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Complex Analysis and Geometry Seminar

"Rigidity theorems for invariant harmonic functions on bounded symmetric domains"

Song-Ying Li , UC Irvine

Location:  HILL 705
Date & time: Friday, 21 April 2017 at 10:30AM -



Time: 10:30 AM
Location: Hill 705
Abstract: Let \(Delta_g\) be the Laplace-Beltrami operator in Bergman metric in the unit ball in \(C^n\). Then the boundary value problem: \[ Delta_g u=0, hbox{ in } B_n; quad u=phi hbox{ on }d B_n \]has a unique solution \[ u(z)=P[phi]=int_{d B_n} {(1-|z|^2)^n over |1-langle z, wrangle|^{2n}} phi(w) dsigma(w) \]It well known that even if \(phiin C^infty(d B_n)\), \(P[phi]\) may not be in \(C^n(overline{B_n})\). A well known theorem of R. Graham says that if \(u\) is invariant harmonic in \(B_n\) and \(C^n(overline{B_n})\), then \(u\) must be pluriharmonic in \(B_n\).

In this talk, I will present a joint work with R-Y. Chen, we try to generalize Graham's theorem to the bounded symmetric domains.


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