On the regularity of solutions to a generalization of the Minkowski problem

Abstract

The Minkowski Problem concerns the existence, uniqueness, and regularity of closed convex hypersurfaces whose Gauss curvature (as a function of the outer normals) is preassigned. Major contributions to this problem were made by Minkowski [28], [29], Aleksandrov [2], [4], Lewy [23], [24], Nirenberg [30], Calabi [9], Pogorelov [34], [35], and Cheng and Yau [10]. Variants of the Minkowski Problem were presented by Gluck [16] and Singer [41], The survey of Gluck [17] still serves as an excellent introduction to the problem. In this article we consider a generalization of the Minkowski Problem. We first recall the analytic formulation of the classical Minkowski Problem. Suppose u = (u , , u n ~x) are smooth local coordinates on the standard unit sphere, S n ~, in Euclidean -space R n , and e = e tj du du j is the first fundamental form of S n ~ . The Einstein convention on summation (over repeated lower and upper indices) is presumed everywhere (with Latin indices running from 1 to n-l). Let ^ denote the Christoffel symbols of the second kind for the metric e. For h e C 2 (S n ~), let where d h k and define the operator V dh du k N by N(h) = = d u h - u' det(V,.,A + det( y d k h, d 2 h du'du J ' ) '