Starshaped compact hypersurfaces in warped product manifolds. II: A class of Hessian type equations (Q6649699)

Let \(\sigma_k:\mathbb R^n \rightarrow \mathbb R\) denote the \(k\)-th elementary symmetric polynomial \N\[\N\sigma_k (x_1, \dots, x_n) = \sum_{1\leq x_{i_1} <\dots<x_{i_k} \leq n}x_{i_1}\dots x_{i_k}. \N\]\NFor a symmetric matrix \(A\), we denote by \(\lambda (A) = (\lambda_1 \dots \lambda_n)\) its eigenvalues and the notation \(\sigma_k (A)\) shall be interpreted as the quantity \(\sigma_k (\lambda_1 \dots \lambda_n)\). When \(A = D^2 u\) is the Hessian matrix of some \(u\in C^2 (\Omega) \cap C(\overline{\Omega})\), then \(\sigma_1 (D^2 u)=\Delta u\) is the Laplacian and \(\sigma_n (D^2 u)=\det (D^2 u)\) is the Monge-Ampère operator. When \(A= (h_{ij})\) is the second fundamental form of a hypersurface \(\Sigma\), then its eigenvalues are the principal curvatures \(\kappa [ \Sigma] = (\kappa_1 \dots \kappa_n)\); in this case \(\sigma_1 ( \kappa [ \Sigma]) = \sum_{i=1}^n \kappa_i\) is the mean curvature, \(\sigma_2 ( \kappa [ \Sigma]) = \sum_{i < j} \kappa_i \kappa_j\) is the scalar curvature, and \(\sigma_n ( \kappa [ \Sigma]) = \kappa_i \dots \kappa_n\) is the Gauss curvature. In general, the quantities \(\sigma_k (D^2 u)\) and \(\sigma_k ( \kappa [ \Sigma])\) are called the \(k\)-Hessian and the \(k\)-curvature, respectively. \N\NEver since the seminal work of Caffarelli-Nirenberg-Spruck [\textit{L. Caffarelli} et al., Acta Math. 155, 261--301 (1985; Zbl 0654.35031)], [\textit{L. Caffarelli} et al., in: Current topics in partial differential equations, Pap. dedic. S. Mizohata Occas. 60th Birthday, 1--26 (1986; Zbl 0672.35027)], [\textit{L. Caffarelli} et al., Commun. Pure Appl. Math. 41, No. 1, 47--70 (1988; Zbl 0672.35028)], there has been a vast literature studying partial differential equations which involve the \(\sigma_k\) operators. Indeed, besides the above simple examples, many problems in analysis and geometry often arise as some type of the \(\sigma_k\) equation. \N\NIn this paper, the authors prove the existence of one particular class of starshaped compact hypersurfaces, by deriving global curvature estimates for such hypersurfaces. This generalizes a recent result on this problem from the Euclidean space to Riemannian warped products. Moreover, they show that interior second order a priori estimates for admissible solutions to the associated fully nonlinear elliptic partial differential equations can be readily established by similar arguments.