This volume is dedicated to the memory of Harry Ernest Rauch, who died suddenly on June 18, 1979. In organizing the volume we solicited: (i) articles summarizing Rauch's own work in differential geometry, complex analysis and theta functions (ii) articles which would give the reader an idea of the depth and breadth of Rauch's researches, interests, and influence, in the fields he investigated, and (iii) articles of high scientific quality which would be of general interest. In each of the areas to which Rauch made significant contribution - pinching theorems, teichmiiller theory, and theta functions as they apply to Riemann surfaces - there has been substantial progress. Our hope is that the volume conveys the originality of Rauch's own work, the continuing vitality of the fields he influenced, and the enduring respect for, and tribute to, him and his accomยญ plishments in the mathematical community. Finally, it is a pleasure to thank the Department of Mathematics, of the Gradยญ uate School of the City University of New York, for their logistical support, James Rauch who helped us with the biography, and Springer-Verlag for all their efforts in producing this volume. Isaac Chavel . Hershel M. Farkas Contents Harry Ernest Rauch - Biographical Sketch. . . . . . . . VII Bibliography of the Publications of H. E. Rauch. . . . . . X Ph.D. Theses Written under the Supervision of H. E. Rauch. XIII H. E. Rauch, Geometre Differentiel (by M. Berger) . . . . . . .
H. E. Rauch, Gรฉomรจtre Diffรฉrentiel -- H. E. Rauch, Function Theorist -- H. E. Rauch, Theta Function Practitioner -- Some loci in Teichmรผller Space for Genus Six Defined by Vanishing Thetanulls -- Mรถbius transformations and Clifford Numbers -- Polynomial Approximation in Quasidisks -- An Inequality for Riemann Surfaces -- Extremal Kรคhler Metrics -- On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume -- Deformation of Surfaces Preserving Principal Curvatures -- One-dimensional Metric Foliations in Constant Curvature Spaces -- The Existence of Three Short Closed Geodesics -- On Lifting Kleinian Groups to SL (2, ?) -- On the Ends of Trajectories -- An Integrability Condition for Simple Lie Groups -- Uniqueness in the Cauchy Problem for a Degenerate Elliptic Second Order Equation -- On the Structure of Complete Manifolds with Positive Scalar Curvature