Area and Gauss–Bonnet inequalities with scalar curvature

The Gauss–Bonnet theorem states for any compact surface (S, g) that the quantity Q^S₆₁ (S) =ₒ Sc (S, s) \, ds+ ₒmean. curv. (S, b) \, db-4 (S) vanishes identically. Let (X, g) be a compact Riemannian manifold of dimension n 3 with smooth boundary, associated with a continuous map f\!=\! (f₁, , f₍-₂) X\!\! 0, 1^{n-2}, where Lipf₈ d₈^-1 for positive constants d₈. For a universal constant C₍ (d₈) depending only on d₈ and n, we show that there is a compact surface homologous to the f -pullback of a generic point such that each component S of satisfies Q₆₁^X (S) C₍ (d₈) (S), where Q^X₆₁ (S) =ₒ Sc (X, s) \, ds+ ₒmean. curv. (X, b) \, db-4 (S). As corollaries, if X has “large positive” scalar curvature, we prove in a variety of cases that if X “spreads” in (n-2) directions “ distance-wise ”, then it cannot much “spread” in the remaining 2-directions “ area-wise ”.