Almost isoclinic Lagrangian submanifolds

Abstract

We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian ${\mathrm{Lag}}^{+}(n)$ of ${\mathbb{C}}^n$, an intrinsic higher-dimensional analog of a natural convex region in ${\mathrm{Lag}}^{+}(2)\simeq {\mathbb{S}}^1\times {\mathbb{S}}^2$. 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 $\Lambda$ on this region and prove that $\log \Lambda$ 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 $\Lambda$ must be a Lagrangian $n$-plane.

36 pages, 4 figures. The main Bernstein theorem has been improved and does not require an asymptotic assumption in this version