非均匀对角膨胀下欧氏极小超曲面的刚性
Rigidity of Euclidean Minimal Hypersurfaces under Nonuniform Diagonal Dilations
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- Chosun University（朝鲜大学）
- Korea Rural Economic Institute（韩国农村经济研究院）
- Gyeongsang National University（庆尚国立大学）
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中文总结 AI 辅助
该研究针对非均匀对角膨胀下的欧氏极小超曲面，通过对和非共振条件与降维论证证明其第二基本形式恒为零，还给出了相关刚性判据与生产函数应用。
中文摘要 AI 辅助
设n≥3，Dₜ=diag(t^g₁,…,t^gₙ)为正对角膨胀族，我们研究其对角像为极小的连通嵌入欧氏超曲面。水平集极小算子可分解为以对和gᵢ+gⱼ为索引的系数；在对和非共振条件下，仅在C(n,2)个不同膨胀参数处满足极小性，就迫使所有对系数消失。随后通过降维论证，无需对法向量的坐标分量作任何假设，即可证明第二基本形式恒为零，由此得到一种仿射刻画。各维数下的重复权重螺旋例及共振二次锥表明，曲率抵消可在真正的非均匀族中存在；其应用给出有限输出水平的刚性判据，以及带极小等值面的加权齐次生产函数的显式表示。
英文摘要
Let $n\ge3$ and $D_t=\operatorname{diag}(t^{g_1},\ldots,t^{g_n})$ be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums $g_i+g_j$. Under pair-sum nonresonance, minimality at only $\binom n2$ distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.
