\(k\)-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds (Q6611127)

Let \((M,g')\) be a compact Riemannian manifold and \(I\) be an open interval in \(\mathbb R\). The warped product manifold \(\overline{M}= I\times_{\lambda} M\) is endowed with the metric \[ {\bar{g}}^2 = dr^2 + \lambda^2 (r) g' ,\] where \(\lambda: I\rightarrow \mathbb R^+\) is a positive \(C^2\) function. Then \(\Sigma\) can be parametrized as a radial graph over \(M\). Specifically there exists a differentiable function \(r: M\rightarrow I\) such that the graph of \(\Sigma\) can be represented by \[ \Sigma =\{X(u)=(r(u), u) : u\in M\}. \] \N\NIn this paper, the authors consider the following prescribed Weingarten curvature equation in a warped product manifold \(\overline{M}\) \[ \sigma_k(\kappa (V))= \psi (V, \nu(V)), \, \forall \, V \in \Sigma, \] where \(V=\lambda \frac{\partial}{\partial r}\) is the position vector field of a hypersurface \(\Sigma\) in \(\overline{M}\), \(\sigma_k\) is the \(k\)-th elementary symmetric function, \(\nu(V)\) is the outward unit normal vector field along the hypersurface \(\Sigma\) and \(\kappa (V)=(\kappa_1, \kappa_2, \dots, \kappa_n)\) are the principle curvatures of the hypersurface \(\Sigma\) at \(V\).\N\NBased on a conjecture proposed by \textit{C. Ren} and \textit {Z. Wang} [''Notes on the curvature estimates for Hessian equations'', Preprint, \url{arXiv:200314234}, which is valid for \(k\geq n-2\), the authors derive curvature estimates for the equation \( \sigma_k(\kappa (V))= \psi (V, \nu(V))\). Furthermore, they also obtain an existence result for a star-shaped compact hypersurface \(\Sigma\) satisfying the above equation under some sufficient conditions via degree theory.