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

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 matters: the geometric thought of compact Riemann surfaces of genus more than one, and the connection of the Laplace operator with the geometry of such surfaces. study employees and graduate scholars attracted to compact Riemann surfaces will locate right here a couple of precious instruments and insights to use to their investigations.

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Our next aim is to define a concept which unites the various configurations of Fig. 3. 3 Definition. Let x and y belong to the set of sides and angles of polygon P. The ordered pair (x, y) is said to be of angle type if one of the following conditions holds: (i) y is the subsequent angle of side x, (ii) (x, y) is a pair of consecutive sides, and y is orthogonal to x, (iii) y is the subsequent side of angle x. * — -. 3 In Fig. 3 all pairs (x, v) and (y, z) are of angle type. 2)). y (ii) If (x, y) is a pair of consecutive sides with angle enl'2, where e - ±1, then we define C L • y JC!

4 Theorem. In a right-angled geodesic hexagon a,y,b, intersecting sides c and y, we have a, c, P with cosh c = sinh a sinh b cosh/ + cosh a cosh b. Proof. 6. Another proof, based on pen­ tagons and trirectangles is as follows. Assume first that the geodesic extensions of p and b have a common per­ pendicular r. , they do not intersect. Hence, we have two right-angled pentagons, one with sides a, y, s, r and t and the other with sides c, a, (b + s), r and (t- p). 4(i) (right-angled pentagons) yields: cosh c = sinh r sinh(& + s) - sinh r sinh b cosh s + sinh r cosh b sinh s = sinh a sinh b cosh y + cosh a cosh b.

Let S be a hyperbolic surface. The universal covering S ofS is isometric to a convex domain in H with piecewise geodesic boundary. IfS is unbordered, then S is isometric to H. Proof. 1 containing S as a subdomain. Then there exists a covering map n: H —» S*. Let S be a connected component of n~l(S). Since all interior angles are less than or equal to n, S is convex and, in particular, simply connected (cf. Beardon [1], p. 140). Hence, S is a universal covering surface. Since all uni­ versal covering surfaces are isometric, this proves the theorem.

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