Local calibration of mass and systolic geometry

Abstract

We prove the simultaneous (k,n-k)-systolic freedom, for a pair of adjacent integers k smaller than n/2, of a simply connected n-manifold X. Our construction, related to recent results of I. Babenko, is concentrated in a neighborhood of suitable k-dimensional submanifolds of X. We employ calibration by differential forms supported in such neighborhoods, to provide lower bounds for the (n-k)-systoles. Meanwhile, the k-systoles are controlled from below by the monotonicity formula combined with the bounded geometry of the construction in a neighborhood of suitable (n-k+1)-dimensional submanifolds, in spite of the vanishing of the global injectivity radius. The construction is geometric, with the algebraic topology ingredient reduced to Poincare duality and Thom's theorem on representing multiples of homology classes by submanifolds. The present result is different from the proof, in collaboration with A. Suciu, and relying on rational homotopy theory, of the k-systolic freedom of X. Our results concerning systolic freedom contrast with the existence of stable systolic inequalities, studied in joint work with V. Bangert.