Research on Monge-Ampère equations part6(Machine Learning)
- Monge-Ampère equation on compact Hermitian manifolds(arXiv)
Author : Genglong Lin, Yinji Li, Xiangyu Zhou
Abstract : Given a cohomology (1,1)-class {β} of compact Hermitian manifold (X,ω) such that there exists a bounded potential in {β}, we show that degenerate complex Monge-Ampère equation (β+ddcφ)n=μ has a unique solution in the full mass class E(X,β), where μ is any probability measure on X which does not charge pluripolar subset. We also study other Monge-Ampère types equations which correspond to λ>0 and λ<0. As a preparation to the λ<0 case, we give a general answer to an open problem about the Lelong number which was surveyed by Dinew-Guedj-Zeriahi \cite[Problem 36]{DGZ16}. Moreover, we obtain more general results on singular space and of the equations with prescribed singularity if the model potential has small unbounded locus. These results generalize much recent work of \cite{EGZ09}\cite{BBGZ13}\cite{DNL18}\cite{LWZ23} etc.
2. On singular strictly convex solutions to the Monge-Ampère equation(arXiv)
Author : Guido De Philippis, Riccardo Tione
Abstract : We show the existence of a strictly convex function u:B1→R with associated Monge-Ampère measure represented by a function f with 0<f<1 a.e. whose Hessian has a singular part. This extends the work [13] and answers an open question of [14,Sec. 6.2(1)].
