The heat flow in nonlinear Hodge theory

Abstract

We study the nonlinear Hodge system dω=0 and δ( ρ(| ω| 2) ω)=0 for an exterior form ω on a compact oriented Riemannian manifold M, where ρ( Q) is a given positive function. The solutions are called ρ- harmonic forms. They are the stationary points on cohomology classes of the functional E(ω)= ∫ M e(|ω| 2) dM with e′( Q)= ρ( Q)/2. The ρ-codifferential of a form ω is defined as δ ρ ω= ρ −1 δ( ρω) with ρ= ρ(| ω| 2). We evolve a given closed form ω 0 by the nonlinear heat flow system ω ̇ =dδ ρω for a time-dependent exterior form ω( x, t) on M. This system is the differential of the normalized gradient flow u ̇ =δ ρω for E( ω) with ω= ω 0+ du. Under a technical assumption on the function 2 ρ′( Q) Q/ ρ( Q), we show that the nonlinear heat flow system ω ̇ =dδ ρω , with initial condition ω(·,0)= ω 0, has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ω 0. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative.