In this paper, we prove for \(n\le 7\) 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 \(n\le 7\) 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 (Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65(1), 45–76 (1979).) and (Schoen, R., Yau, S-T.: Positive scalar curvature and minimal hypersurface singularities, Surveys in differential geometry 2019. Differential geometry, Calabi-Yau theory, and general relativity. Part 2, Surv. Differ. Geom., vol. 24, Int. Press, Boston, MA, pp. 441–480 (2022).). Finally, we investigate Gromov’s fill-in problem and bound the total mean curvature for nonnegative scalar curvature fill-ins of flat 2-tori (an optimal bound is obtained for product 2-tori). This confirms the validity of Mantoulidis-Miao’s definition of generalized Brown–York mass in (Mantoulidis, C., Miao, P.: Total mean curvature, scalar curvature, and a variational analog of Brown-York mass. Comm. Math. Phys. 352(2), 703–718 (2017).) for flat 2-tori.
