Prescribing Gaussian curvature on S2

Abstract

for all v E H 1' 2. This implies that the first non-zero eigenvalue 21 of the Laplacian of the metric Ke2"ds 2 satisfies 21~>2. While the estimate of Hersch [7] says ).1~<2 with equality if and only if the metric Ke2Uds2 has constant curvature one. This coupled with the assumption that K, u satisfy (1. I) is equivalent to K---constant.