Closed hypersurfaces with prescribed Weingarten curvature in Riemannian manifolds (Q368554)

Let \((N,\overline{g})\) be an \((n+1)\)-dimensional Riemannian manifold with a global normal Gaussian coordinate system (for details see [\textit{C. Gerhardt}, Curvature problems. Somerville, MA: International Press (2006; Zbl 1131.53001)]) and let \(M\) be a closed manifold. The main result of this paper is the following. Theorem 1: Given \(T_{-}\) and \(T_{+}\) with \(a<T_{-}<T_{+}<b\), suppose that \(M_t\) is a family of totally umbilical hypersurfaces with positive mean curvature for \(t\in (T_{-},T_{+})\). If the curvature function \(f\) satisfies technical conditions \((F_1)-(F_4)\), then there exists an admissible differentiable function \(u:M\rightarrow I\), solution to the equation \(F(A[u])=f(\kappa [u])=\Psi (u(x),x)\), \(x\in M\), whose graph \(\Sigma\) is contained in the interior of certain region \(N_{-}^{+}=\{\Phi (t,x): T_{-}\leq t \leq T_{+}, x\in M\}\), if one of the following conditions is satisfied: (A) \(M_t\) has constant mean curvature for \(t\in (T_{-},T_{+})\) and \(\lambda \Psi +\partial _t \Psi \leq 0\) for \(T_{-}\leq t \leq T_{+}\) ; (B) \(f\) satisfies the condition [FA] and \(\lambda \Psi +\partial _t \Psi +c_f |D\lambda| \leq 0\) for \(T_{-}\leq t \leq T_{+}\), where \(|D\lambda|^2=\sigma ^{ij}\partial _{x_i}\lambda \partial _{x_j}\lambda\). Here, \(F\) is the function defined on \(n\times n\) symmetric matrices by \(F(A)=f(\mu _1,\dots,\mu _n)\), where \(\mu _1,\dots,\mu _n\) are the eigenvalues of \(A\). The main ingredient used in the proof of this result is the technique developed by \textit{W. Sheng} et al. [Duke Math. J. 123, No. 2, 235--264 (2004; Zbl 1174.35378)].