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Abstract:We address the problem of existence and uniqueness of a Leviflat hypersurface M in C n with prescribed compact boundary S for n ≥ 3. The situation for n ≥ 3 differs sharply from the well studied case n = 2. We first establish necessary conditions on S at both complex and CR points, needed for the existence of M . All CR points have to be nonminimal and all complex points have to be "flat". Then, adding a positivity condition at complex points, which is similar to the ellipticity for n = 2 and excluding the possibility of S to contain complex (n − 2)-dimensional submanifolds, we obtain a solution M to the above problem as a projection of a possibly singular Levi-flat hypersurface in R × C n . It turns out that S has to be a topological sphere with two complex points and with compact CR orbits, also topological spheres, serving as boundaries of the (possibly singular) complex leaves of M . There are no more global assumptions on S like being contained in the boundary of a strongly pseudoconvex domain, as it was in case n = 2. Furthermore, we show in our situation that any other Levi-flat hypersurface with boundary S must coincide with the constructed solution.

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