Endlichkeitssatz von Cheeger

Cheegers Endlichkeitssatz ist ein Lehrsatz aus dem mathematischen Teilgebiet der Differentialgeometrie, der eine Aussage über die Anzahl (und insbesondere die Endlichkeit) von Mannigfaltigkeiten mit vorgegebenen Durchmesser-, Volumen- und Krümmungsschranken macht.

Cheegers Endlichkeitssatz

Zu vorgegebenen positiven Zahlen $n,D,V,K$ gibt es nur eine endliche Anzahl $C(n,D,V,K)$ von Diffeomorphietypen $n$ -dimensionaler riemannscher Mannigfaltigkeiten $M$ mit

\mathrm {diam} (M)\leq D,\mathrm {vol} (M)\geq V,-K\leq \mathrm {sec} (M)\leq K.

Hierbei bezeichnen $\operatorname {vol} (M)$ das Volumen, $\operatorname {diam} (M)$ den Durchmesser und $\sec(M)$ die Schnittkrümmungen der riemannschen Mannigfaltigkeit $M$ .

Geschichte

Der Endlichkeitssatz erschien zuerst in Cheegers Dissertation,^[1] damals zunächst für Endlichkeit von Homöomorphietypen und nur unter zusätzlichen Voraussetzungen auch für Diffeomorphietypen. Der Endlichkeitssatz in seiner obigen Form erscheint bei Cheeger-Ebin^[2] und mit Beweis sowie einer expliziten Abschätzung für $C(n,D,V,K)$ bei Peters.^[3] Einen weiteren Beweis sowie zahlreiche Verallgemeinerungen und Anwendungen findet man in Kapitel 8 von Gromovs Buch.^[4]

Anmerkungen

Cheegers Beweis beruhte wesentlich auf einer unteren Schranke für den Injektivitätsradius. Er bewies, dass unter den Voraussetzungen $\operatorname {diam} (M)\leq D,-K\leq \sec(M)\leq K$ eine untere Schranke für das Volumen äquivalent zu einer unteren Schranke für den Injektivitätsradius ist.

Das Beispiel der Linsenräume $S^{3}/\mathbb {Z} _{p}$ mit $\sec(M)\equiv 1,\operatorname {diam} (M)\leq 1$ zeigt, dass auf eine untere Schranke für Volumen (oder Injektivitätsradius) nicht verzichtet werden kann.

Einzelnachweise

↑ Jeff Cheeger: Comparison and finiteness theorems for Riemannian manifolds. Thesis (Ph.D.)–Princeton University. 1967
↑ Jeff Cheeger, David G. Ebin: Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975.
↑ Stefan Peters: Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds. J. Reine Angew. Math. 349 (1984), 77–82.
↑ Michail Leonidowitsch Gromow: Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. ISBN 0-8176-3898-9

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