Convex hypersurfaces of prescribed curvatures

For a smooth strictly convex closed hypersurface Σ in R, the Gauss map n : Σ → S is a diffeomorphism. A fundamental question in classical differential geometry concerns how much one can recover through the inverse Gauss map when some information is prescribed on S ([27]). This question has attracted much attention for more than a hundred years. The most notable example is probably the Minkowski problem of finding a closed convex hypersurface in R whose Gauss curvature is prescribed as a positive function defined on S. This problem has been solved due to the work of Minkowski [18], Alexandrov [1], Lewy [17], Nirenberg [19], Pogorelov [21], [22], Cheng-Yau [6] and others. In particular, the analytic approach of Nirenberg, Pogorelov and Cheng-Yau to the problem has inspired significant development of the theory of Monge-Ampère equations. Besides the Gauss curvature, there are other important Weingarten curvature functions such as, for example, the mean and scalar curvatures. In the 1950s, A. D. Alexandrov [2] and S.-s. Chern [8], [9] raised questions regarding prescribing Weingarten curvatures. So far, a large part of the problem has not received much consideration. Apart from the Gauss curvature case (the Minkowski problem), very little is known except a uniqueness result for the case n = 2 (see [2] and [13]). In this paper, we initiate an investigation of problems in this direction. Specifically, we consider the problem of finding closed, strictly convex hypersurfaces in R whose Weingarten curvatures is prescribed as a function defined on S in terms of the inverse Gauss map. We first recall the definition of Weingarten curvatures for hypersurfaces. Let Sk(λ1, . . . , λn) be the kth elementary symmetric function normalized so that