On a twisted conical Kähler–Ricci flow

Abstract

In this paper, we discuss diameter bound and Gromov–Hausdorff convergence of a twisted conical Kähler–Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical Kähler metric with possibly unbounded scalar curvature, the conical Kähler–Ricci flow will instantly have bounded scalar curvature for $$t>0$$ , and the bound is of the form $$\frac{C}{t}$$ . Several key results will be obtained by direct arguments on the conical equation without passing to a smooth approximation. In the last section, we present several remarks on a twisted Kähler–Ricci flow and its convergence.