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Anti-de Sitter 3-manifolds with non-trivial Killing field. (Variétés anti-de Sitter de dimension 3 possédant un champ de Killing non trivial.) (French. Abridged English version) Zbl 0896.53043
This note classifies all compact 3-dimensional Lorentz manifolds of constant negative curvature which possess a nontrivial Killing field. Such manifolds are locally isometric to anti-de Sitter space $$X$$, a convenient model of which is the universal covering group $$\widetilde{PSL} (2, \mathbb{R})$$, with isometries given by the group $$G = \widetilde{PSL} (2, \mathbb{R})\times\widetilde{PSL}(2, \mathbb{R})$$ where one factor acts by left-multiplication and the other by right-multiplication. R. S. Kulkarni and F. Raymond [J. Differ. Geom. 21, 113-138 (1985; Zbl 0563.57004)] gave the first examples of compact anti-de Sitter manifolds. Their examples are quotients of anti-de Sitter space by discrete groups of the form $$\text{graph}(\rho)$$, where $$\rho:\Gamma\longrightarrow\widetilde{PSL}(2, \mathbb{R})$$ is a homomorphism taking values in a compact subgroup of $$\widetilde{PSL}(2, \mathbb{R})$$, and $$\Gamma\subset\widetilde{PSL}(2, \mathbb{R})$$ is a discrete group. (This condition is useful since it implies that the holonomy group lies in a subgroup $$H\subset G$$ which acts transitively and properly on $$X$$.) Furthermore, Kulkarni and Raymond showed that every properly discontinuous action with compact quotient is equivalent to the graph of a homomorphism $$\rho:\Gamma\longrightarrow\widetilde{PSL}(2,\mathbb{R})$$ where $$\Gamma\subset\widetilde{PSL}(2,\mathbb{R})$$ is a discrete cocompact subgroup. Subsequent further examples were given by W. M. Goldman [J. Differ. Geom. 21, 301-308 (1985; Zbl 0591.53051)] and E. Ghys [Ann. Sci. Ècole Norm. Sup. 20, 251-270 (1987; Zbl 0663.58025)], all of which possess nontrivial Killing vector fields, but the holonomy does not lie in a properly acting transitive subgroup of $$G$$. The difficulty in these examples is that, in general, compactness of an indefinite metric does not imply completeness. Recently, B. Klingler [Math. Ann. 306, 353-370 (1996; Zbl 0862.53048)] proved that compact anti-de Sitter manifolds are geodesically complete; thus any compact Lorentzian manifold of constant negative curvature is a quotient of anti-de Sitter space. It follows that whenever $$\rho$$ is sufficiently close to the trivial representation, $$\text{graph}(\rho)$$ acts properly and cocompactly on $$X$$.
The paper under review determines explicitly how small $$\rho$$ must be for $$\text{graph}(\rho)$$ to act properly, in the special case that $$\rho$$ takes values in an abelian subgroup of $$\widetilde{PSL}(2, \mathbb{R})$$. If $$\rho$$ takes values in an elliptic or parabolic one-parameter subgroup, then $$\text{graph}(\rho)$$ acts properly. Thus, suppose that $$\rho$$ takes values in a hyperbolic one-parameter subgroup $$H$$. For $$\gamma\in\Gamma$$, let $$t(\gamma)$$ denote the minimum displacement of $$\gamma$$ acting on the hyperbolic plane. Then the stable norm on $$H^1(\Gamma,\mathbb R)$$ is defined as the the function $\| h\| = \lim_{n\to\infty}{\textstyle \frac 1n} \inf \{t(\gamma_n) \mid [\gamma_n] = n h \},$ where $$[\gamma]$$ denotes the equivalence class of $$\gamma\in\Gamma$$ in $$H^1(\Gamma,\mathbb R)$$. Identifying the $$H$$-valued representations $$\rho$$ with $$H^1(\Gamma,\mathbb R)$$, the author proves that $$\text{graph}(\rho)$$ acts properly if and only if $$\|\rho\| < 1$$. The author investigates sufficient conditions for the existence of oscillatory solutions of equations of the form $u''' + p_1(t)u'' + p_2(t)u'' + p_3(t)u = 0.$ This interesting survey includes explicitly recent results on oscillatory solutions. It is a continuation of an article of K.-H. Mayer [J. Oscillation 136, No. 5, 35-102 (1987; per bibl.)].

MSC:
 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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