Let $\psi$ be a given function defined on a Riemannian space. Under whatconditions does there exist a compact starshaped hypersurface $M$ for which$\psi$, when evaluated on $M$, coincides with the $m-$th elementary symmetricfunction of principal curvatures of $M$ for a given $m$? The correspondingexistence and uniqueness problems in Euclidean space have been investigated byseveral authors in the mid 1980's. Recently, conditions for existence wereestablished in elliptic space and, most recently, for hyperbolic space.However, the uniqueness problem has remained open. In this paper we investigatethe problem of uniqueness in hyperbolic space and show that uniqueness (up to ageometrically trivial transformation) holds under the same conditions underwhich existence was established.
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机译：令$ \ psi $是在黎曼空间上定义的给定函数。在什么条件下，是否存在一个紧凑的星形超曲面$ M $，在给定的$ m $上对$ \ psi $进行评估时，$ \ psi $与给定的$ m $的主曲率的$ m-th个基本对称函数重合？欧几里德空间中相应的存在性和唯一性问题已由1980年代中期的几位作者进行了研究。最近，在椭圆空间中建立了生存条件，而在最近的双曲空间中建立了生存条件。但是，唯一性问题仍然存在。在本文中，我们研究了双曲空间中的唯一性问题，并证明了唯一性（直至年龄上的琐碎变换）在建立存在的相同条件下成立。
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