Uniform asymptotic expansion on Riemann surfaces (Q2786821)

The Bergman kernel associated to high powers of a line bundle over a Kähler manifold has received much attention since the fundamental works of \textit{G. Tian} [J. Differ. Geom. 32, No. 1, 99--130 (1990; Zbl 0706.53036)], \textit{D. Catlin} [in: Analysis and geometry in several complex variables. Proceedings of the 40th Taniguchi symposium, Katata, Japan, June 23--28, 1997. Boston, MA: Birkhäuser. 1--23 (1999; Zbl 0941.32002)], and \textit{S. Zelditch} [Int. Math. Res. Not. 1998, No. 6, 317--331 (1998; Zbl 0922.58082)].NEWLINENEWLINEThe authors have previously studied uniform upper and lower bounds for the Bergman kernel in the setting of a degenerating family of compact Riemann surfaces of genus at least \(2\) with constant scalar curvature \(-1\), polarized by the canonical bundle [J. Fixed Point Theory Appl. 10, No. 2, 327--338 (2011; Zbl 1253.30054)]. Now they obtain a uniform asymptotic expansion for the Bergman kernel in the same setting. The expression is substantially more complicated than in the case considered by Catlin and Zelditch because the injectivity radius approaches zero as the family approaches a singular limit.NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].