对称极小曲面方程

IF 1.2 2区 数学 Q1 MATHEMATICS
K. Fouladgar, L. Simon
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引用次数: 3

摘要

并且,在几何上,A(u)表示S(u)的面积泛函;也就是说,A(u)是(n+m−1)维Hausdorff测度Hn+m−1(S(u))。这是清楚的,因为A(u)的被积函数√1+|Du|2 um−1是映射(x,ω)∈Ω×Sm−1 7的雅可比矩阵→ (x,u(x)ω)∈Ω×R,并且该映射是对称图S(u)的局部坐标表示。由于1.1 1.1表达了u相对于A是静止的这一事实,我们看到S(u)相对于光滑对称变形是静止的,因此根据众所周知的原理(例如,参见[Law72]),相对于所有变形都是静止的。(这里的后一个原理只是以下事实的自然推广:如果光滑超曲面∑关于轴旋转对称,并且如果∑相对于光滑旋转对称紧支撑扰动是静止的,那么∑是最小的——即相对于所有光滑紧支撑扰动(无论对称与否)是静止的。)因此,光滑子流形S(u)是Ω×(R\{0})中多重数为1的变倍的平稳子流形,因此是Ω×。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The symmetric minimal surface equation
and, geometrically, A (u) represents the area functional for S(u); that is, A (u) is the (n+m−1)-dimensional Hausdorff measure H n+m−1(S(u)). This is clear because the integrand √ 1+|Du|2 um−1 for A (u) is the Jacobian of the map (x,ω) ∈Ω×Sm−1 7→ (x, u(x)ω)∈Ω×R, and this map is a local coordinate representation for the symmetric graph S(u). Since 1.1 1.1 expresses the fact that u is stationary with respect to A , we see that S(u) is stationary with respect to smooth symmetric deformations, and hence stationary with respect to all deformations by a well-known principle (see e.g. [Law72]). (The latter principle here is just the natural generalization of the fact that if a smooth hypersurface Σ is rotationally symmetric about an axis and if Σ is stationary with respect to smooth rotationally symmetric compactly supported perturbations, then Σ is minimal—i.e. stationary with respect to all smooth compactly supported perturbations whether symmetric or not.) Thus the smooth submanifold S(u) is stationary as a multiplicity 1 varifold in Ω× (R \ {0}) and hence is a smooth minimal submanifold of Ω× (R \{0}) as claimed.
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来源期刊
CiteScore
2.10
自引率
0.00%
发文量
52
审稿时长
4.5 months
期刊介绍: Information not localized
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