Hilbert's fourth problem: the constant curvature case

Speaker：Benling Li (Ningbo University)

Time：2026-6-9 10:00

Location：Conference Room S204 at Experiment Building at Haiyun Campus

Abstract:

Hilbert's fourth problem asks for the characterization of metric geometries in which straight line segments are shortest paths. Its regular case is to classify projectively flat Finsler metrics. In this talk, we discuss recent progress on this problem within the framework of constant curvature — a setting where the global structure has long remained a subtle and challenging topic. Instead of presenting only final results, the focus is on the ideas and developments that have led to a better global understanding. We explain how explicit distance formulas emerge, describe the classification achieved in the non-positive curvature case, and discuss why in the positive curvature case the metric completion must be a sphere. An unexpected link to the nonlinearity of Sobolev spaces is also highlighted, together with several new examples of exotic metrics defined on evolving domains.