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 $G=\psi (\frac{X}{|X|})⟨X,\nu {⟩}^{\frac{1}{k}}|X{|}^{-\frac{n+1}{k}}$. This result will be generalized to $G=\psi (\frac{X}{|X|})⟨X,\nu {⟩}^{\frac{1-p}{k}}|X{|}^{\frac{q-k-1}{k}}$ with $p>q$ for arbitrary $k$ by a suitable auxiliary function. The uniqueness of the solutions in some cases is also studied.