We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate $L_p$-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the $L_p$-Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli et al. (2007). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures.

我们使用没有全局项的曲率流来寻找简单连接的Riemannian空间形式和Lorentz de de Sitter空间中的一类椭圆规定曲率方程的严格凸的类空解，其中规定的函数可能取决于位置和法线向量。特别是在欧几里得空间中，我们解决了一类规定的曲率测度问题${\mathrm{大号}}_p$-Aleksandrov和对偶Minkowski问题，以及它们的对应问题，即 ${\mathrm{大号}}_p$-Christoffel-Minkowski类型的问题。在某些情况下，除了正性外，我们不对各向异性施加任何条件，而在其他情况下，由于Caffarelli等人，我们的条件类似于恒秩定理/凸原理。（2007）。我们的方法不依赖于单调熵函数，它适合于处理不具有变结构的曲率问题。