A Prescribed Gauss-kronecker Curvature Problem on the Product of Unit Spheres
نویسنده
چکیده
Prescribed Gauss-Kronecker curvature problems are widely studied in the literature. Famous among them is the Minkowski problem. It was studied by H. Minkowski, A.D. Alexandrov, H. Lewy, A.V. Pogorelov, L. Nirenberg and at last solved by S.Y. Cheng and S.T. Yau [CY]. After that, V.I.Oliker [O] researched the arbitrary hypersurface with prescribed Gauss curvature in Euclidean space. On the other hand, L.A. Caffarelli, L. Nirenberg, and J. Spruck studied the boundary-value problem of prescribedWeingarten curvature of graphs over some Euclidean domain in [CNS2], [CNS3], [CNS4]. Then B. Guan and J. Spruke [GS] studied the boundary-value problem in the case of hypersurfaces that can be represented as a radial graph over some domain on some unit sphere. But on the product unit spheres, the similar problem has not been studied systematically. The present paper tries to ask and partly solve a problem of this kind. Let S ⊂ R, S ⊂ R, and S ⊂ R ⊕ R are three unit spheres. ~γ, ~ ρ are position vectors of S, S respectively, and u is a smooth function defined on S×S. Consider a hypersurfaceM ⊂ S defined by a natural embedding ~ X
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