Compactification of a class of conformally flat 4-manifold
Let \((M,g)\) be a complete non-compact \(4\)-manifold such that \(M=N\cup \{ \bigcup_{i=1}^{k}E_{i}\} \), where \((N,g)\) is a compact Riemannian manifold with boundary \(\partial N=\bigcup_{i=1}^{k}\partial E_{i}\), and each \(E_{i}\) is a conformally flat simple end of \(M\) (i.e. \((E_{i},g)=(\mathbb{R}^{4}\backslash B,\) \(e^{2\omega i}|dx|^{2})\) for some function \(\omega _{i}\), where \(B\) is the unit ball in \(\mathbb{R}^{4}\). Then we say \((M,g)\) is a complete 4-manifold with a finite number of conformally flat simple ends. According to the authors there are many examples of such manifolds, and they give some. This paper contains the detailed proof of three results that generalise Huber's result on complete surfaces of finite total curvature. Following the introduction, let us recall that in the theory of complete surfaces of finite total curvature, \textit{S. Cohn-Vossen} [Compos. Math. 2, 69-133 (1935; Zbl 0011.22501)] showed that if the Gauss curvature of a complete analytic metric is absolutely integrable, then \(\int KdA\leqq 2\pi \chi \), where \(\chi \) is the Euler number of the surface. \textit{A. Huber} [Comment. Math. Helv. 32, 13-72 (1957; Zbl 0080.15001)] extended this inequality to metrics with weaker regularity and proved that such surfaces can be conformally compactified by adjoining a finite number of points. For dimension four \textit{R. E. Greene} and \textit{H. Wu} [Acta Math. 137, 209-245 (1976; Zbl 0372.53019)] obtained a generalization of Cohn-Vossen's inequality to complete manifolds of positive sectional curvature outside a compact set. In a previous paper, the authors [Duke Math. J. 103, 523-544 (2000; Zbl 0971.53028)] proved the following result: suppose that \(e^{2\omega }|dx|^{2}\) on \(\mathbb{R}^{4}\) is a complete metric with its \(Q\)-curvature (Paneitz curvature) absolutely integrable, and suppose that its scalar curvature is negative at infinity, then \[ 1-\frac{1}{4\pi ^{4}}\int_{\mathbb{R}^{4}}Qe^{4\omega} dx=\lim_{r\to \infty }\frac{(\text{vol}(\partial B_{r}(0))^{\frac{4}{3}}}{4(2\pi ^{2})^{\frac{1}{3}} \text{vol}(B_{r}(0))}\geqq 0. \] Recall that \(Q=\frac{1}{12}(-\Delta R+\frac{1}{4}R^{2}-3|E|^{2})\), where \(E\) is the traceless Ricci tensor and \(R\) is the scalar curvature. \(Q\) is a fourth order curvature invariant defined by the Paneitz operator and \(Q\) is related to the Chern-Gauss-Bonnet formula. This article generalises the previous result proved by the same authors. It is divided in three sections and in each of them is given the proof of one generalization. In Section one the authors localise their arguments given in [loc. cit.] to an end corresponding to a puncture and obtain the theorem: Suppose \((\mathbb{R}^{4}\backslash B,\) \(e^{2\omega i}|dx|^{2})\) is a complete conformal metric with nonnegative scalar curvature at infinity. If in addition \(\int |Q|dV<\infty \), then \[ \lim_{r\to \infty }\frac{(\text{vol}(\partial B_{r})^{\frac{4}{3}}}{4(2\pi ^{2})^{\frac{1}{3}} \text{vol}(B_{r}\backslash B)}=\frac{1}{4\pi ^{4}}\int_{\partial B}Te^{3\omega } dx-\frac{1}{4\pi ^{4}}\int_{\mathbb{R}^{4}\backslash B}Qe^{4\omega} dx. \] This formula makes it possible to extend the previous result to allow the domain to have a finite number of punctures. (\(T\) is a boundary curvature invariant associated to a boundary operator defined by \textit{S.-Y. Chang} and \textit{J. Qing} [J. Funct. Anal. 147, 327-362 (1997; Zbl 0914.58039)] whose expression involves the mean curvature of the boundary, the second fundamental form on the boundary, the unit inward normal derivative and the boundary Laplacian.) In Section two the authors study the simply connected case using the Chern-Gauss-Bonnet formula to obtain a compactification criterium analogous to Huber's two-dimensional result. The result is the following theorem: Let \((\Omega \subset S^{4},g=e^{2\omega }g_{0})\) be a complete conformal metric satisfying the following conditions: a) The scalar curvature is bounded between two positive constants, and \(|\nabla _{g}R|\) is bounded; b) The Ricci curvature of the metric \(g\) has a lower bound, c) the Paneitz curvature is absolutely integrable, i.e., \(\int_{\Omega }|Q|dv_{g}<\infty \). Then \(\Omega =S^{4}\backslash \{ p_{1},\dots,p_{k}\} \). According to the authors an essential ingredient in the above finiteness result is to view the boundary integral as measuring the growth of volume. The finiteness of the \(Q\) integral implies a control on the growth of volume, which can only accommodate the growth of a finite number of ends. As a consequence, the authors can classify solutions of the equation \(Q=\text{Const.}\) in the following way: Let \((\Omega \subset S^{4},g=e^{2\omega }g_{0})\) be a complete conformal metric satisfying conditions a), b) and c) of the previous theorem and in addition let us suppose \(Q\) is constant, then there are only two possibilities: 1) \((\Omega \subset S^{4},g=e^{2\omega }g_{0})=(S^{4},g_{0})\), corresponding to \(Q=3\); 2) \((\Omega \subset S^{4},g=e^{2\omega }g_{0})=(R^{4}\backslash \{0\},\frac{|dx|^{2}}{|x|^{2}})\), corresponding to \(Q=0\). The criterium of compactification given in Section 2 applies immediately to simply connected manifolds for which conditions a), b) and c) hold, through \textit{R. Schoen} and \textit{S.-T. Yau} [``Lectures on differential geometry'', International Press (1994; Zbl 0830.53001)]. In Section three the non-simply connected case is studied. Provided the natural notion of geometric finiteness in Kleinian groups is assumed it is true that: Let \((M^{4},g)\) be a locally conformally flat complete 4-manifold satisfying the following conditions: a) The scalar curvature is bounded between two positive constants, and \(|\nabla _{g}R|\) is bounded; b) The Ricci curvature of the metric \(g\) has a lower bound; c) the Paneitz curvature is absolutely integrable, i.e., \(\int_{\Omega }|Q|dv_{g}<\infty \); d) The fundamental group of \(M\) acting as deck transformation group is a geometrically finite Kleinian group without torsion. Then \(M=\overline{M}\backslash \{ p_{i}\} _{i=1}^{k}\) where \(\overline{M}\) is a compact manifold with a locally conformally flat structure. \(\overline{M}\) is sometimes called the conformal compactification of \(M\). If the manifold satisfies conditions a), b), c) and does not satisfy d) we can still apply the theorem to the finite covering of \(M\).
scientific article; zbMATH DE number 1533361