Disks with Boundaries in Totally Real and Lagrangian Manifolds

1. Introduction. An analytic disk in C n is a map f : D → C n , where D is the closed unit disk, which is holomorphic on the open disk D o and smooth up to the boundary. One can also consider disks that are not smooth at every point of bD. An H ∞-disk is a bounded holomorphic map f : D o → C n. One says that the boundary of f is contained in a compact set X ⊆ C n if the almost-everywhere defined boundary values f * (e iθ) are contained in X for almost all θ. The maximum principle shows that f (D o) is contained in the polynomially convex hull of X. There is also an intermediate notion of a nearly smooth analytic disk (n.s.a.d.); this is an H ∞-disk f : D o → C n that extends to be smooth on all of D except for (at most) a single point of bD (usually taken to be 1 ∈ bD). One says that the boundary of f lies in X if the image by f of bD with the single point deleted is contained in X. The main result of [A] is that if L is an n-dimensional compact totally real manifold in C n , then there exists a nonconstant n.s.a.d. with boundary in L. This was proved by adapting an argument of Gromov [G] that was a cornerstone in his theory of pseudoholomorphic curves. Gromov's theorem, for the special case of C n , is that if L is an n-dimensional compact Lagrangian manifold in C n , then there exists a nonconstant analytic disk with boundary in L. The object here is to obtain more information about these disks. Our motivation comes from the simple example of the standard torus T 3 in C 3 , where there are analytic disks in each of the three " faces " { Our result is that, analogously, contained in each of the faces (appropriately defined) associated with an n-dimensional totally real L in C n , there is a nonconstant n.s.a.d. with boundary in L. Likewise, in the Lagrangian case considered by Gromov, there exists a nonconstant analytic disk in each face. As with T 3 , the faces associated with L are related to the images of L under complex linear maps. Let φ be a complex …