Incompressible hypersurface, positive scalar curvature and positive mass theorem

In this paper, we prove for n ≤ 7 that if a differentiable nmanifold contains a relatively incompressible essential hypersurface in some class Cdeg, then it admits no complete metric with positive scalar curvature. Based on this result, we show for n ≤ 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 [52] and [55]. 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 [39] for flat 2-toruses.

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