Prescribed curvature measure problem in hyperbolic space
نویسندگان
چکیده
The problem of the prescribed curvature measure is one important problems in differential geometry and nonlinear partial equations. In this paper, we consider hyperbolic space. We obtain existence star-shaped k-convex bodies with (n-k)-th measures (k<n) by establishing crucial C^2 regularity estimates for solutions to corresponding fully PDE
منابع مشابه
Hypersurfaces 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 Weyl Problem with Nonnegative Gauss Curvature in Hyperbolic Space
In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to ndimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.
متن کاملSTARSHAPED COMPACT HYPERSURFACES WITH PRESCRIBED k-TH MEAN CURVATURE IN HYPERBOLIC SPACE
In this paper we consider the problem of finding a star-shaped compact hypersurface with prescribed k-th mean curvature in hyperbolic space. Under some sufficient conditions, we obtain an existence result by establishing a priori estimates and using degree theory arguments.
متن کاملSTARSHAPED COMPACT HYPERSURFACES WITH PRESCRIBED m−TH MEAN CURVATURE IN HYPERBOLIC SPACE
Let S be the unit sphere in the Euclidean space R, and let e be the standard metric on S induced from R. Suppose that (u, ρ) are the spherical coordinates in R, where u ∈ S, ρ ∈ [0,∞). By choosing the smooth function φ(ρ) := sinh ρ on [0,∞) we can define a Riemannian metric h on the set {(u, ρ) : u ∈ S, 0 ≤ ρ < ∞} as follows h = dρ + φ(ρ)e. This gives the space form R(−1) which is the hyperboli...
متن کاملUniqueness of Starshaped Compact Hypersurfaces With Prescribed m-th Mean Curvature in Hyperbolic Space
Let ψ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurfaceM for which ψ, when evaluated onM , coincides with them−th elementary symmetric function of principal curvatures of M for a given m? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980’s. Rec...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Communications on Pure and Applied Mathematics
سال: 2023
ISSN: ['1097-0312', '0010-3640']
DOI: https://doi.org/10.1002/cpa.22160