几乎等斜拉格朗日子流形
Almost isoclinic Lagrangian submanifolds
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中文总结 AI 辅助
该研究在定向拉格朗日格拉斯曼流形中引入几乎等斜区域，构造典范正函数$\u039b$并证明其对数的凹性，为极小拉格朗日量等提供相关公式，最终证明了刚性与伯恩斯坦型结果。
中文摘要 AI 辅助
我们在复欧几里得空间$\boldsymbol{\rm Lag}^+(n)$的定向拉格朗日格拉斯曼流形中引入几乎等斜区域，这是$\boldsymbol{\rm Lag}^+(2) \backsimeq \boldsymbol{\rm S}^1\times \boldsymbol{\rm S}^2$中自然凸区域的高维内在类比。若拉格朗日子流形的高斯映射取值于该区域，则称其为几乎等斜的，这扩展了切平面特征角保持一致接近的图条件。我们在该区域构造了一个典范正函数$\boldsymbol{\rm \u039b}$，并证明$\boldsymbol{\rm log \u039b}$关于不变格拉斯曼度量是凹的。该性质为极小拉格朗日量和拉格朗日平均曲率流提供了次调和性与单调性公式。作为应用，我们证明了刚性与伯恩斯坦型结果，包括：具有$\boldsymbol{\rm \u039b}$正下界的完备连通几乎等斜极小拉格朗日必为拉格朗日$n$-平面。
英文摘要
We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian ${\rm Lag}^+(n)$ of $\mathbb{C}^n$, an intrinsic higher-dimensional analog of a natural convex region in ${\rm Lag}^+(2) \simeq \mathbb S^1\times \mathbb S^2$. A Lagrangian submanifold is called almost isoclinic if its Gauss map takes values in this region, extending the graphical condition that the characteristic angles of the tangent plane remain uniformly close. We construct a canonical positive function $Λ$ on this region and prove that $\log Λ$ is concave with respect to the invariant Grassmannian metric. This property yields subharmonicity and monotonicity formulas for minimal Lagrangians and Lagrangian mean curvature flow. As applications, we prove rigidity and Bernstein-type results, including that a complete connected almost isoclinic minimal Lagrangian with a positive lower bound for $Λ$ must be a Lagrangian $n$-plane.
