, {τk}∞k=1 is increasing and tends to infinity, and µ̂k(−τk) = +∞ and µ̂k(0) = 3− k−1. Now, choose ak, regular values of ρ, such that τk ≤ ak < min{τk+1, τk + 1}, and then define Ek := ρ−1 ( [−ak, 0] ) ⊂M . Denote ∂−k := ρ−1(−ak), which are smooth hypersurfaces of M . This makes (Ek, g\|Ek ; ∂−k , ∂M) a Riemannian band. Finally, let ρk := (τk/ak)ρ, and define µk := µ̂k ◦ ρk. By this arrangement, µk\|∂− k = ∞. 5.2 µk-bubbles in Ek For each fixed k, consider (Ek, g\|Ek ; ∂−k , ∂M). Note that H∂M ≥ 3; by construction, µk satisfies the barrier condition (see Definition A.1). By Fact A.2, a smooth µk-bubble Ωk exists. Define Σk := ∂Ωk \ ∂−k , which is smooth, closed, and separates ∂−k from ∂M . The following lemma shows that all Σk must meet a fixed compact subset of M . Lemma 5.1. Let K := {x ∈M : distg(x, ∂M) ≤ 10}. Then Σk ∩ K ̸= ∅. Proof. Suppose on the contrary that Σk∩K = ∅. This implies that η◦ρk = 0 on Σk. Moreover, by assumption, Rg ≥ −12, and by construction, \|dρk\|g < 1. Thus, we have (see (A.1)) Rµk + > −12 + 4 3 [3 coth(2(ρk + τk))] 2 − 12[sinh(2(ρk + τk)] −2 = 0 on Σk. This, along with Fact A.5, implies that Σk admits a PSC metric; since Σk is separating, we get a contradiction, by Lemma A.11. ■ 5.3 Convergence of Σk By using [37, Theorem 3.6], one can show that the second fundamental form IIΣk is uniformly bounded within any compact subset of M . Thus, by Lemma 5.1, Σk subconverges to a smooth hypersurface Σ in M (for convenience, denote the subsequence by the same symbol Σk). Within compact subsets ofM , the convergence is uniform and has multiplicity one; moreover, Σ bounds a ‘minimizing 3-bubble’ for which minimality is interpreted with respect to compactly supported perturbations (cf. [25, Lemma 4.10]). Depending on whether Σ is compact, we consider the two cases below. Case 1: Σ is compact. By minimality, we have (see Fact A.3) HΣ = 3 and LΣ = −∆Σ + 1 2 ( RΣ −R3 + ) ≥ 0. (5.1) Since R3 + = Rg+12 ≥ 0, (5.1) implies that −∆Σ+ 1 2RΣ ≥ 0; thus, there exists a smooth function u > 0 defined on Σ and a constant λ ≥ 0 such that( −∆Σ + 1 2 RΣ ) u = λu. (5.2) 22 T. Hao, Y. Hu, P. Liu and Y. Shi Define g̃Σ = ugΣ where gΣ is the metric on Σ induced by g. We have Rg̃Σ = u−1 ( RΣ + 3 2 ∣∣∣∣∇uu ∣∣∣∣2 − 2 ∆u u ) = u−1 ( 2λ+ 3 2 ∣∣∣∣∇uu ∣∣∣∣2 ) ≥ 0. (5.3) Since each Σk is separating, so is Σ. By Lemma A.11, Σ admits no PSC metric; then by (5.3) and the trichotomy theorem of Kazdan and Warner, Rg̃Σ = 0. Thus, λ must vanish, and u must be a constant; (5.2) in turn implies that RΣ = 0. Then by Bourguignon’s theorem (see [27, Lemma 5.2]), gΣ is Ricci-flat, which must be flat since dimΣ = 3. Now we prove that a neighborhood of Σ splits. When Σ∩∂M = ∅, since Σ is the boundary of minimizing 3-bubble, [2, Theorem 2.3] implies that there exists an open neighborhood of Σ that is isometric to a warped product ( (−ϵ, ϵ)× Σ, dt2 + e2tgΣ ) , where t is the coordinate on (−ϵ, ϵ) and Σ corresponds to t = 0. When Σ ∩ ∂M ̸= ∅, we must have Σ = ∂M , by the maximum principle. In this case, the proof of [2, Theorem 2.3] still applies and gives an open neighborhood of Σ that is isometric to a warped product ( (−ϵ, 0]× Σ,dt2 + e2tgΣ ) . Thus, a neighborhood of Σ is foliated by the t-level sets. Note that moving along the foliation leaves the energy functional invariant; thus, each t-slice also bounds a minimizing 3-bubble, to which the same analysis above applies. This implies that a maximal neighborhood U of Σ on which the metric splits as( I × Σ,dt2 + e2tgΣ ) must be both open and closed in M . By connectedness, U = M , and I must be of the form (−∞, c]. This achieves the desired splitting. Case 2: Σ is non-compact. By finding a contradiction, we prove that this case does not occur. The argument largely follows the proof of [38, Theorem 1.1], so we only sketch the steps. Let (Mk, gk) = ( Σk × S1, gΣk + u2kdt 2 ) , where uk is the first eigenfunction of LΣk ; that is, LΣk uk = λkuk with λk ≥ 0. Since dimΣk = 3, [9, Corollary 1.10] implies that Mk admits no PSC metric. Now Rgk = RgΣk − 2 ∆gΣk uk uk = Rµk + + 2λk. (5.4) By construction, Rµk + ≥ 0 outside K, and there exist δk > 0, satisfying lim δk = 0, such that Rµk + ≥ −δk on M . Since Rgk cannot be positive and λk ≥ 0, by (5.4), we must have limλk = 0. Next, choose qk ∈ Σk ∩ K so that lim qk = q ∈ Σ, and let pk = (qk, t0) ∈ Σk × S1 and p = (q, t0) ∈ M̃ = Σ × S1. Normalize uk such that uk(qk) = 1. By the Harnack inequality, uk converges smoothly to a positive function u on Σ with u(q) = 1. Thus, (Mk, gk) converges in the pointed smooth topology to ( M̃, g̃ ) , where g̃ = gΣ + u2dt2. Now one can follow the proof 8 of [38, Proposition 3.2] to show that Ricg̃ = 0, and then follow the proof 9 of [38, Theorem 1.1] to show that u is constant, which implies RicgΣ = 0. In summary, (Σ, gΣ) is complete, non-compact, Ricci-flat, and with finite area; this contradicts [32, p. 25, Theorem 4.1]. ■ 8The proof of [38, Proposition 3.2] only relies on M̃ admitting no PSC metric and the properties of R µk + mentioned above. 9In particular, the boundedness of area(Σ) follows from Aµk Ω0 (Ωk) ≤ Aµk Ω0 (Ek) and µk > 0. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 23 Remark 5.2. The PSC obstruction, provided by [9, Corollary 1.10], for manifolds of the form Σ×S1 only works when dimΣ ̸= 4. On the other hand, if Σ (2 ≤ dimΣ ≤ 6) is closed, orientable, and if it admits a map of nonzero degree to some Σ′ ∈ Cdeg, then by a similar argument as [11, Theorem 1.1], one can show that Σ× S1 admits no PSC metric. Proof of Theorem 1.10. The inequality inf∂M H ≤ n−1 follows directly from Proposition 4.1. To prove the second part of the theorem, first obtain a covering ( M̂, ĝ ) of (M, g) as in the proof of Proposition 4.1, and then apply (essentially) the same proof of Theorem 1.7 to ( M̂, ĝ ) ; to assist the reader, we list a few points that may need attention. � ∂M̂ may not be connected, but Riemannian bands can still be constructed in a similar manner as in the proof of Proposition 4.1. To avoid clash of symbols, denote S := ∂M and let Ŝ be a fixed lifting of S in M̂ . Thus ∂+ = Ŝ and ∂⋆ ⊂ ∂−; µk > 0 can be defined such that µk\|∂⋆ = ∞ and µk\|Ŝ = (n−1)−1/k; on ∂− \∂⋆ (if nonempty) we have H ≥ n−1; one can check that the barrier condition is satisfied, and the Σks exist; restricting j : M̂ → S to Σk yields a map Σk → Ŝ of nonzero degree. � An adapted version of Lemma 5.1 holds; in the proof, invoke Lemma A.8 instead of Lemma A.11. It follows that Σk converges to some Σ. � When Σ is compact, the corresponding part in Section 5.3 applies, apart from dimensional adjustments and the fact that Ricci-flatness may no longer imply flatness. � When Σ is non-compact, we need to argue, without relying on [9, Corollary 1.10], that Mk = Σk × S1 admits no PSC metric, and this is already addressed by Remark 5.2. The consequence is that ( M̂, ĝ ) is of the form( (−∞, 0]× Σ,dt2 + e2tgΣ ) , where gΣ is Ricci-flat. In particular, the covering M̂ → M is 1-fold and hence an isometry. Since Σ = ∂M is assumed to be aspherical, gΣ must be flat, which can be seen by applying the Cheeger–Gromoll splitting theorem to the universal cover; for details, see the beginning paragraph of [12, Section 6]. ■ A µ-bubbles This section collects some ‘definitions’ and ‘facts’ concerning the µ-bubble technique, about which we make no claim to originality. For detailed expositions and proofs, the reader may con- sult [9, 12, 37, 39] and [22, Section 5]. This section also includes three supplementary ‘lemmas’. A common setting for µ-bubbles is a Riemannian band, namely a compact, connected Rie- mannian manifold ( Mn, g ) whose (nonempty) boundary is expressed as a disjoint union ∂M = ∂− ⊔ ∂+, where each of ∂± is a smooth, closed and possibly disconnected (n− 1)-manifold. Given a Riemannian band ( Mn, g; ∂−, ∂+ ) and a function µ ∈ C∞(M̊), consider the following variational problem: Let Ω0 be a smooth open neighborhood of ∂−; among all Caccioppoli sets Ω ⊂M that satisfy ∂− ⊂ Ω and Ω∆Ω0 ⋐ M̊ , seek a minimizer of the functional Aµ Ω0 (Ω) = Hn−1(∂Ω)−Hn−1(∂Ω0)− ∫ M (χΩ − χΩ0)µdHn, where Hk is the induced k-dimensional Hausdorff measure, and χΩ, χΩ0 are characteristic func- tions. Such a minimizer is called a µ-bubble. Existence and regularity of µ-bubbles are well-established when µ satisfies the following ‘bar- rier condition’. 24 T. Hao, Y. Hu, P. Liu and Y. Shi Definition A.1. Let ( Mn, g; ∂−, ∂+ ) be a Riemannian band. A function µ ∈ C∞(M̊) is said to satisfy the barrier condition if, for each connected component S ⊂ ∂+ (resp., S ⊂ ∂−), � either µ smoothly extends to S and satisfies HS > µ\|S (resp., HS > −µ\|S), where HS is the mean curvature of S with respect to the outward normal; � or µ→ −∞ (resp., µ→ +∞) towards S. Fact A.2. For 3 ≤ n ≤ 7, if µ ∈ C∞(M̊) satisfies the barrier condition, then there exists a smooth µ-bubble Ω. In particular, ∂Ω \ ∂− is homologous to ∂+ and is separating (see Defini- tion A.6 below). Also well-known are the following variational properties. To fix notation, let Σ denote the hypersurface ∂Ω \ ∂− with outward unit normal ν; let RΣ and ∆Σ be, respectively, the scalar curvature and the Laplacian along Σ (with the induced metric); let HΣ and II be, respectively, the mean curvature and the second fundamental form of Σ, computed with respect to ν; define the operators JΣ = −∆Σ + 1 2 ( RΣ −Rg − µ2 − \|II\|2 ) − ν(µ) LΣ = −∆Σ + 1 2 ( RΣ −Rµ + ) , where Rµ + = Rg + n n− 1 µ2 − 2\|dµ\|g. (A.1) Fact A.3. Suppose that Ω is a smooth µ-bubble. We have (a) HΣ = µ\|Σ; (b) LΣ ≥ JΣ ≥ 0. The semi-positivity of LΣ has several applications, and we shall list a few. To start with, let u > 0 be an eigenfunction associated to the first eigenvalue λ ≥ 0 of LΣ. Consider the warped-product metric ĥ := gΣ + u2dθ2 defined on Σ̂ := Σ× S1, where θ ∈ S1. Fact A.4. Suppose that Ω is a smooth µ-bubble. The scalar curvature of ( Σ̂, ĥ ) is Rĥ = RΣ − 2u−1∆Σu = Rµ + + 2λ. In particular, if Rµ + > 0 on Σ, then Σ× S1 admits a PSC metric. Alternatively, one can compare LΣ with the conformal Laplacian on Σ and obtain the fol- lowing. Fact A.5. For n ≥ 3, suppose that Ω is a smooth µ-bubble on which Rµ + > 0. Then Σ admits a PSC metric. With additional topological assumptions onM , Fact A.5 can be used to prove width estimates for (M, g). To be precise, we start by recalling the following notion (cf. [9, Property A]). Definition A.6. Given a (topological) band ( Mn; ∂−, ∂+ ) , we say that a (closed) hypersurface S in M is separating, if all paths connecting ∂− and ∂+ must intersect S. A band is said to satisfy the NSep+property if no separating hypersurface admits a PSC metric. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 25 Remark A.7. If S ⊂ Mn is a separating hypersurface, then there exists a minimal list of connected components Si (i = 1, . . . , k) of S such that their union S ′ remains separating. For details, see [9, Lemma 2.2]. Using intersection theory, one can show that [S ′] ̸= 0 ∈ Hn−1(M ;Z). Moreover, with suitable orientation, S ′ is homologous to ∂+ in M . Lemma A.8. Let ( Mn, g; ∂−, ∂+ ) be a Riemannian band, and let ι : ∂+ ↪→ M be the inclu- sion map. Suppose that ∂+ ∈ Cdeg (see Definition 1.9) and that there exists a continuous map f : M → ∂+ such that f ◦ ι is homotopic to id∂+. Then (M, g) satisfies the NSep+property. Proof. Suppose that S is a separating hypersurface in M , and let S ′ be as in Remark A.7; in particular, S ′ is homologous to ∂+ in M . Now since f ◦ ι is homotopic to id∂+ , it is easy to see that the restriction f \|S′ : S ′ → ∂M has degree 1. Since ∂+ ∈ Cdeg, S ′ admits no PSC metric. ■ The next lemma is a variant of Gromov’s band-width estimate [22, Section 5.3]. Lemma A.9. For 3 ≤ n ≤ 7, let ( Mn, g; ∂−, ∂+ ) be a Riemannian band that satisfies the NSep+property, and let ∂⋆ ⊂ ∂− be a compact subset without boundary. Suppose that (a) Rg ≥ −n(n− 1); (b) H∂−\∂⋆ ≥ −(n− 1); (c) H∂+ ≥ (n− 1)(1 + δ) for some constant δ > 0. Then there exists a constant Tδ > 0, depending only on δ, such that distg(∂⋆, ∂+) ≤ Tδ. Proof. Set ϵ = δ/3, and define Cδ, Tδ > 0 by coth(Cδ/2) = 1 + δ/2 1 + ϵ and Tδ = Cδ n(1 + ϵ) . For the sake of deriving a contradiction, suppose that distg(∂⋆, ∂+) > Tδ. By the proof of [39, Lemma 4.1], there exists a smooth, proper function ρ : M → [−Tδ, 0] such that ρ−1(−Tδ) = ∂⋆, ρ−1(0) = ∂+, and \|dρ\|g < 1. (A.2) Now consider the function h(t) = (n− 1)(1 + ϵ) coth ( n(1 + ϵ)t+ Cδ 2 ) , t ∈ (−Tδ, 0]. By construction, h is decreasing, strictly greater than n− 1, and satisfies h(0) < H∂+ , lim t→−Tδ h(t) = ∞, n n− 1 h(t)2 + 2h′(t) ≡ n(n− 1)(1 + ϵ)2. (A.3) Combining (A.2), (A.3), and the assumptions (a), (b), (c), one can easily check that the function µ := h ◦ ρ, defined on M \∂⋆, satisfies both the barrier condition and the inequality Rµ + > 0. By Facts A.2 and A.5, there exists a separating hypersurface Σ in (M ; ∂−, ∂+) that admits a PSC metric. This contradicts the NSep+hypothesis. ■ Remark A.10. We mention two variants of Lemma A.9, both of which can be obtained by slightly modifying the proof above. (A) For 3 ≤ n ≤ 7, no Riemannian band can simultaneously satisfy theNSep+property and the conditions Rg ≥ −n(n−1), H∂− ≥ −(n−1) andH∂+ > n−1. (B) For 3 ≤ n ≤ 7, let (Mn, g) be a complete, non-compact Riemannian manifold with compact boundary ∂M . Suppose that M satisfies the NSep+property (see below); then (M, g) cannot satisfy the conditions Rg ≥ −n(n− 1) and H∂M > n− 1 simultaneously. 26 T. Hao, Y. Hu, P. Liu and Y. Shi The concept of separating hypersurfaces can also be defined for complete, non-compact Rie- mannian manifolds (M, g) with compact boundary—just require that S intersects with all paths connecting ∂M and infinity. The NSep+property can be extended to such manifolds. Lemma A.11. Let ( M4, g ) be a complete, non-compact Riemannian 4-manifold with compact (nonempty) boundary ∂M . Suppose that the homotopy groups π2(M) = π3(M) = 0. Then (M, g) satisfies the NSep+property. Proof. Suppose that S ⊂ M is a (closed) separating hypersurface that admits a PSC metric, and let S ′ ⊂ S be as indicated in Remark A.7. In particular, S ′ admits a PSC metric, and [S ′] ̸= 0 ∈ H3(M,Z). Since π2(M) is trivial, the topological classification of closed 3-manifolds admitting a PSC metric implies that S ′ is homologous to a spherical class in H3(M,Z) (see [35, p. 112]). Since π3(M) is also trivial, this violates Lemma B.1 below. ■ B Topological lemmas Lemma B.1. Let M be a non-compact 4-manifold satisfying π3(M) = 0. Then H3(M,Z) contains no nontrivial spherical class (i.e., classes of the form [ S3/Γ ] ). Proof. Let [β] denote the fundamental class of S3/Γ where Γ is a discrete subgroup of O(4). Let i : S3/Γ → M be a continuous map. The goal is to prove that i∗[β] = 0 ∈ H3(M,Z). Now let [α] be the fundamental class of S3. The composition S3 π−→ S3/Γ i−→M induces a map at the level of H3(·,Z), such that [α] π∗−→ d[β] i∗−→ di∗[β] where d is the degree of π. Since π3(M) = 0, Hurewicz homomorphism implies that di∗[β] = (i ◦ π)∗[α] = h([i ◦ π]) = 0 ∈ H3(M,Z), where h : π3(M) → H3(M,Z) is the Hurewicz map. Thus, in order to show that i∗[β] = 0, it suffices to show that H3(M,Z) is torsion free, and this follows from M being non-compact (see [7, Corollary 7.12]). ■ Lemma B.2. For 1 ≤ k ≤ n−2, letM1 = ( R×Tn−k−1−B ) ×Tk, where B is an embedded (n−k)- ball in R × Tn−k−1. Let M2 be a smooth, possibly non-compact, manifold with boundary ∂M2. Suppose that Φ: ∂M1 → ∂M2 is a diffeomorphism, and let M := M1 ⊔Φ M2 be the manifold obtained by identifying ∂M1, ∂M2 via Φ. Let t ∈ R be such that {t}×Tn−k−1 is disjoint from B. Then the hypersurface Σ = {t} × Tn−1 is incompressible in M if and only if the Tk-factor 10 of ∂M2 is incompressible in M . Proof. In M , the Tk-factor of Σ is homotopic to that of ∂M1 and hence to that of ∂M2. Thus, (⇒) is clear. For (⇐), we prove its contrapositive. Suppose that L ⊂ Σ is a non-contractible loop that is contractible in M . Write [L] = (miαi, njβj) ∈ π1 ( Tn−k−1 ) × π1 ( Tk ) ∼= π1(Σ), where αi generates the fundamental group of the i-th S1-factor in Tn−k−1 and mi ∈ Z, similarly for βj and nj . Let us write α̂i, β̂j for the corresponding elements in the homology class H1(Σ;Z). It will be convenient to view the R-factor in R×Tn−k−1 as S1 minus a point, and to view M as a subset of M̂ := M̂1 ⊔Φ M2, where M̂1 := ( S1 × Tn−k−1 −B ) × Tk. 10Note that ∂M2 has the product structure Sn−k−1 × Tk induced by Φ. Rigidity and Non-Rigidity of Hn/Zn−2 with Scalar Curvature Bounded from Below 27 Let ι : Σ ↪→ M̂ be the inclusion map. For 1 ≤ i ≤ n − k − 1, let θi be the coordinate on the i-th S1-factor of Tn−k−1. By construction, there exists ti ∈ S1 such that θi = ti defines a hypersurface Si in M̂ that is ‘dual’ to ι∗α̂i, in the sense that the intersection products [Si] · ι∗α̂i = 1 and [Si] · ι∗α̂i′ = [Si] · ι∗β̂j = 0, i′ ̸= i. Since L is contractible in M ⊂ M̂ , we have∑ i miι∗α̂i + ∑ j njι∗β̂j = 0 ∈ H1(M̂ ;Z); by taking intersection products with [Si], we see that mi = 0 for all i = 1, . . . , n− k− 1, so L is homotopic to a loop in the Tk-factor of Σ. Thus, the Tk-factor of Σ is not incompressible in M . By homotopy, the same is true for the Tk-factor of ∂M2. This completes the proof. ■ Acknowledgements We thank Shihang He for kindly sharing his proof of Lemma B.1. We also thank the anonymous referees for carefully reading the manuscript and offering suggestions, which has led to improved exposition and a more direct argument now included in Remark 4.4. Research leading to this work was supported by the National Key R&D Program of China Grant 2020YFA0712800 (T. Hao, P. Liu and Y. Shi) and the China Postdoctoral Science Foundation Grant 2021TQ0014 (Y. Hu). 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Not. 2023 (2023), 9880–9900, arXiv:2204.05491. https://doi.org/10.1090/surv/144 https://doi.org/10.1090/gsm/077 https://doi.org/10.1088/1361-6382/ac1fd1 https://arxiv.org/abs/2107.05603 https://doi.org/10.1088/1361-6382/aabed1 https://arxiv.org/abs/1801.03442 https://doi.org/10.1215/S0012-7094-92-06723-8 https://doi.org/10.1007/s10107-010-0354-x https://doi.org/10.1007/s00039-018-0453-z https://arxiv.org/abs/1710.04655 https://arxiv.org/abs/1811.04311 https://doi.org/10.1007/10.1142/9789811273223_000 https://arxiv.org/abs/1908.10612 https://doi.org/10.1007/BF02953774 https://doi.org/10.2140/pjm.2023.323.89 https://arxiv.org/abs/2205.08428 https://doi.org/10.1090/tran/8755 https://arxiv.org/abs/2108.12887 https://doi.org/10.1007/s00220-015-2421-x https://arxiv.org/abs/1310.3002 https://arxiv.org/abs/2103.02744 https://doi.org/10.1007/BF01452046 https://doi.org/10.1007/978-1-4612-4176-8 https://doi.org/10.4310/jdg/1214443292 https://doi.org/10.1515/crelle-2021-0072 https://arxiv.org/abs/2007.06756 https://theses.hal.science/tel-02953229 https://doi.org/10.4310/ATMP.2001.v5.n4.a4 https://doi.org/10.4310/ATMP.2001.v5.n4.a4 https://arxiv.org/abs/math/0109053 https://doi.org/10.4310/CJM.2020.v8.n2.a2 https://doi.org/10.4310/CJM.2020.v8.n2.a2 https://arxiv.org/abs/1808.03527 https://arxiv.org/abs/2008.07028 https://doi.org/10.1090/tran/8263 https://arxiv.org/abs/2003.01315 https://doi.org/10.1093/imrn/rnac117 https://arxiv.org/abs/2204.05491 1 Introduction 2 Non-rigidity of H^n/Z^(n-2) 2.1 First preliminary construction 2.2 Second preliminary construction 2.3 The gluing step 2.4 Further remarks 2.4.1 Surgery applied to H^n/Z^(n-1) 2.4.2 A note on topology 3 ALH manifolds, mass and deformations 3.1 ALH manifolds and mass 3.2 NRF deformations 3.3 Conformal deformations 4 Two rigidity results 5 Two splitting results of `cuspidal-boundary' type 5.1 Specification of mu_k and E_k 5.2 mu_k-bubbles in E_k 5.3 Convergence of Sigma_k A mu-bubbles B Topological lemmas References
id	nasplib_isofts_kiev_ua-123456789-212048
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spelling	Hao, Tianze Hu, Yuhao Liu, Peng Shi, Yuguang 2026-01-23T10:12:41Z 2023 Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below. Tianze Hao, Yuhao Hu, Peng Liu and Yuguang Shi. SIGMA 19 (2023), 083, 28 pages 1815-0659 2020 Mathematics Subject Classification: 53C21; 53C24 arXiv:2303.15752 https://nasplib.isofts.kiev.ua/handle/123456789/212048 https://doi.org/10.3842/SIGMA.2023.083 We show that the hyperbolic manifold ℍⁿ/ℤⁿ⁻² is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound −𝑛(𝑛 − 1), and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related splitting results. We thank Shihang He for kindly sharing his proof of Lemma B.1. We also thank the anonymous referees for carefully reading the manuscript and offering suggestions, which have led to improved exposition and a more direct argument now included in Remark 4.4. Research leading to this work was supported by the National Key R&D Program of China Grant 2020YFA0712800 (T. Hao, P. Liu, and Y. Shi) and the China Postdoctoral Science Foundation Grant 2021TQ0014 (Y. Hu). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below Article published earlier
spellingShingle	Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below Hao, Tianze Hu, Yuhao Liu, Peng Shi, Yuguang
title	Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below
title_full	Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below
title_fullStr	Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below
title_full_unstemmed	Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below
title_short	Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below
title_sort	rigidity and non-rigidity of ℍⁿ/ℤⁿ⁻² with scalar curvature bounded from below
url	https://nasplib.isofts.kiev.ua/handle/123456789/212048
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Rigidity and Non-Rigidity of ℍⁿ/ℤⁿ⁻² with Scalar Curvature Bounded from Below

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