An Eigenvalue Problem for Prescribed Curvature Equations

Abstract

Abstract We study an eigenvalue problem for prescribed $\sigma _{k}$-curvature equations of star-shaped, $k$-convex, closed hypersurfaces. We establish the existence of a unique eigenvalue and its associated hypersurface, which is also unique, provided that the given data is even. Moreover, we show that the hypersurface must be strictly convex. A crucial aspect of our proof involves deriving uniform estimates in $p$ for $L_{p}$-type prescribed curvature equations.