Title: Isoparametric foliations and the Pompeiu problem Show Chinese title

Title: 等参叶状结构与庞佩乌问题

Abstract: A bounded domain $\Omega$ in a Riemannian manifold $M$ is said to have the Pompeiu property if the only continuous function which integrates to zero on $\Omega$ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $M$ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $M$ under which the level domains of an isoparametric function fail the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out. Show Chinese abstract

Abstract: 一个黎曼流形$M$中的有界区域$\Omega$被称为具有庞佩乌性质，如果在$\Omega$及其所有合同像上积分都为零的连续函数只能是零函数。 在某些方面，庞佩乌性质可以看作是一个超定问题，因为它与施费弗问题有关。 众所周知，每个欧几里得球体都不满足庞佩乌性质，而球面球体在几乎所有半径下都具有该性质（翁格的怪异定理）。 在本文中，我们讨论当$M$是紧致且具有等参叶状结构时的庞佩乌性质。 特别是，我们在$M$上拉普拉斯算子的谱上确定了精确条件，使得等参函数的水平区域不满足庞佩乌性质。 当环境流形是单位球时进行了具体的计算，并得出了一些结论。 此外，还对翁格的怪异定理及其推广进行了详细讨论。