Characterizations of monotonicity of vector fields on metric measure space

Bang-Xian Han

Published 2017-10-22Version 1

There are two main aims of this paper. The first aim is to characterize the convexity of functions on metric measure space, so that we could link the existence of some special K-convex functions to the particular metric structure of the space, which is a new approach to deal with some rigidity theorems such as `splitting theorem' and `volume cone implies metric cone theorem'. The second aim is to study the convexity/monotonicity of non-smooth vector fields on metric measure space. We introduce the notion of K-monotonicity which is stable under measured Gromov-Hausdorff convergence, then characterize the K-monotone vector fields in several equivalent ways.