Marcel Schmidt

Abstract

Cheeger energies provide a way to define a “Dirichlet integral” and, derived from it, a notion of “Laplacian” on metric measure spaces. These objects are central tools in modern analysis on non-smooth geometries, especially on spaces that arise as limits of sequences of smooth spaces subject to lower curvature bounds.

In this talk, we discuss basic properties and phenomena related to Cheeger energies, including:

• the definition of the Cheeger energy and the role of geometric assumptions in making the theory useful;
• properties of the heat flow generated by the Cheeger energy, and why this heat flow may be nonlinear;
• the relation to relaxed upper gradients and the geometric conditions that ensure a robust first-order calculus.