A flow method for curvature equations

We consider a general curvature equation $F\left(k\right)=G(X,\nu \left(X\right))$, where $k$ 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 et al. (2015) 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 $\forall 1⩽k⩽n-1$, the existence result has been derived in the famous work Guan et al. (2012) with $G=\psi \left(\frac{X}{\left|X\right|}\right){〈X,\nu 〉}^{\frac{1}{k}}{\left|X\right|}^{-\frac{n+1}{k}}$. This result will be generalized to $G=\psi \left(\frac{X}{\left|X\right|}\right){〈X,\nu 〉}^{\frac{1-p}{k}}{\left|X\right|}^{\frac{q-k-1}{k}}$ with $p>q$ for arbitrary $k$ by a suitable auxiliary function. The uniqueness of the solutions in some cases is also studied.