Area and Gauss–Bonnet inequalities with scalar curvature

The Gauss–Bonnet theorem states for any compact surface (S,g) that the quantity QSGB(S)=∫S Sc(S,s)\,ds+∫∂ Smean.curv.(∂ S,b)\,db-4πχ(S) vanishes identically. Let (X,g) be a compact Riemannian manifold of dimension n≥ 3 with smooth boundary, associated with a continuous map {f\!=\!(f_1,…,fₙ₋₂) X\!→\! [0,1]ⁿ⁻²} , where Lipfᵢ≤ dᵢ⁻¹ for positive constants dᵢ . For a universal constant Cₙ(dᵢ) depending only on dᵢ and n , we show that there is a compact surface Σ homologous to the f -pullback of a generic point such that each component S of Σ satisfies QGBX(S)≤ Cₙ(dᵢ)(S) , where QXGB(S)=∫S Sc(X,s)\,ds+∫∂ Smean.curv.(∂ X,b)\,db-4πχ(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 ”.