A flow method for curvature equations

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.