By Peter Buser (auth.)

This vintage monograph is a self-contained advent to the geometry of Riemann surfaces of continuing curvature –1 and their size and eigenvalue spectra. It makes a speciality of topics: the geometric concept of compact Riemann surfaces of genus more than one, and the connection of the Laplace operator with the geometry of such surfaces. the 1st a part of the publication is written in textbook shape on the graduate point, with in basic terms minimum requirements in both differential geometry or complicated Riemann floor thought. the second one a part of the e-book is a self-contained creation to the spectrum of the Laplacian in line with the warmth equation. Later chapters take care of fresh advancements on isospectrality, Sunada’s building, a simplified facts of Wolpert’s theorem, and an estimate of the variety of pairwise isospectral non-isometric examples which relies merely on genus. Researchers and graduate scholars drawn to compact Riemann surfaces will locate this booklet an invaluable reference. *Anyone acquainted with the author's hands-on method of Riemann surfaces might be gratified by way of either the breadth and the intensity of the subjects thought of the following. The exposition can also be super transparent and thorough. an individual no longer accustomed to the author's technique is in for a true deal with. **— Mathematical Reviews**This is a thick and leisurely ebook with a view to pay off repeated examine with many friendly hours – either for the newbie and the professional. it really is thankfully roughly self-contained, which makes it effortless to learn, and it leads one from crucial arithmetic to the “state of the artwork” within the conception of the Laplace–Beltrami operator on compact Riemann surfaces. even though it isn't really encyclopedic, it's so wealthy in info and ideas … the reader may be thankful for what has been incorporated during this very pleasing e-book. **—Bulletin of the AMS **The e-book is especially good written and relatively available; there's an outstanding bibliography on the finish. **—Zentralblatt MATH*

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It is not difficult to see, for example, that if C is a set of disjoint simple closed geodesies on a hyperbolic surface 5 then S can be cut open along C. In a similar way we shall frequently cut open surfaces along sets of piecewise geodesic curves. 4 The Universal Covering Every complete unbordered surface of constant curvature - 1 is universally covered by the hyperbolic plane (Cheeger-Ebin [1] or Klingenberg [1, 2]). In this section we adapt this to the complete hyperbolic surfaces with boundary.

9 Theorem. , we have sinh Ka sinh Kb > cosh KC, cosh coc > sinh coa sinh cob. Proof. 4) for T = fcand r= co. ' and a' at the endpoints a and b. In (M, ds1) the curves p' and a' are generally not geodesies but they are orthogonal to a and b with respect to both metrics. 2). We check that Fig. 1 is drawn correctly in the sense that p' and a' do not intersect the open strip between (3 and a. For this we let p' e p' (respectively, p' e a') and consider the geodesic arc (with respect to either metric) w from p0 to p'.

Let S be a compact hyperbolic surface and let L > 0. Only finitely many closed geodesies on S have length < L. Proof. By compactness we may cover S with finitely many coordinate neighborhoods. Hence, there exists a constant r > 0 such that for any x e S the points at distance smaller than 4r form a convex neighborhood. Assume now that there exists an infinite sequence of pairwise different closed geodesies yx, y2, ... on S of length £< L, say parametrized on the interval [0, 1]. Then we may extract a subsequence such that the initial points, the initial tangent vectors and the lengths converge.