Asymptotic Invariants of Smooth Manifolds.

摘要

The purpose of the paper is the definition and study of new homotopical invariants of smooth manifolds which are called 'absolute asymptotic volumes'. They are defined by means of volume growth functions v(t;q,g) and have to do with different problems of topology and geometry of manifolds, e.g. such as a number of closed geodesics, systolic constants, minimum surfaces. Chapter headings include: The Homotopy Invariance and Properties of Absolute Volumes; The Extremal Problem of the Convex Geometry, Universal Constant Construction; The Total Volume and Total Mass of Polyvectors in a Linear Space; The Homological Absolute Volume of Smooth Manifolds; Manifolds with Almost Abelian Fundamental Group; Closed Geodesics; Homological and Systolic Stiffness; Minimum Surfaces in Jacobian and the Absolute Homological Volume.