Energy-minimizing maps from manifolds with nonnegative Ricci curvature

TL;DR

This paper establishes lower bounds for the energy of maps from manifolds with nonnegative Ricci curvature into manifolds without conjugate points, characterizing when maps are totally geodesic based on energy minimization.

Contribution

It proves a new lower bound for the energy of maps under nonnegative Ricci curvature and characterizes energy-minimizing maps as totally geodesic, extending previous results to broader classes of manifolds.

Findings

Energy bounds depend on the asymptotic geometry of the target.

Equality in energy bounds characterizes totally geodesic maps.

Results extend to manifolds with universal covers splitting as a product.

Abstract

The energy of any $C^1$ representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger-Gromoll splitting theorem.