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Incompressible hypersurface, positive scalar curvature and positive mass theorem
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Incompressible hypersurface, positive scalar curvature and positive mass theorem
abstract
In this paper, we prove for $n\leq 7$ that if a differentiable $n$-manifold contains a relatively incompressible essential hypersurface in some class $\mathcal C_{deg}$, then it admits no complete metric with positive scalar curvature. Based on this result, we show for $n\leq 7$ that surgeries between orientable $n$-manifolds and $n$-torus along incompressible sub-torus with codimension no less than $2$ still preserve the obstruction for complete metrics with positive scalar curvature. As an application, we establish positive mass theorem with incompressible conditions for asymptotically flat/conical manifolds with flat fiber $F$ (including ALF and ALG manifolds), which can be viewed as a generalization of the classical positive mass theorem from \cite{SY79PMT} and \cite{SY2017}. Finally, we investigate Gromov's fill-in problem and bound the total mean curvature for nonnegative scalar curvature fill-ins of flat $2$-toruses (an optimal bound is obtained for product $2$-toruses). This confirms the validity of Mantoulidis-Miao's definition of generalized Brown-York mass in \cite{MM2017} for flat $2$-toruses.
Forward citations
Cited by 4 Pith papers
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A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds
The mass of toric ALE or ALF 4-manifolds with nonnegative scalar curvature is at least the mass of the corresponding toric gravitational instanton plus a term from its conical defects, with equality only when the mani...
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A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds
The mass of any toric ALE/ALF 4-manifold with nonnegative scalar curvature is at least the mass of its corresponding toric gravitational instanton, corrected by conical angle defects, and equality holds only for the i...
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Mass Lower Bounds for Asymptotically Locally Flat Manifolds
For ALF 4-manifolds with an almost free circle symmetry and nonnegative scalar curvature, mass is nonnegative and at least (ℓ/16) times the degree of the asymptotic circle bundle; in the AF case, homology-trivial coor...
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Charged parallel spinors and applications to mass--charge inequalities
Equality in the spin mass–charge inequality holds precisely when the manifold carries a charged parallel spinor and is isometric to extremal Reissner–Nordström (connected boundary or one cylindrical end).
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