Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature

Abstract

We give a simple proof of an extension of the existence results of Ricci flow of Giesen and Topping (2010, 2011) [15,20], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi’s existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of Perelman [7] on Ricci flow. We will also give a simple proof of a special case of the existence theorem of Topping (2010) [16] without using the existence theorem of Shi (1989) [9].