On the fill-in of nonnegative scalar curvature metrics
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Mathematische Annalen
On the fill-in of nonnegative scalar curvature metrics Yuguang Shi1
· Wenlong Wang2 · Guodong Wei1,3 · Jintian Zhu1
Received: 29 August 2019 / Revised: 29 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data (Σ, γ , H ). We prove that given a metric γ on Sn−1 (3 ≤ n ≤ 7), (Sn−1 , γ , H ) admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if γ is a positive scalar curvature (PSC) metric isotopic to the standard metric on Sn−1 , then the much weaker condition that the total mean curvature Sn−1 H dμγ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the θ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins. Mathematics Subject Classification 53C20 · 83C99
Communicated by F.C. Marques. Yuguang Shi, Guodong Wei and Jintian Zhu are partially supported by NSFC 11671015 and 11731001. Wenlong Wang is partially supported by NSFC 11671015, 11701326 and BX201700007.
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Yuguang Shi [email protected] Wenlong Wang [email protected] Guodong Wei [email protected] Jintian Zhu [email protected]
1
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
2
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
3
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082, People’s Republic of China
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Y. Shi et al.
1 Introduction A triple of (generalized) Bartnik data (Σ n−1 , γ , H ) consists of an orientable closed null-cobordant Riemannian manifold (Σ n−1 , γ ) and a given smooth function H on it. One basic problem in Riemannian geometry is to study (see [11]): under what conditions does the Bartnik data (Σ n−1 , γ , H ) admit a fill-in with scalar curvature bounded below by a given constant? That is, there is a compact Riemannian manifold (Ω n , g) with boundary that has scalar curvature Rg ≥ σ > −∞ and an isometry X : (Σ n−1 , γ ) → (∂Ω n , g|∂Ω n ) so that H = Hg ◦ X on Σ, where Hg is the mean curvature of ∂Ω n in (Ω n , g) with respect to the outward normal. Note that the above definition of fill-in is different from that in [15]. In our case, if (Ω n , g, X ) is a fill-in of (Σ n−1 , γ , H ), we have ∂Ω n = X (Σ n−1 ) rather than X (Σ n−1 ) ⊂ ∂Ω n and ∂Ω n \X (Σ n−1 ) is allowed to be a closed (possibly disconnected) minimal hypersurface of (Ω n , g). By the gluing arguments in [13,25], it is easy to see our definition is more restrictive than that in [15]. On the other hand, in [27] (also see an improvement in [29]), L.-F. Tam and the first author proved the positivity of B
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