Gradient flows and a Trotter--Kato formula of semi-convex functions on CAT(1)-spaces

TL;DR

This paper extends the theory of gradient flows for semi-convex functions from CAT(0)-spaces to CAT(1)-spaces, introducing new tools and results that deepen understanding of geometric analysis in curved metric spaces.

Contribution

It generalizes gradient flow theory to CAT(1)-spaces using commutativity and semi-convexity, and establishes a Trotter--Kato formula for such functions.

Findings

Convergence of discrete schemes to gradient curves

Establishment of contraction properties and evolution variational inequalities

Development of a Trotter--Kato product formula for semi-convex functions

Abstract

We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savar\'e, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativity with semi-convex squared distance functions. Our approach combining the semi-convexity of the squared distance function with a Riemannian property of the space seems to be of independent interest, and can be compared with Savar\'e's work on the local angle condition under lower curvature bounds. Applications include the convergence of the discrete variational scheme to a unique gradient curve, the contraction property and the evolution variational inequality of the gradient flow, and a Trotter--Kato product formula for pairs of semi-convex functions.