On the maximal rank problem for the complex homogeneous Monge–Ampère equation

Abstract

We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Ampère Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank.

Complex Brunn–minkowski Theory and Positivity of Vector Bundles

“…In our terminology in section 2, this means that c(φ) = 0, and in general, c(φ) can be interpreted as the geodesic curvature of the curve. Since honest geodesics are scarce, see [42], [27], [59], [60] for problems encountered in solving the homogenous complex Monge-Ampère equation involved, we will also use 'generalized geodesics'. These are metrics on L as above that are only assumed to be locally bounded, such that i∂ ∂φ ≥ 0 and (i∂ ∂φ) n+1 = 0 in the sense of pluipotential theory ( [4]).…”

An Obstacle for Higher Regularity of Geodesics in the Space of Kähler Potentials

“…However these assumptions may not be satisfied in general as shown by [16][8] [21] and [14]. So the computations are formal.…”

“…This is shown to be wrong for general solutions. In [21], when the Riemann surface is a disc, a solution Φ to Problem 1.2 is constructed, which satisfies…”

The Dirichlet problem for the complex homogeneous Monge-Ampère equation

“…Proof. We shall prove the slightly weaker statement that for each τ ∈ D × the current ω + dd cΦ (·, τ ) vanishes on some non-empty open subset of P 1 (and the reader is referred to [104] for the proof of the full statement). As Ω 1 = P 1 \ γ, and γ passes through the point w = 0, we see Ω 1 is a simply connected proper subset of C z .…”

“…Final Bibliographical Remarks. The final example is new, but the first three are taken from [103], [102] and [104] respectively, and the reader will find stronger statements in these cited papers. For instance in [103] one can find an area form whose Hele-Shaw flow develops self-tangency along any given finite collection of smooth points and nonselfintersecting curve segments.…”