On the Geodesics in the space of K\"ahler metrics with prescribed singularities

TL;DR

This paper extends the existence of weak geodesics in the space of Kähler metrics to cases with singularities and semi-definite metrics, building on recent estimates and previous theorems.

Contribution

It introduces new methods to prove the existence of weak $C^{1,1}$ geodesics between singular Kähler potentials, broadening the scope of prior results.

Findings

Existence of weak $C^{1,1}$ geodesics with singular endpoints

Extension of previous theorems to semi-definite metrics

Application of new estimates by W. He

Abstract

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 K\"ahler potentials to the case where the metrics on the end points may have singularities on some analytic set and may be positive semi-definite.