Solutions of some Monge-Amp\`ere equations with isolated and line singularities

摘要

In this paper, we study existence, regularity, classification, andasymptotical behaviors of solutions of some Monge-Amp\`ere equations withisolated and line singularities. We classify all solutions of $\det \nabla^2u=1$ in $\R^n$ with one puncture point. This can be applied to characterizeellipsoids, in the same spirit of Serrin's overdetermined problem for theLaplace operator. In the case of having $k$ non-removable singular points for$k>1$, modulo affine equivalence the set of all generalized solutions can beidentified as an explicit orbifold of finite dimension. We also establishexistence of global solutions with general singular sets, regularityproperties, and optimal estimates of the second order derivatives ofgeneralized solutions near the singularity consisting of a point or a straightline. The geometric motivation comes from singular semi-flat Calabi-Yaumetrics.