Laplace算子是重要的微分算子，且应用广泛。物理学中描述实物理系统的许多微分方程中都能看到其身影。其他学科如经济学生态学等也需要用Laplace算子对于演化现象进行描述。在几何与分析学中，我们知道黎曼流形上Laplace算子的特征值包含了流形的很多几何信息，故黎曼流形上Laplace算子的特征值问题一直以来也是数学家们研究的重点。

在几何学中的子流形几何领域，等参超曲面作为一类较为规范且相对简单的子流形，为子流形几何的研究提供了大量的例子，这自然也让其备受数学家们的关注。如此，数学家们自然将目光放在了等参领域有关Laplace特征值的问题，希望知道在这样典型的例子上Laplace算子会有什么样的表现。

由于相关问题的重要性，本文将梳理Laplace算子特征值及等参超曲面与焦流形上Laplace算子特征值的研究历史，同时也将介绍一些该领域的最新的进展。希望文章内容能对相关领域的学习与研究提供帮助。

本文中第一章将作为引言导入问题。

第二章将介绍Laplace特征值相关的基本性质与定理，其中包括重要的特征值估计以及逆谱问题，我们将梳理一些重要的结果并给出这一领域的一些最新进展。

第三章将介绍等参超曲面的基本知识，包括球面上等参超曲面的分类问题。

第四章将对等参超曲面与焦流形的特征值问题进行阐释，呈现目前这一领域的已有结果。

Laplace operator is an important differential operator and is widely used. It can be seen in many differential differential equations that describe real physical systems in physics. Other disciplines, such as economics and ecology, also use Laplace operator to describe evolutionary phenomena. In the field of geometry and analysis, we know that the eigenvalues of Laplace operators on Riemannian manifold contain a lot of geometric information of the manifold, so the problem of Laplace operators on Riemannian manifold has always been the focus of mathematicians' research.

In the field of submanifold geometry, as a kind of normal and simple submanifold, isoparametric hypersurface provides a large number of examples for the study of submanifold geometry. So it become focus of mathematicians' research. Therefore, mathematicians study the problem about the eigenvalues of Laplace in the field of isoparmatric field, hoping to know the behavior of the Laplace operator in such a typical example.

Due to the importance of relevant problems, the research history of Laplace operator eigenvalues on isoparametric hypersurfaces and focal manifolds will be reviewed in this paper, and the latest progress in this field will also be introduced.I hope that the content of this article can provide help to the study and research in related fields.

The first chapter of this article will be the introduction of the problem.

In Chapter 2, the fundamental properties and theorems related to the eigenvalues of Laplace will be introduced, including some important eigenvalue estimation and inverse spectrum problem. Then, the latest progress in this field will be presented.

The third chapter will introduce the basic knowledge of isoparametric hypersurfaces, including the classification of spherical isoparametric hypersurfaces.

In Chapter 4, the eigenvalue problems of hypersurfaces and focal submanifolds will be introduced, and the existing results in this field will be presented.