螺旋极小乘积的几何与动力学
The Geometry and Dynamics of Spiral Minimal Products
- Tsinghua University（清华大学）
- Capital Normal University（首都师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究由球形全实浸入与S³中轮廓构成的螺旋极小乘积，证明其极小性条件、轮廓流的可积性，推导指数下界并构造特殊勒让德乘积与德洛内型极小拉格朗日浸入等相关几何对象。
AI中文摘要：
我们研究螺旋乘积 $G_\gamma(t,x,y)=(z_1(t)f_1(x),z_2(t)f_2(y))$，其由球形 $\mathscr{C}$-全实浸入 $f_i:M_i^{k_i}\to S^{2n_i+1}$（满足 $k_1+k_2>0$）与 $S^3$ 中的轮廓 $\gamma=(z_1,z_2)$ 构成。该乘积为极小的充要条件是：各因子均为极小，且轮廓 $\gamma$ 在重新参数化后为 $\bar g=|z_1|^{2k_1}|z_2|^{2k_2}g_{S^3}$ 正定区域内的测地线。轮廓流是刘维尔可积的。在双螺旋参数域的每个正则双转向分量上，全胞相位映射为实解析映射，且存在开稠密的满秩轨迹；因此，任意大本原阶的普通闭合轮廓均稠密。在固定正则非零动量下，精确的劳斯完备化将轮廓黑塞矩阵约化为标量斯特姆型加上两个非负平方项。对于紧致极小输入与本原阶为 $m_\gamma$ 的普通闭合轮廓，$G_\gamma$ 满足 $\operatorname{Ind}(G_\gamma)\geq\operatorname{Ind}(f_1)+\operatorname{Ind}(f_2)+2m_\gamma-3$。在接触动量层级，因子适配选择可从任意给定的一对紧致连通嵌入特殊勒让德流形，生成任意足够大素数闭合阶的紧致嵌入特殊勒让德乘积；其第二基本形式一致有界，而体积与法莫尔斯指数随阶至少线性增长。实球形极小嵌入可得到类似族，应用于有限和乐水平提升时，可得到复射影空间中的德洛内型极小拉格朗日浸入。对于紧致嵌入输入的典范提升与正则普通闭合接触轮廓，本原球形商为嵌入；其霍普夫商为嵌入当且仅当约化相对环绕数为1，且满足加性哈密顿-指数界。
英文摘要：
We study the spiral product $G_γ(t,x,y)=(z_1(t)f_1(x),z_2(t)f_2(y))$, which couples two spherical $\mathscr C$-totally real immersed factors through a profile curve $γ=(z_1,z_2)\subset\mathbb S^3$. Its minimality is governed by a weighted-geodesic system: $G_γ$ is minimal precisely when both factors are minimal and $γ$ is an unparametrized geodesic of $|z_1|^{2k_1}|z_2|^{2k_2}g_{\mathbb S^3}$. This flow is Liouville integrable and, when $k_1+k_2>0$, its phase map has an open dense full-rank locus. Consequently, closed profiles of arbitrarily large primitive order occur densely. For compact minimal factors, Routh reduction and Sturm oscillation give $\text{Ind}(G_γ)\geq \text{Ind}(f_1)+\text{Ind}(f_2)+2m_γ-3$ for a closed oscillatory profile of primitive closing order $m_γ$. On the contact level, a factor-adapted choice of profiles produces, from any prescribed pair of compact connected embedded special Legendrians, embedded special Legendrian products of every sufficiently large prime closing order. Finally, canonical finite horizontal lifts and Hopf projection give Delaunay-type minimal Lagrangians in complex projective spaces. For compact embedded inputs and ordinarily closed profiles, the primitive spherical quotient is embedded, while its projective quotient is embedded exactly when the reduced relative winding is one.
