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## Über dieses Buch

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 accom­ plishments in the mathematical community. Finally, it is a pleasure to thank the Department of Mathematics, of the Grad­ 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) . . . . . . . .

## Inhaltsverzeichnis

### H. E. Rauch, Géomètre Différentiel

Résumé
En 1951 paraissait aux Annals of Mathematics un article intitulé «A contribution to differential geometry in the large», sous la signature de H. E. Rauch. Un texte de dix-huit pages et au titre modeste. Ce n’était pas le premier travail de l’auteur, qui avait fait auparavant un PhD intitulé «Generalizations of some classical theorems to the case of functions of several variables» sous la direction de Salomon Bochner (Princeton 1947).
Par Marcel Berger

### H. E. Rauch, Function Theorist

Abstract
H. E. Rauch made important contributions to the theory of closed Riemann surfaces throughout his mathematical career. For example his 1954 papers [9] and [10] with M. Gerstenhaber propose a forward-looking method for using the then undeveloped theory of harmonic maps to prove Teichmüller’s theorem about extremal quasi-conformal maps. His 1979 paper [14] with L. Keen and A. T. Vasquez sheds interesting light on the accessory parameter problem in the uniformization of punctured tori. His work thus covers far too much ground to be surveyed in one article, but his books and papers reveal a striking consistency of purpose and point of view. They deal with central questions and they show his knowledge of the classical literature and his love of concrete examples and explicit computation, traits that he successfully transmitted to his graduate students.
Clifford J. Earle

### H. E. Rauch, Theta Function Practitioner

Abstract
In 1.3 of [E] it was stated that “no adequate discussion of Rauch’s contributions to Riemann surface theory can ignore his work on theta functions”. In complete agreement with this statement, we have tried to include in this article a brief description of the most important aspects of Rauch’s own work in this area and some of the work he inspired and continues to inspire in others.
Hershel M. Farkas

### Some loci in Teichmüller Space for Genus Six Defined by Vanishing Thetanulls

Abstract
In H. F. Baker’s monumental tomb on Abelian varieties [3, p. 273] there is an exercise which must surely catch the eyes of those few readers whose fortitude carries them that far.
Robert D. M. Accola

### Möbius Transformations and Clifford Numbers

Abstract
The theory of Möbius transformations in ℝ n can be treated in various ways. One way is to use the projective model of hyperbolic geometry which expresses the Möbius transformations in terms of the matrix group O(n + 1,1). While very satisfactory from a theoretical point of view it leads quickly to overly complicated formulas, and I have therefore advocated an approach which works directly in ℝ n and uses formulas strikingly analogous to those in the complex case [1].
Lars V. Ahlfors

### Polynomial Approximation in Quasidisks

Abstract
Let B denote the open unit disk in the complex plane ℂ and D a bounded Jordan domain in ℂ. We say that D is an open k-quasidisk, 0 ≦ k < 1, if one and hence each conformai mapping g: ℂ̄\B̄ → ℂ̄\D̄ can be extended to a K-quasiconformal mapping of the extended complex plane ℂ̄ where K = (1 + k)/(1 ‒ k). A continuum E ⊂ ℂ is said to be a closed k-quasidisk if E = D̄where D is as above.
J. M. Anderson, F. W. Gehring, A. Hinkkanen

### An Inequality for Riemann Surfaces

Abstract
This note contains a proof of an elementary but useful inequality for Riemann surfaces with a finitely generated non-Abelian fundamental group.
Lipman Bers

### Extremal Kähler Metrics II

Abstract
Given a compact, complex manifold M with a Kähler metric, we fix the deRham cohomology class Ω of the Kahler metric, and consider the function space ℊΩ of all Kahler metrics in M in that class. To each (g) ∈ GΩ we assign the non-negative real number $$\Phi (g) = \int\limits_{M} {R_{g}^{2}d{V_{g}}}$$ (R g = scalar curvature, d V g = volume element).
Aiming to find a (g) ∈ ℊΩ that minimizes the function Φ, we study the geometric properties in M of any (g) ∈ ℊΩ that is a critical point of Φ, with the following results.
1) Any metric (g) that is a critical point of Φ is necessarily invariant under a maximal compact subgroup of the identity component ℌ0(M) of the complex Lie group of all holomorphic automorphisms of M.
2) Any critical metric (g) ∈ ℊΩ of Φ achieves a local minimum value of Φ in ℊΩ; the component of (g) in the critical set of Φ coincides with the orbit of Φ under the action of the group ℌ0(M), it is diffeomorphic to an open euclidean ball, and the critical set is always non-degenerate in the sense of ℌ0(M)-equivariant Morse theory.
3) If there exists a (g) ∈ ℊΩ with constant scalar curvature R, then it achieves an absolute minimum value of Φ; furthermore every critical metric in ℊΩ has constant R, and achieves the same value of Φ.
4) Whenever the existence of a critical Kahler metric (g) can be guaranteed (i.e., always, according to a conjecture 2), then Futaki’s obstruction determines a necessary and sufficient condition for the existence of a (g) ∈ ℊΩ with constant scalar curvature.
Eugenio Calabi

### On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume

Abstract
Let M n be a non-compact complete Riemannian manifold, whose sectional curvature K, and volume Vol(M) satisfy
$$\left| K \right| \leqq 1$$
(0.1)
$$Vol(M) < \infty$$
(0.2)
Jeff Cheeger, Mikhael Gromov

### Deformation of Surfaces Preserving Principal Curvatures

Abstract
The isometric deformation of surfaces preserving the principal curvatures was first studied by O. Bonnet in 1867. Bonnet restricted himself to the complex case, so that his surfaces are analytic, and the results are different from the real case. After the works of a number of mathematicians, W. C. Graustein took up the real case in 1924-, without completely settling the problem. An authoritative study of this problem was carried out by Elie Cartan in [2], using moving frames. Based on this work, we wish to prove the following: Theorem: The non-trivial families of isometric surfaces having the same principal curvatures are the following:
1)
a family of surfaces of constant mean curvature;

2)
a family of surfaces of non-constant mean curvature. Such surfaces depend on six arbitrary constants, and have the properties: a) they are W-surfaces; b) the metric
$$d{s^2} = {\left( {gradH} \right)^2}d{s^2}/\left( {{H^2} - K} \right)$$
, where d s 2 is the metric of the surface and H and K are its mean curvature and Gaussian curvature respectively, has Gaussian curvature equal to — 1.

Shiing-shen Chern

### One-Dimensional Metric Foliations in Constant Curvature Spaces

Abstract
Let $$Q_c^{n + 1}$$ be a connected space of constant curvature c. In this note we will discuss the structure of 1-dimensional bundlelike Riemannian foliations T of Q, which we call metric foliations for short. The leaves of T are locally fibers of Riemannian submersions, and thus everywhere equidistant. Such foliations T will turn out to be either flat or homogeneous. As a global application we obtain that the Hopf fibrations S2m + 1 → ℂ P m are the only metric fibrations of euclidean spheres with fiber dimension 1.
Detlef Gromoll, Karsten Grove

### The Existence of Three Short Closed Geodesics

Abstract
A famous theorem of Lusternik and Schnirelmann [LS] states that, for a Riemannian manifold M given by an arbitrary Riemannian metric on the differentiate 2-sphere, there are at least three closed geodesics without self-intersections. See [Ly] for a more complete proof.
Wilhelm Klingenberg

### On Lifting Kleinian Groups to SL (2, ℂ)

Abstract
The exact sequence of groups and group homomorphisms
$$1 \to \{ \pm I\} \to SL(2,\mathbb{C})\xrightarrow{\mathcal{P}}PSL(2,\mathbb{C}) \to 1$$
(0.1)
does not split. If in this sequence we identify PSL(2, ℂ) with the Möbius group, then for $$A = \left( {\begin{array}{*{20}{c}} a & b \\ c & d \\ \end{array} } \right) \in SL(2,\mathbb{C}),\mathcal{P}(A)$$ is the Möbius transformation $$z \mapsto \frac{{az + b}}{{cz + d}}$$. A lift of an element α ∈ PSL (2, ℂ) is an element A ∈ SL (2, ℂ) with P (A) = α, while a lift of a subgroup Γ of PSL(2, ℂ) is an isomorphism i: Γ → SL(2, ℂ) such that P ° i is the identity.
Irwin Kra

### On the Ends of Trajectories

Abstract
The purpose of this note is to establish a certain basic property concerning in particular the “ends” of the trajectories and more generally of the geodesics of a (holomorphic) quadratic differential φ d z 2 of finite norm
$$\left\| \varphi \right\| = \int {\int {\left| \varphi \right|dxdy} }$$
on an arbitrary Riemann surface R. In the remainder of this section we will briefly recall certain aspects of the geometry associated with these differentials. The theorem is stated in Sect. 2.
Albert Marden, Kurt Strebel

### An Integrability Condition for Simple Lie Groups

Abstract
In [6] H. E. Rauch pointed out that “the symmetric manifolds, far from being isolated phenomena of a special nature, derive their structure from certain parallelism and curvature properties which, when satisfied to a certain degree of approximation, delimit a general class of Riemannian manifolds with the same structure”. In addition, Rauch observed that these curvature properties can be viewed as the integrability condition of a certain set of partial differential equations. Rauch beautifully motivated the following comparison theorem and proved it in the case where the model symmetric space is of rank one, the general manifold simply connected, and equivalence is proved up to homeomorphism. The result envisioned by Rauch was finally proved in Min-Oo, Ruh [4], where the approximate integrability condition is formulated in terms of the curvature of an appropriate Cartan connection. The following theorem states the final result for small deviations from the standard geometry.
Min-Oo, Ernst A. Ruh

### Uniqueness in the Cauchy Problem for a Degenerate Elliptic Second Order Equation

Abstract
There is by now a large literature devoted to the question of local uniqueness in the Cauchy problem for linear partial differential equations (or having linear leading part) — assuming the boundary is non-characteristic. The lecture notes by C. Zuily [4] give an excellent survey and presentation of many of the recent results, as well as very complete references. Various classes of sufficient conditions have been proved and under certain circumstances these are close to necessary. In addition to the references in [4] there is soon to appear [1] by S. Alinhac with further counterexamples and results.
Louis Nirenberg

### On the Structure of Complete Manifolds with Positive Scalar Curvature

Abstract
One of the greatest contributions of Rauch in differential geometry is his famous work on manifolds with positive curvature. His comparison theorems, which are needed for his proof of the pinching theorem, are fundamental for later developments in Riemannian geometry. His work initiated a systematic research developed by Klingenberg, Berger, Gromoll, Meyer, Cheeger, Gromov, Ruh, Shio-hama, Karcher, etc. This work depends heavily on how a length-minimizing geodesic behaves under the influence of the curvature. Since geodesic is one-dimensional, the information we need from the curvature tensor is the curvature of the two planes which are tangential to the geodesic. This means that we need to know the behavior of the sectional curvature or the Ricci curvature of the manifold. Therefore, it seems very unlikely that arguments based only on length-minimizing geodesics can be used to deal with problems related to scalar curvature. The problem of scalar curvature, however, has drawn a lot of attention of the differential geometers in the late sixties and the seventies, partly because of its interest in general relativity.
Shing Tung Yau

### Backmatter

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