Download Geometry and Spectra of Compact Riemann Surfaces (Modern by Peter Buser PDF

By Peter Buser

This monograph is a self-contained creation to the geometry of Riemann Surfaces of continuous curvature –1 and their size and eigenvalue spectra. It makes a speciality of topics: the geometric conception of compact Riemann surfaces of genus more than one, and the connection of the Laplace operator with the geometry of such surfaces.The first a part of the ebook is written in textbook shape on the graduate point, with few standards except historical past in both differential geometry or advanced Riemann floor concept. It starts with an account of the Fenchel–Nielsen method of Teichmüller area. Hyperbolic trigonometry and Bers’ partition theorem (with a brand new evidence which yields specific bounds) are proven to be easy yet robust instruments during this context. the second one a part of the ebook is a self-contained advent to the spectrum of the Laplacian in response to head equations. The strategy selected yields an easy facts that compact Riemann surfaces have an identical eigenvalues if and provided that they've got an analogous size spectrum. Later chapters take care of contemporary advancements on isospectrality, Sunada’s development, a simplified evidence of Wolpert’s theorem, and an estimate of the variety of pairwise isospectral non-isometric examples which relies in basic terms on genus.Research employees and graduate scholars attracted to compact Riemann surfaces will locate right here a couple of invaluable instruments and insights to use to their investigations.

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