The Christoffel-minkowski Problem Ii: Weingarten Curvature Equations
نویسندگان
چکیده
In [12], we treated the Christoffel-Minkowski problem as a convexity problem of a spherical hessian equation on S via Gauss map. In this paper, we study the curvature equations of radial graphs over Sn. Our main concern is the existence of hypersurface with prescribed Weingarten curvature on radial directions. For a compact hypersurface M in Rn+1, the kth Weingarten curvature at x ∈ M is defined as Wk(x) = Sk(κ1(x), κ2(x), · · · , κn(x)) where κ = (κ1, κ2, ..., κn) the principal curvatures of M , and Sk is the kth elementary symmetry function. If the surface is starshaped about the origin, it follows that the surface can be parametrized as a graph over Sn: X = ρ(x)x, x ∈ S, (1.1)
منابع مشابه
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...
متن کاملThe Christoffel-minkowski Problem Iii: Existence Problem for Curvature Measures
Curvature measure is one of the basic notion in the theory of convex bodies. Together with surface area measures, they play fundamental roles in the study of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. Let Ω is a bounded convex body in R with C2 boundary M , the corresponding curvature measures and surface area measures of ...
متن کاملA Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz–Minkowski space
In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz–Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, p...
متن کاملHypersurfaces of Prescribed Curvature Measure
We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped (n − k)-convex bodies with prescribed k-th curvature measures (k > 0) has been a longstanding problem. This is settled in this paper through the establishment of a crucial C a priori estimate for the corresponding curvature equation on S.
متن کاملThe structural equations of surfaces associated with CPN−1 sigma models on a Minkowski space
The objective of this paper is to construct and investigate smooth orientable surfaces in RN −1 by analytical methods. The structural equations of surfaces in connection with CPN−1 sigma models on Minkowski spacetime are studied in detail. This is carried out using a moving frame adapted to the surface immersed in su(N) algebra. The first and second fundamental forms of the surface as well as t...
متن کامل