Dept of Mathematics Colloquium - Zheng-Chao Han

About this event

Classification of Isolated Singularities of Solutions to the Yamabe Equation in Higher Dimensions

Prof. Zheng-Chao Han, Rutgers University

Abstract: In this talk, we report on our recent joint work on the classification of isolated singularities of positive solutions to the Yamabe equation in dimensions 3 ≤ n ≤ 29 in the general Riemannian setting, initiated in the flat case by Caffarelli, Gidas, and Spruck in 1989. We prove that every positive solution to the Yamabe equation with an isolated singularity either extends smoothly across the singularity or is asymptotic to a Fowler solution there. The main new ingredient is a proof of the critical upper bound for arbitrary smooth background metrics using an adjoint integral identity. We correct the Euclidean bubble profile arising in the blow-up analysis by terms determined by the Taylor coefficients of the metric, canceling the leading linear geometric errors in the rescaled equation. An adapted adjoint test function relates the remaining geometric and nonlinear terms to a positive boundary flux. We also prove, in every dimension n ≥ 7, that the critical upper bound implies the corresponding lower bound at a non-removable singularity. In a companion paper, we establish sharpness by constructing, in every dimension n ≥ 30, positive solutions with isolated singularities that violate the critical upper bound u(x) ≤ C d g (x, o) −(n−2)/2 near its isolated singularity.

Tea and refreshments available from 3:00-3:25 p.m. in the Assmus Conference Room (CU 212).