Buckley, Stephen M.
Nonpositive curvature and complex analysis.
Five lectures in complex analysis : second Winter School on Complex Analysis and Operator Theory, February 5-9, 2008, University of Sevilla, Sevilla, Spain.
American Mathematical Society, Providence, R.I., pp. 43-83.
We discuss a few of the metrics that are used in complex analysis and
potential theory, including the Poincaré, Carathéodory, Kobayashi, Hilbert, and quasihyperbolic
metrics. An important feature of these metrics is that they are quite often
negatively curved. We discuss what this means and when it occurs, and proceed to
investigate some notions of nonpositive curvature, beginning with constant negative
curvature (e.g. the unit disk with the Poincaré metric), and moving on to CAT(k) and
Gromov hyperbolic spaces. We pay special attention to notions of the boundary at
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