Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions

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.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.

$(1,1)$ forms with specified Lagrangian phase: <i>a priori</i> estimates and algebraic obstructions

“…The main difficulty in the proof of Theorem 1.1 is the C 2 estimate which is rather delicate owing to the lack of concavity. In the real case, a priori second order estimates for graphical solutions of the special Lagrangian equation with constant and critical phase are proved by Wang-Yuan [50]. By contrast, the complex setting studied here introduces several new negative terms into the estimate, which together with the non-constant phase, further complicate the analysis.…”

“…Again, by specifying Arg as a map from C → R (rather than S 1 ), so that the argument of the constant function 1 is zero, we defineΘ to be Arg(Z [ω] ). Following [26] and [50], we say that an angle is supercritical if it is larger than (n − 2) π 2 , and hypercritical if it is larger than (n − 1) π 2 . For further discussion and background we refer to reader to [26].…”

“…Written in terms of the eigenvalues of the relative endomorphism Λ j k = α jl ω kl , equation (1.1) can be written as [26] (1. Equation (1.2) is the natural generalization to compact Kähler manifolds of the special Lagrangian equation with potential introduced by Harvey-Lawson [24] and since studied extensively; see, for instance, [4,37,38,50,51,52,57,58] and the references therein. The third author, with Leung and Zaslow [31], showed that when Ω = c 1 (L) for a holomorphic line bundle L → X, and X is a torus fibration over a torus, solutions of equation (1.2) are related via the Fourier-Mukai transform to special Lagrangian sections of the dual torus fibration.…”

Gradient Bounds for Almost Complex Special Lagrangian Equation with Supercritical Phase

“…After that, the special Lagrangian equation with supercritical phase has also been studied extensively in the past few years. For instance, Warren-Yuan [26] considered the interior gradient estimates, and the interior second order estimates were obtained by Wang-Yuan [27]. For special Lagrangian equations with more general phases, one can refer a serious works of Harvey-Lawson [11,12,14] et al and references therein.…”

The Neumann Problem of Complex Special Lagrangian Equations with Supercritical Phase

“…For the special Lagrangian equations with supercritical phase, Yuan obtained the interior C 1 estimate with Warren in [29] and the interior C 2 estimate with Wang in [28]. Recently Collins-Picard-Wu [8] obtained the existence theorem of the Dirichlet problem.…”