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| 摘要: |
| 本文考虑欧氏空间中一种余一维的高维旋转曲面，通过发展出一种全新的复合映射、维数分解与分块矩阵递推法，我们系统性地研究了同它的面积和曲率有关的一系列问题.当母函数是多元函数时，这种高维旋转曲面的概念尚属首次提出.我们给出了这种高维旋转曲面的面积公式以及它的一些简单应用.我们发现：在任一直径方向上，单位球面的面积分布和低一维单位球体的体积分布完全相同，并且当维数趋于无穷时它们的密度函数的极限都是狄拉克函数.通过研究相应面积泛函的变分问题，我们得到了所谓的极小旋转曲面方程.我们证明了：满足极小旋转曲面方程的母函数对应的旋转曲面的平均曲率等于零.这种极小旋转曲面方程推广了传统的极小曲面方程，并且为非参数极小曲面理论提供了新的更一般的研究框架；通过计算径向对称解对应的常微分方程，我们研究了它的一些简单的特解.我们也简单讨论了相应的预定平均曲率和预定高斯曲率问题. |
| 关键词: 高维旋转曲面 极小旋转曲面方程 单位球面的面积分布 平均曲率 极小曲面方程 |
| DOI： |
| 分类号:O176.1;O175.29;O186.1;O172 |
| 基金项目: |
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| HIGH-DIMENSIONAL SURFACE OF REVOLUTION |
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LI Ying,LI Zhi-su
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| Abstract: |
| In this paper, we consider a co-dimensional one high-dimensional surface of revolution in Euclidean space. By developing a new kind of function composition, dimensional decomposition and block matrix recursion argument, we study systematically a series of problems related to the area and curvature of this new kind of surface. While the generating function is multi-variable, the concept of this high-dimensional surface of revolution is proposed for the first time. We give the area formula of the high-dimensional surface of revolution and some of its simple applications. We find that, along with the direction of any diameter, the area distribution of the unit sphere is exactly as the same as the volume distribution of the one-dimensional lower unit sphere, and the limit of their density functions is the Dirac function when the dimension tends to infinity. By studying the variational problem of the corresponding area functional, we obtain the so-called minimal surface of revolution equation. We prove that the mean curvature of a surface of revolution corresponding to the generating function satisfying the minimal surface of revolution equation is equal to zero. The minimal surface of revolution equation is a generalization of the traditional minimal surface equation, which provides a new and more general research framework for the theory of the nonparametric minimal surface; by calculating the ordinary differential equation corresponding to the radially symmetric solution, we study some of its simple special solutions. We also briefly discuss the corresponding prescribed mean curvature and prescribed Gaussian curvature problems. |
| Key words: high-dimensional surface of revolution minimal surface of revolution equation the area distribution of the unit sphere mean curvature minimal surface equations |
