arXiv subjects

S. Ali Aleyasin

Publications and source records attributed to S. Ali Aleyasin.

3 recordsLinked to original sources

arXiv2018

Calabi problem for manifolds with edge-cone singularities

In this note, we propose a new approach to solving the Calabi problem on manifolds with edge-cone singularities of prescribed angles along complex hypersurfaces. It is shown how the classical approach of Aubin-Yau in derving {\it a priori} estimates for the complex hessian can be made to work via adopting a \emph{good reference metric} and studying equivalent equations with different referrence metrics. This further allows extending much of the methods used in the smooth setting to the edge setting. These results generalise to the case of multiple hypersufaces with possibly normal crossing.

math.DG↗

arXiv2014

The space of Kähler potentials on an asymptotically locally euclidean Kähler manifold

In this note, we shall prove geodesic convexity of the space of Kähler potentials on an ALE Kähler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness from below of the Mabuchi energy, and give an alternative proof for the uniqueness of scalar-flat metrics in the category of ALE Kähler manifold.

math.DG↗

arXiv2013

On the Geodesics in the space of Kähler metrics with prescribed singularities

Motivated by the results of B. Berndtsson, in this memoir we use the new estimates developed by W. He to extend a theorem of the second author on the existence of weak $C^{1,1}$ geodesics between two smooth non-degenerate Kähler potentials to the case where the metrics on the end points may have singularities on some analytic set and may be positive semi-definite.

math.DG↗