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By Michael Spivak

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One way to look at the metric is that it assigns a length to ∂/∂r and to ∂/∂θ at each point of the annulus. Consider the set M of points where the Bergman length of ∂/∂θ is minimal. Such points exist just because the Bergman metric blows up at the boundary of the domain. What geometric properties will the set M have? , the rotations are conformal self-maps of A). Thus M is a union of circles centered at the origin. And M is certainly a closed set by the continuity of the metric. The set has no interior because the Bergman metric ρ is given by a real analytic function (it is the second derivative of the logarithm of the real analytic Bergman kernel), and the zero set of a nontrivial real analytic function can have no interior—see [KRP1].

What kernel results? 15. Let f be a one-to-one, onto map of the Riemann sphere to itself that maps lines and great circles to lines and great circles. Prove that f is either a linear fractional transformation or the conjugate of a linear fractional transformation. 2 Variations on the Theme of the Schwarz Lemma Genesis and Development The Schwarz lemma is one of the simplest results in all of complex function theory. A direct application of the maximum principle, it is merely a statement about the rate of growth of holomorphic functions on the unit disk.

44 2 Variations on the Theme of the Schwarz Lemma Q P Fig. 3. Calculation of geodesic arcs. 17. First notice that since the curve γP,Q is the image under a linear fractional transformation of a part of a line, the trace of γP,Q is, therefore, either a line segment or an arc of a circle. In fact, if P and Q are collinear with 0, then the formula for γP,Q quickly reduces to that for a segment; otherwise γP,Q traces an arc of a Euclidean circle. Which circle is it? Matters are simplest if we let t range over the entire real line and look for the whole circle.

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