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).
Funding source: National Key R&D Program of China
Award Identifier / Grant number: SQ2020YFA070059
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12001292
Award Identifier / Grant number: 12101619
Award Identifier / Grant number: 11731001
Award Identifier / Grant number: 63201151
Funding source: Natural Science Foundation of Tianjin
Award Identifier / Grant number: 20JCQNJC02100
Funding source: Science and Technology Projects of Guangzhou
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].
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Monodromy of elliptic curve convolution, seven-point sheaves of G2 type, and motives of Beauville type
- Long time behavior of discrete volume preserving mean curvature flows
- Morse quasiflats I
- Numerical equivalence of ℝ-divisors and Shioda–Tate formula for arithmetic varieties
- Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups
- The inertial Jacquet–Langlands correspondence
- Total mean curvature of the boundary and nonnegative scalar curvature fill-ins
- The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups
Artikel in diesem Heft
- Frontmatter
- Monodromy of elliptic curve convolution, seven-point sheaves of G2 type, and motives of Beauville type
- Long time behavior of discrete volume preserving mean curvature flows
- Morse quasiflats I
- Numerical equivalence of ℝ-divisors and Shioda–Tate formula for arithmetic varieties
- Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups
- The inertial Jacquet–Langlands correspondence
- Total mean curvature of the boundary and nonnegative scalar curvature fill-ins
- The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups