Home Total mean curvature of the boundary and nonnegative scalar curvature fill-ins
Article
Licensed
Unlicensed Requires Authentication

Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

  • Yuguang Shi ORCID logo EMAIL logo , Wenlong Wang ORCID logo and Guodong Wei ORCID logo
Published/Copyright: January 23, 2022

Abstract

In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, completely solving an open problem due to Gromov (see Question 1.1). Then we introduce a fill-in invariant (see Definition 1.2) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds via this fill-in invariant. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromov’s two conjectures formulated in [M. Gromov, Four lectures on scalar curvature, preprint 2019] (see Conjecture 1.1 and Conjecture 1.2 below).

Award Identifier / Grant number: SQ2020YFA070059

Award Identifier / Grant number: 12001292

Award Identifier / Grant number: 12101619

Award Identifier / Grant number: 11731001

Award Identifier / Grant number: 63201151

Award Identifier / Grant number: 20JCQNJC02100

Award Identifier / Grant number: 202102020743

Funding statement: Yuguang Shi is partially supported by National Key R&D Program of China (SQ2020YFA070059) and NSFC (11731001). Wenlong Wang is partially supported by NSFC (12001292), Fundamental Research Funds for the Central Universities Nankai University (63201151) and Natural Science Foundation of Tianjin (20JCQNJC02100). Guodong Wei is partially supported by NSFC (12101619, 11731001) and Science and Technology Projects of Guangzhou (202102020743).

Acknowledgements

We would like to express our sincere gratitude to Misha Gromov for his interest, comments and suggestions on this work. We would like to thank Luen-Fai Tam and Pengzi Miao for their interest in this work, especially, for Luen-Fai Tam for informing us about [31, Theorem 1.4] and suggesting us to clarify the constant in Theorem 1.2. We are also very grateful to Roman Prosanov for the discussion about Pogorelov’s rigidity theorem [40].

References

[1] A. D. Alexandrov, A. D. Alexandrov selected works. Part II. Intrinsic geometry of convex surfaces, Chapman & Hall/CRC, Boca Raton 2006. Search in Google Scholar

[2] M. T. Anderson and J. Cheeger, Cα-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 (1992), no. 2, 265–281. 10.4310/jdg/1214448075Search in Google Scholar

[3] L. Andersson, M. Cai and G. J. Galloway, Rigidity and positivity of mass for asymptotically hyperbolic manifolds, Ann. Henri Poincaré 9 (2008), no. 1, 1–33. 10.1007/s00023-007-0348-2Search in Google Scholar

[4] R. Arnowitt, S. Deser and C. W. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. (2) 122 (1961), 997–1006. 10.1103/PhysRev.122.997Search in Google Scholar

[5] R. H. Bamler, A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature, Math. Res. Lett. 23 (2016), no. 2, 325–337. 10.4310/MRL.2016.v23.n2.a2Search in Google Scholar

[6] S. Bando, A. Kasue and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313–349. 10.1007/BF01389045Search in Google Scholar

[7] R. Bartnik, Quasi-spherical metrics and prescribed scalar curvature, J. Differential Geom. 37 (1993), no. 1, 31–71. 10.4310/jdg/1214453422Search in Google Scholar

[8] V. Bonini and J. Qing, A positive mass theorem on asymptotically hyperbolic manifolds with corners along a hypersurface, Ann. Henri Poincaré 9 (2008), no. 2, 347–372. 10.1007/s00023-008-0358-8Search in Google Scholar

[9] J. D. Brown and J. W. York, Jr., Quasilocal energy in general relativity, Mathematical aspects of classical field theory, Contemp. Math. 132, American Mathematical Society, Providence, (1992), 129–142. 10.1090/conm/132/1188439Search in Google Scholar

[10] J. D. Brown and J. W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D (3) 47 (1993), no. 4, 1407–1419. 10.1103/PhysRevD.47.1407Search in Google Scholar

[11] D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Grad. Stud. Math. 33, American Mathematical Society, Providence 2001. 10.1090/gsm/033Search in Google Scholar

[12] P. Burkhardt-Guim, Pointwise lower scalar curvature bounds for C0 metrics via regularizing Ricci flow, Geom. Funct. Anal. 29 (2019), no. 6, 1703–1772. 10.1007/s00039-019-00514-3Search in Google Scholar

[13] A. J. Cabrera Pacheco and P. Miao, Higher dimensional black hole initial data with prescribed boundary metric, Math. Res. Lett. 25 (2018), no. 3, 937–956. 10.4310/MRL.2018.v25.n3.a10Search in Google Scholar

[14] P. T. Chruściel and E. Delay, The hyperbolic positive energy theorem, preprint (2019), https://arxiv.org/abs/1901.05263v2. Search in Google Scholar

[15] P. T. Chruściel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), no. 2, 231–264. 10.2140/pjm.2003.212.231Search in Google Scholar

[16] T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), no. 3, 477–501. 10.2307/2951841Search in Google Scholar

[17] X. Dai and L. Ma, Mass under the Ricci flow, Comm. Math. Phys. 274 (2007), no. 1, 65–80. 10.1007/s00220-007-0275-6Search in Google Scholar

[18] D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), no. 1, 157–162. 10.4310/jdg/1214509286Search in Google Scholar

[19] M. Eichmair, L.-H. Huang, D. A. Lee and R. Schoen, The spacetime positive mass theorem in dimensions less than eight, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 1, 83–121. 10.4171/JEMS/584Search in Google Scholar

[20] X.-Q. Fan, Y. Shi and L.-F. Tam, Large-sphere and small-sphere limits of the Brown–York mass, Comm. Anal. Geom. 17 (2009), no. 1, 37–72. 10.4310/CAG.2009.v17.n1.a3Search in Google Scholar

[21] M. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734. 10.1070/IM1969v003n04ABEH000796Search in Google Scholar

[22] M. Gromov, Dirac and Plateau billiards in domains with corners, Cent. Eur. J. Math. 12 (2014), no. 8, 1109–1156. 10.2478/s11533-013-0399-1Search in Google Scholar

[23] M. Gromov, Four lectures on scalar curvature, preprint (2019), https://arxiv.org/abs/1908.10612v3. 10.1142/9789811273223_0001Search in Google Scholar

[24] M. Gromov, Scalar curvature of manifolds with boundaries: Natural questions and artificial constructions, preprint (2019), https://arxiv.org/abs/1811.04311v2. Search in Google Scholar

[25] M. Gromov, Four lectures on scalar curvature, preprint (2021), https://arxiv.org/abs/1908.10612v5. 10.1142/12644Search in Google Scholar

[26] P. Guan and Y. Y. Li, The Weyl problem with nonnegative Gauss curvature, J. Differential Geom. 39 (1994), no. 2, 331–342. 10.4310/jdg/1214454874Search in Google Scholar

[27] J. Hong and C. Zuily, Isometric embedding of the 2-sphere with nonnegative curvature in 𝐑3, Math. Z. 219 (1995), no. 3, 323–334. 10.1007/BF02572368Search in Google Scholar

[28] X. Hu and Y. Shi, NNSC-cobordism of Bartnik data in high dimensions, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 30. 10.3842/SIGMA.2020.030Search in Google Scholar

[29] J. L. Jauregui, Fill-ins of nonnegative scalar curvature, static metrics, and quasi-local mass, Pacific J. Math. 261 (2013), no. 2, 417–444. 10.2140/pjm.2013.261.417Search in Google Scholar

[30] J. L. Jauregui, P. Miao and L.-F. Tam, Extensions and fill-ins with non-negative scalar curvature, Classical Quantum Gravity 30 (2013), no. 19, Article ID 195007. 10.1088/0264-9381/30/19/195007Search in Google Scholar

[31] J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113–134. 10.4310/jdg/1214432678Search in Google Scholar

[32] J. Lohkamp, Scalar curvature and hammocks, Math. Ann. 313 (1999), no. 3, 385–407. 10.1007/s002080050266Search in Google Scholar

[33] C. Mantoulidis and P. Miao, Total mean curvature, scalar curvature, and a variational analog of Brown–York mass, Comm. Math. Phys. 352 (2017), no. 2, 703–718. 10.1007/s00220-016-2767-8Search in Google Scholar

[34] C. Mantoulidis, P. Miao and L.-F. Tam, Capacity, quasi-local mass, and singular fill-ins, J. reine angew. Math. 768 (2020), 55–92. 10.1515/crelle-2019-0040Search in Google Scholar

[35] P. Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, Adv. Theor. Math. Phys. 6 (2002), no. 6, 1163–1182. 10.4310/ATMP.2002.v6.n6.a4Search in Google Scholar

[36] P. Miao, Nonexistence of NNSC fill-ins with large mean curvature, Proc. Amer. Math. Soc. 149 (2021), no. 6, 2705–2709. 10.1090/proc/15400Search in Google Scholar

[37] P. Miao, Y. Shi and L.-F. Tam, On geometric problems related to Brown–York and Liu–Yau quasilocal mass, Comm. Math. Phys. 298 (2010), no. 2, 437–459. 10.1007/s00220-010-1042-7Search in Google Scholar

[38] P. Miao and X. Wang, Boundary effect of Ricci curvature, J. Differential Geom. 103 (2016), no. 1, 59–82. 10.4310/jdg/1460463563Search in Google Scholar

[39] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. 10.1002/cpa.3160060303Search in Google Scholar

[40] A. V. Pogorelov, Extrinsic geometry of convex surfaces, Transl. Math. Monogr. 35, American Mathematical Society, Providence 1973. 10.1090/mmono/035Search in Google Scholar

[41] X. Rong, Convergence and collapsing theorems in Riemannian geometry, Handbook of geometric analysis. No. 2, Adv. Lect. Math. (ALM) 13, International Press, Somerville (2010), 193–299. Search in Google Scholar

[42] L. A. Santaló, Integral geometry and geometric probability, 2nd ed., Cambridge Math. Libr., Cambridge University, Cambridge 2004. 10.1017/CBO9780511617331Search in Google Scholar

[43] R. Schneider, Convex bodies: The Brunn–Minkowski theory, 2nd expanded ed., Encyclopedia Math. Appl. 151, Cambridge University, Cambridge 2014. Search in Google Scholar

[44] R. Schoen and S. T. Yau, Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity, Proc. Natl. Acad. Sci. USA 76 (1979), no. 3, 1024–1025. 10.1073/pnas.76.3.1024Search in Google Scholar PubMed PubMed Central

[45] R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. 10.1007/BF01940959Search in Google Scholar

[46] R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1–3, 159–183. 10.1016/B978-0-12-195255-6.50019-2Search in Google Scholar

[47] R. Schoen and S. T. Yau, The energy and the linear momentum of space-times in general relativity, Comm. Math. Phys. 79 (1981), no. 1, 47–51. 10.1007/BF01208285Search in Google Scholar

[48] R. Schoen and S. T. Yau, Positive scalar curvature and minimal hypersurfaces singularities, preprint (2017), https://arxiv.org/abs/1704.05490. 10.4310/SDG.2019.v24.n1.a10Search in Google Scholar

[49] Y. Shi and L.-F. Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), no. 1, 79–125. 10.4310/jdg/1090425530Search in Google Scholar

[50] Y. Shi and L.-F. Tam, Quasi-spherical metrics and applications, Comm. Math. Phys. 250 (2004), no. 1, 65–80. 10.1007/s00220-004-1118-3Search in Google Scholar

[51] Y. Shi and L.-F. Tam, Rigidity of compact manifolds and positivity of quasi-local mass, Classical Quantum Gravity 24 (2007), no. 9, 2357–2366. 10.1088/0264-9381/24/9/013Search in Google Scholar

[52] Y. Shi and L.-F. Tam, Scalar curvature and singular metrics, Pacific J. Math. 293 (2018), no. 2, 427–470. 10.2140/pjm.2018.293.427Search in Google Scholar

[53] Y. Shi, W. Wang, G. Wei and J. Zhu, On the fill-in of nonnegative scalar curvature metrics, Math. Ann. 379 (2021), no. 1–2, 235–270. 10.1007/s00208-020-02087-1Search in Google Scholar

[54] M. Simon, Deformation of C0 Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074. 10.4310/CAG.2002.v10.n5.a7Search in Google Scholar

[55] M. Taylor, Existence and regularity of isometries, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2415–2423. 10.1090/S0002-9947-06-04090-6Search in Google Scholar

[56] P. Topping, Lectures on the Ricci flow, London Math. Soc. Lecture Note Ser. 325, Cambridge University, Cambridge 2006. 10.1017/CBO9780511721465Search in Google Scholar

[57] P. Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv. 83 (2008), no. 3, 539–546. 10.4171/CMH/135Search in Google Scholar

[58] M.-T. Wang and S.-T. Yau, A generalization of Liu–Yau’s quasi-local mass, Comm. Anal. Geom. 15 (2007), no. 2, 249–282. 10.4310/CAG.2007.v15.n2.a2Search in Google Scholar

[59] X. Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), no. 2, 273–299. 10.4310/jdg/1090348112Search in Google Scholar

[60] E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. 10.1007/BF01208277Search in Google Scholar

Received: 2021-06-13
Published Online: 2022-01-23
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2021-0072/html?lang=en
Scroll to top button