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Energy-minimizing maps from manifolds with nonnegative Ricci curvature

James Dibble

TL;DR

The paper establishes a sharp energy lower bound for $C^1$ maps in a fixed homotopy class from a compact manifold with nonnegative Ricci curvature into a complete manifold with no conjugate points, with equality realized only by totally geodesic maps. It constructs a flat semi-Finsler torus via an NNRC diagram and a totally geodesic surjection, linking energy to an asymptotic norm and a length-intersection quantity, and proving a sequence of inequalities that connect energy, length, and intersection. The results extend to domains finitely covered by products through the Cheeger--Gromoll splitting framework and rely on Santalo-type averaging and the asymptotic norm of abelian covers. In the special case that the target has no conjugate points, the bounds become even more explicit, with a canonical totally geodesic map $S$ achieving the bound and providing a precise equality characterization for all minimizers.

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.

Energy-minimizing maps from manifolds with nonnegative Ricci curvature

TL;DR

The paper establishes a sharp energy lower bound for maps in a fixed homotopy class from a compact manifold with nonnegative Ricci curvature into a complete manifold with no conjugate points, with equality realized only by totally geodesic maps. It constructs a flat semi-Finsler torus via an NNRC diagram and a totally geodesic surjection, linking energy to an asymptotic norm and a length-intersection quantity, and proving a sequence of inequalities that connect energy, length, and intersection. The results extend to domains finitely covered by products through the Cheeger--Gromoll splitting framework and rely on Santalo-type averaging and the asymptotic norm of abelian covers. In the special case that the target has no conjugate points, the bounds become even more explicit, with a canonical totally geodesic map achieving the bound and providing a precise equality characterization for all minimizers.

Abstract

The energy of any 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.

Paper Structure

This paper contains 13 sections, 22 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Energy and length
  4. Asymptotic semi-norm of a periodic metric on $\mathbb{Z}^m$
  5. Totally geodesic maps into manifolds with no conjugate points
  6. Commutative diagrams
  7. Beta and gamma functions
  8. Proof of the main theorems
  9. Statement of the main technical result
  10. The semi-Finsler torus and totally geodesic surjection associated to $[F]$
  11. Length and intersection
  12. Main inequalities
  13. Targets with no conjugate points

Key Result

Theorem 1.1

Let $M$ be a compact and connected $C^2$ Riemannian manifold with nonnegative Ricci curvature, $N$ a complete Riemannian manifold with no conjugate points, and $[F]$ a homotopy class of maps from $M$ to $N$. Then, for the flat semi-Finsler manifold $K$ and totally geodesic surjection $S : M \to K$ c

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2: Croke--Fathi
  • Theorem 1.3
  • Lemma 2.1: Croke
  • Theorem 2.2: Santaló's formula
  • Lemma 2.3: Croke
  • Theorem 2.4: Burago--Burago--Ivanov
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 28 more