Geometry And Spectra Of Compact Riemann Surfaces

Download or read book Geometry and Spectra of Compact Riemann Surfaces written by Peter Buser and published by Springer Science & Business Media. This book was released on 2010-10-29 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

Download or read book Spectral Theory of Infinite-Area Hyperbolic Surfaces written by David Borthwick and published by Birkhäuser. This book was released on 2016-07-12 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added. Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution. The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields. Review of the first edition: "The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)

Download or read book Lectures on Riemann Surfaces written by Otto Forster and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."—-MATHEMATICAL REVIEWS

Download or read book The Laplacian on a Riemannian Manifold written by Steven Rosenberg and published by Cambridge University Press. This book was released on 1997-01-09 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Download or read book Geometry of Riemann Surfaces written by William J. Harvey and published by Cambridge University Press. This book was released on 2010-02-11 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: Original research and expert surveys on Riemann surfaces.

Download or read book Spectral Geometry written by Pierre H. Berard and published by Springer. This book was released on 2006-11-14 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Progress in Inverse Spectral Geometry written by Stig I. Andersson and published by Birkhäuser. This book was released on 2012-12-06 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t> O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x, O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e- ; namely, u(·, t) = V(t)uoU· Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt, E* ®E), locally given by 00 K(x, y; t) = L>-IAk(~k ® 'Pk)(X, y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g., the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.