An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds

TL;DR

This paper proves a new geometric inequality linking the Dirichlet energy and bienergy of smooth maps between Riemannian manifolds, revealing a connection between Ricci curvature and higher-order energies with rigidity conditions.

Contribution

It establishes the first known inequality relating Dirichlet energy and bienergy, involving Ricci curvature and providing rigidity results for equality cases.

Findings

Inequality: E2(f) ≥ Ric_min * E1(f)

Equality holds iff f is totally geodesic with constant rank

Applications to maps into Hadamard manifolds

Abstract

We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected Riemannian manifold whose Ricci curvature has global minimum $\operatorname{Ric}_{\min}$, and that the target manifold $(\overline{M},\overline{g})$ has non-positive sectional curvature along $f(M)$. We prove that \[ E_2(f) \ge \operatorname{Ric}_{\min}\, E_1(f). \] We further analyze the equality case and obtain rigidity results: equality holds if and only if $f$ is totally geodesic and of constant rank. Applications to maps into Hadamard manifolds are also presented. To the best of our knowledge, this is the first geometric inequality directly relating the Dirichlet energy and the bienergy of smooth maps. This result establishes a direct connection between the Ricci curvature of the domain and higher-order variational energies.