A flow method for curvature equations

Abstract

We consider a general curvature equation $F(\kappa)=G(X,\nu(X))$, where $\kappa$ is the principal curvature of the hypersurface $M$ with position vector $X$. It includes the classical prescribed curvature measures problem and area measures problem. However, Guan-Ren-Wang \cite{GRW} proved that the $C^2$ estimate fails usually for general function $F$. Thus, in this paper, we pose some additional conditions of $G$ to get existence results by a suitably designed parabolic flow. In particular, if $F=\sigma_{k}^\frac{1}{k}$ for $\forall 1\le k\le n-1$, the existence result has been derived in the famous work \cite{GLL} with $G=\psi(\frac{X}{|X|})\langle X,\nu\rangle^{\frac1k}{|X|^{-\frac{n+1}{k}}}$. This result will be generalized to $G=\psi(\frac{X}{|X|})\langle X,\nu\rangle^\frac{{1-p}}{k}|X|^\frac{{q-k-1}}{k}$ with $p>q$ for arbitrary $k$ by a suitable auxiliary function. The uniqueness of the solutions in some cases is also studied.