The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider the prescribed curvature measure problem in the hyperbolic space. We obtain the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures by establishing crucial C² regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.

中文翻译：

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双曲空间中的规定曲率测度问题

规定曲率测度问题是微分几何和非线性偏微分方程中的重要问题之一。在本文中，我们考虑双曲空间中的规定曲率测量问题。我们得到了具有指定 (nk) 次曲率测量的星形 k-凸体的存在性通过关键地建立C ²正则估计来求解双曲空间中相应的完全非线性 PDE。