Area and Gauss-Bonnet Inequalities with Scalar Curvature

Let X be an n-dimensional Riemannian manifold with "large positive" scalar curvature. In this paper, we prove in a variety of cases that if X "spreads" in (n-2) directions "distance-wise", then it can't much "spread" in the remaining 2-directions "area-wise". Here is a geometrically transparent example of what we plan prove in this regard that illustrates the idea. Let g be a Riemannin metric on X= S^2×ℝ^n-2, for which the submanifolds are mutually orthogonal at all intersection points x=(s,y)∈ X=ℝ_s^n-2∩ S^2_y. (An instance of this is g=g(s,y)=ϕ(s,y)^2ds^2+ψ(s,y)^2dy^2.) Let the Riemannian metric on ℝ_s^n-2 induced from (X,g), that is g|_ℝ_s^n-2, be greater than the Euclidean metric on ℝ_s^n-2 =ℝ^n-2 for all s∈ S^2. (This is interpreted as "large spread" of g in the (n-2) Euclidean directions.) If the scalar curvature of g is strictly greater than that of the unit 2-sphere, Sc(g) ≥ Sc(S^2)+ε=2+ε, ε>0, then, provided n≤ 7, (this, most likely, is unnecessary) there exists a smooth non-contractible spherical surface S⊂ X, such that area(S)更多