The fractional Liouville equation in dimension 1 Geometry, Compactness and quantization (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

We discuss some recent results about the fractional Liouville equation in dimension 1 and related questions. These results, resting on a geometric interpretation in terms of holomorphic maps, on the study of geodesics in a conformal metric and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis-Merle and Li-Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates under weak assumptions which are not known in dimension 2.