The heat flow in nonlinear Hodge theory under general growth

Abstract

We study the nonlinear Hodge system $d\omega ={\delta }_{\rho }\omega =0$ for an exterior form ω on a compact oriented Riemannian manifold M. Its solutions are called ρ-harmonic forms. Here the ρ-codifferential of ω is defined as ${\delta }_{\rho }\omega ={\rho }^{-1}\delta \left(\rho \omega \right)$ with a given positive function $\rho =\rho \left(\right|\omega \left|\right)$.

We evolve a given closed form ${\omega }_0$ by the nonlinear heat flow system $\dot{\omega }=d{\delta }_{\rho }\omega$ for a time dependent exterior form $\omega (x,t)$ on M. Under an ellipticity condition on the function ρ, we show that the nonlinear heat flow system with initial condition $\omega (\cdot ,0)={\omega }_0$ has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ${\omega }_0$. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative.