[Paper Review] A Bernstein theorem for special Lagrangian graphs

Jürgen Jost, Yuanlong Xin|ArXiv.org|Jan 16, 2001

Geometric Analysis and Curvature Flows7 references4 citations

TL;DR

This paper establishes a Bernstein-type theorem for entire special Lagrangian graphs in $$\mathbb{C}^n$$ by proving that any such graph with bounded slope must be an affine $n$-plane, extending previous results by removing quantitative curvature bounds and leveraging harmonic maps into the Lagrangian Grassmannian. The key contribution is a Liouville-type theorem for harmonic Gauss maps under convexity and boundedness conditions, leading to the conclusion that the graph is planar if the Hessian satisfies a uniform bound on the determinant of $I + (\text{Hess } F)^2$. The result holds for all $n \geq 2$ without requiring convexity or growth restrictions beyond bounded slope.

ABSTRACT

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.

Motivation & Objective

• To establish a sharp Bernstein-type theorem for entire special Lagrangian graphs in $\mathbb{C}^n$ without requiring quantitative curvature bounds.
• To extend known results on minimal graphs to the special Lagrangian setting by exploiting the Lagrangian condition and harmonic Gauss maps.
• To prove that bounded slope and convexity imply that the graph is an affine $n$-plane, even without growth or curvature restrictions.
• To analyze the asymptotic behavior of such graphs at infinity via tangent cones and Legendrian links in $S^{2n-1}$.

Proposed method

• Use the Ruh-Vilms theorem to show that the Gauss map of a minimal Lagrangian submanifold is harmonic.
• Restrict the Gauss map to the Lagrangian Grassmannian $LG_n$, which is totally geodesic in the Grassmannian $G_{n,n}$, preserving harmonicity.
• Construct geodesically convex neighborhoods in $LG_n$ around a reference $n$-plane using normal coordinates and the condition $\langle P, P_0 \rangle \geq \delta > 0$, ensuring the image of the Gauss map lies in such a set.
• Apply a Liouville-type theorem for harmonic maps into geodesically convex sets to conclude the Gauss map is constant, implying the graph is affine.
• Analyze the tangent cone at infinity of the graph to obtain a special Lagrangian cone whose link is a minimal Legendrian submanifold in $S^{2n-1}$.
• Use the condition $\Delta_F \leq \beta$ to derive a uniform lower bound $\langle P, P_0 \rangle \geq \beta^{-1}$ on the angle between the Gauss map and a fixed plane, enabling application of Theorem 2 on Legendrian submanifolds.

Experimental results

Research questions

• RQ1 Under what conditions is an entire special Lagrangian graph in $\mathbb{C}^n$ necessarily an affine $n$-plane?
• RQ2 Can the bounded slope condition alone, without curvature or growth bounds, force a special Lagrangian graph to be planar?
• RQ3 How does the geometry of the Lagrangian Grassmannian $LG_n$ enable stronger Liouville-type theorems for harmonic Gauss maps compared to the full Grassmannian?
• RQ4 What role does the Legendrian link of the tangent cone at infinity play in classifying special Lagrangian cones?
• RQ5 Can the condition $\Delta_F \leq \beta$ be used to uniformly bound the angle between the Gauss map and a fixed $n$-plane, leading to rigidity?

Key findings

• Any smooth function $F: \mathbb{R}^n \to \mathbb{R}$ whose graph of $\nabla F$ is a special Lagrangian submanifold in $\mathbb{C}^n$ must be a quadratic polynomial if $F$ is convex and $\Delta_F \leq \beta < \infty$.
• The Gauss map of such a special Lagrangian graph is harmonic and takes values in the Lagrangian Grassmannian $LG_n$, which is totally geodesic in $G_{n,n}$, enabling stronger geometric control.
• The existence of a geodesically convex neighborhood in $LG_n$ where the Gauss map lies allows the application of a Liouville-type theorem, forcing the Gauss map to be constant.
• The tangent cone at infinity of the graph is a special Lagrangian cone whose link is a compact minimal Legendrian submanifold in $S^{2n-1}$, and under the angle condition $\langle P, P_0 \rangle \geq \delta$, this link lies in a totally geodesic subsphere.
• The condition $\Delta_F \leq \beta$ implies $\langle P, P_0 \rangle \geq \beta^{-1}$, which, combined with Theorem 2, forces the Legendrian link to be totally geodesic, hence the cone is an $n$-plane.
• By Allard’s regularity theorem, the original graph $M$ must be an affine $n$-plane, so $F$ is a quadratic polynomial.