The heat flow in nonlinear Hodge theory

We evolve a given closed form ω0 by the nonlinear heat flow systemω̇=dδρω for a time-dependent exterior form ω(x,t) on M. This system is the differential of the normalized gradient flow u̇=δρω for E(ω) with ω=ω0+du. Under a technical assumption on the function 2ρ′(Q)Q/ρ(Q), we show that the nonlinear heat flow system ω̇=dδρω, with initial condition ω(·,0)=ω0, has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ω0. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative.