Preserving Positive Intermediate Curvature

Abstract

AbstractConsider a compact manifold N (with or without boundary) of dimension n. Positive m-intermediate curvature interpolates between positive Ricci curvature ($$m = 1$$ m = 1 ) and positive scalar curvature ($$m = n-1$$ m = n - 1 ), and it is obstructed on partial tori $$N^n = M^{n-m} \times \mathbb {T}^m$$ N n = M n - m × T m . Given Riemannian metrics $$g, {\bar{g}}$$ g , g ¯ on $$(N, \partial N)$$ ( N , ∂ N ) with positive m-intermediate curvature and m-positive difference $$h_g - h_{{\bar{g}}}$$ h g - h g ¯ of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive m-intermediate curvature interpolating between g and $${\bar{g}}$$ g ¯ . Moreover, we apply this result to prove a non-existence result for partial torical bands with positive m-intermediate curvature and strictly m-convex boundaries.