We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.Comment: 9 pages, submitted on December 10, 200

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C m with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C m with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the Kähler Ricci flow.

We derive a priori interior Hessian estimates for the special Lagrangian equation 2 D 1 in dimension 3.

We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.

We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.Abstract. We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.

We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.

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We derive an a priori C 2,α estimate for solutions of the fully non-linear elliptic equation F(D 2 u) = 0, provided the level set Σ = {M | F(M) = 0} satisfies: (a) Σ ∩ {M | Tr M = t} is strictly convex for all constants t; (b) the angle between the identity matrix I and the normal F ij to Σ is strictly positive on the non-convex part of Σ. Moreover, we do not need any convexity assumption on F in the course of the proof for the two dimensional case, as the classical result indicates.. II ≥ ω(Λ) > 0, (2) (Non-convex part, strict transversality) The angle between I and the normal (F ij) to Σ, (I, F ij) ≥ Θ(Λ) > 0, provided that M ≤ Λ.