A parabolic flow approach to generalized Gauduchon metrics

Abstract

We give a parabolic proof of the Fu-Wang-Wu theorem for semilinear elliptic equations associated with generalized Gauduchon metrics. Fixing a Hermitian metric $ω$, we introduce a flow $\partial_t v-Δv=ψ(\vert\nabla v\vert^2)+\langle B,dv\rangle-c(t)-f(x)$ for the conformal factor, whose stationary points solve the semilinear elliptic equation of Fu-Wang-Wu. Under the growth condition $\liminf_{t\to+\infty}ψ(t)/t^μ>0$ for some $μ>\tfrac{1}{2}$, we prove that the flow exists uniquely for all time and converges; the limit solves the elliptic equation and is unique up to addition of a constant. Our convergence argument does not rely on a gradient-flow structure or a monotone energy functional. In particular, this recovers, by a flow method rather than the continuity method, the existence and uniqueness theorem of Fu-Wang-Wu on $k$-th generalized Gauduchon metrics, providing a new parabolic approach to their elliptic theory.