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Metrics with Λ1(−∆ + Kr) ≥ 0 and Flexibility in the Riemannian

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Metrics with Λ1(−∆ + Kr) ≥ 0 and Flexibility in the Riemannian Metrics with λ1( ∆ + kR) 0 and flexibility in the Riemannian− Penrose≥ Inequality Chao Li and Christos Mantoulidis Abstract. On a closed manifold, consider the space of all Riemannian metrics for which ∆+ kR is positive (nonnegative) definite, where k> 0 and R is the scalar curvature.− This spectral generalization of positive (nonnegative) scalar curvature arises naturally for different values of k in the study of scalar curvature in dimension via minimal hypersurfaces, the Yamabe problem, and Perelman’s Ricci flow with surgery. When k =1/2, the space models apparent horizons in time-symmetric ini- tial data to the Einstein equations. We study these spaces in unison and generalize Cod´aMarques’s path-connectedness theorem. Applying this with k =1/2, we com- pute the Bartnik mass of 3-dimensional apparent horizons and the Bartnik–Bray mass of their outer-minimizing generalizations in all dimensions. Our methods also yield efficient constructions for the scalar-nonnegative fill-in problem. Contents 1. Introduction 2 2. Proof of Theorem 1.2 8 arXiv:2106.15709v2 [math.DG] 5 Jul 2021 3. Monotone PSC almost-cobordances 12 4. Proof of Theorem 1.4 19 5. Proof of Theorem 1.11 20 6. General Relativity, I: Bartnik mass 24 7. General Relativity, II: Bartnik–Bray mass 25 Appendix A. Some curvature formulas 30 Appendix B. Round normal foliations 31 M >0 M ≥0 Appendix C. More facts and formulas regarding k , k 34 M >0 Appendix D. A connected sum operation on k (M) 37 Appendix E. Some results from Kleiner–Lott’s notes 44 Appendix F. A refined gluing argument by B¨ar–Hanke 45 References 50 1 2 CHAOLIANDCHRISTOSMANTOULIDIS 1. Introduction 1.1. The spaces. In all that follows, M denotes a closed n-manifold and Met(M) denotes the space of smooth Riemannian metrics on M. Definition 1.1. For k (0, ), we define ∈ ∞ M ≥0(M) := g Met(M): λ ( ∆ + kR ) 0 , (1.1) k { ∈ 1 − g g ≥ } where λ ( ∆ + kR ) is the first eigenvalue of the operator ∆ + kR on M, and 1 − g g − g g Rg is the scalar curvature of g. We also define M ≥0(M) := g Met(M): R 0 . (1.2) ∞ { ∈ g ≥ } Finally, we define M >0(M), k (0, ], as above with all “ ” replaced by “>.” k ∈ ∞ ≥ These spaces are not generally encountered in the literature in this level of gener- ality, so some remarks are in order about their actual geometric significance. First, and crucially, these spaces of metrics are closed under scaling and diffeomorphisms. M >0 M ≥0 Second, k (M) and k (M) are descending filtrations in the space of metrics on M, i.e., for 0 <k<k′ , ≤∞ M >0 M ≥0 k′ (M) k′ (M) ⊂ (1.3) ∩≥0 ∩ M >0(M) M (M). k ⊂ k Their geometric interest is due to: M >0 M ≥0 When k = , k (M) and k (M) denote the spaces of metrics with • positive and∞ nonnegative scalar curvature on M. The study of these spaces goes back several decades and is still an active research area, even insofar as determining necessary and sufficient conditions for the non-emptiness of these spaces. When k = n−2 and n = dim M 3, M >0(M) and M ≥0(M) encode the • 4(n−1) ≥ k k conformal Laplacian, which appears in the Yamabe problem. Specifically, the spaces encode whether our metric is conformal to one of positive or nonnegative scalar curvature, respectively. See Lemma C.1 in the overview Appendix C. When k = 1 , M >0(M) plays a crucial role in Perelman’s work on Ricci flow • 4 k with surgery [Per02, Per03b]. For the purposes of Ricci flow, λ1( ∆g + 1 − 1 4 Rg) plays the role of a replacement for a suitably normalized minM Rg. 1 Treating λ1( ∆g + kRg), k [ 4 , ), as a replacement for Rg with better monotonicity− properties under∈ Ricci∞ flow strongly informs our work. 1 M >0 M ≥0 1 When k = 2 , k (M) and k (M) encode the operator ∆g + 2 Rg that • shows up in the study of positive or nonnegative scalar curvature− in one dimension higher than M via minimal hypersurfaces. See [SY79b, GL83], and Lemma C.6. A perhaps less well known but related instance of these spaces is when k ( 1 , 1 ] and dim M = 2, where M >0(M) plays a crucial role ∈ 4 2 k 1For k (0, ), min R k−1λ ( ∆ + kR ) vol (M)−1 R dµ . ∈ ∞ M g ≤ 1 − g g ≤ g M g g R METRICS WITH λ1 ≥ 0 3 in providing a priori diameter and area estimates in [SY79b, SY83, GL83] 1 The threshold k = 4 plays a special role for us too, but due to Ricci flow. In this paper, we put these spaces on common footing by incorporating them into a single filtration that interpolates between them. 1.2. Tools. Our starting point is a generalization of a theorem of Cod´aMarques [Mar12], who proved that the ultimate space in the filtration has a connected moduli space, i.e., M >0 ∞ (M)/ Diff+(M) is path-connected, when M is a closed orientable 3-manifold. He exploited Hamilton–Perelman’s Ricci flow with surgery [Per03b], which he combined with the Gromov–Lawson [GL80a] construction of the PSC-connected-sum procedure (cf. Schoen–Yau [SY79a]). We prove: Theorem 1.2. Let M be a closed orientable topologically PSC2 3-manifold. Then M >0(M)/ Diff (M), M ≥0(M)/ Diff (M) are both path connected for all k [ 1 , ]. k + k + ∈ 4 ∞ M >0 Note that, by Remark C.2, if M is a closed 3-manifold and k (M) is nonempty for some k [ 1 , ], then M is automatically topologically PSC. ∈ 8 ∞ Theorem 1.2 strictly generalizes [Mar12] from k = to k [ 1 , ], and implies ∞ ∈ 4 ∞ that the inclusion in (1.3) induces a map which is an isomorphism on the level of π0 1 when k [ 4 , ). Once we have Theorem 1.2, the recent breakthrough theorem of Bamler–Kleiner∈ ∞ [BK19] proving that M >0 ∞ (M) is contractible when nonempty (and thus path-connected) combines with Theorem 1.2 to give: Theorem 1.3. Let M be a closed orientable topologically PSC 3-manifold. Then M >0(M), M ≥0(M) are both path connected for all k [ 1 , ]. k k ∈ 4 ∞ Naturally, it would be interesting to know what happens in Theorems 1.2, 1.3 in 1 the regime k (0, 4 ), particularly given that the 3-dimensional conformal Laplacian 1 1∈ has k = 8 < 4 . To that end, we note the following special companion result: Theorem 1.4. Let M be as in Theorem 1.3. Then, >0 (M) is contractible and M1/8 ≥0 (M) is weakly contractible. M1/8 It would also be interesting to understand further topological properties of the inclusion in (1.3) besides path-connectedness, along the lines of the Bamler–Kleiner result. We do not pursue this. See also Appendix C and Remark C.5. To prove Theorem 1.2 we needed a suitable generalization of the Gromov–Lawson M >0 M >0 surgery process (cf. Schoen–Yau’s [SY79a]) from ∞ (M) to k (M). Theorem 1.5. Let M be any closed n-dimensional manifold, n 3, k (0, ), M >0 ′ ≥ ∈ ∞ and g k (M). Any manifold M which can be obtained from M by performing ∈ M >0 ′ surgeries in codimension 3 also carries a metric in k (M ) that coincides with g away from the surgery region.≥ 2This is sometimes referred to as having positive Yamabe invariant in the literature. 4 CHAOLIANDCHRISTOSMANTOULIDIS Such a surgery was carried out by B¨ar–Dahl in [BD03, Theorem 3.1], who in fact also dealt with higher eigenvalues. Since we need to embed the codimension-3 surgery into a continuous Ricci-flow-with-surgery, we need a slightly more refined statement. See Theorem D.1, Corollary D.3. 1 M ≥0 1.3. Bartnik mass of apparent horizons. When k = 2 , k (M) relates to the space of apparent horizons (quasilocal black hole boundaries, diffeomorphic to M) of time-symmetric (n+1)-dimensional initial data to the Einstein equations satisfying the dominant energy condition. To see why, let us recall the setting. For general n-dimensional closed orientable (M n,g), the apparent horizon Bartnik mass is defined as3 m (M,g,H = 0) = inf m (M, g):(M, g) (M,g,H = 0) , (1.4) B { ADM ∈EB } where B(M,g,H = 0) is the set of complete, connected, asymptotically flat (M, g) with nonnegativeE scalar curvature, no closed interior minimal hypersurfaces, and minimal (H = 0) boundary isometric to (M,g). The quantity mADM (M, g) is the ADM energy/mass of the time-symmetric initial data set [ADM60, ADM59]. Such (M, g) are initial data sets for solutions of Einstein’s equations with the dominant energy condition. The Bartnik mass is difficult to compute because: the set B(M,g,H = 0) is difficult to control (minimal surfaces abound • in RiemannianE manifolds [IMN18, MNS19, CM20b, GG19, Zho20, Son18]), and the quantity m (M, g) is difficult to compute. • ADM Nonetheless, there exists a very nontrivial lower bound by Bray [Bra01] and Bray– Lee’s [BL09] Riemannian Penrose Inequality (“RPI”), a refinement of the Schoen– Yau Positive Energy Theorem [SY79c, Sch89] (cf. Witten [Wit81]), which says: (M, g) (M,g,H =0) = m (M, g) 1 (σ−1 vol (M))(n−1)/n, (1.5) ∈EB ⇒ ADM ≥ 2 n g and thus obviously m (M,g,H = 0) 1 (σ−1 vol (M))(n−1)/n, B ≥ 2 n g n where σn is the volume of the standard round S , when 2 n 6; see also Huisken– Ilmanen [HI01] in case n = 2 and M is connected.
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