Comments on the Green’s function of a planar domain
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We study several quantities associated to the Green’s function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and \(L^2\)-cohomology of the capacity metric and critical points of the Green’s function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green’s function and related objects.
KeywordsGreen’s function Critical points Capactity metric Geodesics Curvature
Mathematics Subject Classification30C40 30F45 31A15 31C15 32F45 32Q05 32Q15 32Q45 34B27 53C22 53C23
The first named author was supported by the DST-INSPIRE Grant IFA-13 MA-21. The last named author was supported by the DST Swarnajayanti Fellowship 2009–2010 and a UGC–CAS Grant.
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Conflict of interest
The authors declare that they have no competing interests.
- 14.Minda, D.: Inequalities for the Hyperbolic Metric and Applications to Geometric Function Theory, Complex Analysis, I (College Park, Md., 1985–86). Lecture Notes in Mathematics, 1275th edn, pp. 235–252. Springer, Berlin (1987)Google Scholar