Compact Riemann Surfaces: An Introduction to Contemporary by Jürgen Jost PDF

By Jürgen Jost

Even supposing Riemann surfaces are a time-honoured box, this e-book is novel in its vast viewpoint that systematically explores the relationship with different fields of arithmetic. it may function an creation to modern arithmetic as a complete because it develops heritage fabric from algebraic topology, differential geometry, the calculus of diversifications, elliptic PDE, and algebraic geometry. it truly is certain between textbooks on Riemann surfaces in together with an creation to Teichm?ller idea. The analytic technique is also new because it is predicated at the idea of harmonic maps.

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Def. 2 below), it provides the easiest method of studying the transformation behaviour of the metric. 3 The Laplace-Beltrami operator with respect to the metric >. 2 8z Oz = ;2 (8~2 + 8~2 ), z= x + iy. 1 log>.. Remark. With z = x + iy, we have Thus the metric differs from the Euclidian metric only by the conformal factor >. 2. In particular, the angles with respect to >. 2dzdz are the same as those with respect to the Euclidian metric. 5 A bijective map h : El --. 2 dzdz and (P dwdw respectively, is called an isometry if it preserves angles and arc-lengths.

A metric is most simply described by means of a potential. Since a potential is invariant under coordinate transformations (and hence also under isometries, cf. Def. 2 below), it provides the easiest method of studying the transformation behaviour of the metric. 3 The Laplace-Beltrami operator with respect to the metric >. 2 8z Oz = ;2 (8~2 + 8~2 ), z= x + iy. 1 log>.. Remark. With z = x + iy, we have Thus the metric differs from the Euclidian metric only by the conformal factor >. 2. In particular, the angles with respect to >.

L) constitutes a system of two ordinary differential equations satisfying the assumptions of the Picard-Lindel6f theorem. l Let M be a metric surface with a coordinate chart cp : U ---+ V c C. 2(z) dzdz. Let p E V, v E C. e. l}) "/ : [0, c] ---+ M with ,,/(0) = p 1'(0) = v. 2) o "/ depends smoothly on p and v. We denote this geodesic by ,,/p,v. t) for constant ,\ E R Thus "/p,v(t) t = ,,/p,,\v(-:\) for'\ > 0, t E [O,c]. 3) In particular, "/p,Av is defined on the interval [0, f]. Since "/p,v depends smoothly on vas noted in the lemma and since {v E C : Ilvll~ := ,\2(p) vv = I} is compact, there exists co > 0 with the property that for any v with Ilv lip = 1, "/p,v is defined on the interval [0, co].

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