Geometric operators and conformal geometry. (French) Zbl 1179.53047

Djadli, Zindine

Gaz. Math., Soc. Math. Fr. 109, 37-68 (2006).

Conformal changes of metric have played an important role in surface theory. For example, one consequence of the famous uniformization theorem of complex analysis is the fact that every surface has a conformal metric of constant Gaussian curvature. In higher dimensions this result leads naturally to the Yamabe problem, namely to seek a conformal change of metric that makes the scalar curvature constant. The solution of this problem has been a milestone in the development of the theory of nonlinear partial differential equations. The value of the Yamabe invariant, a conformal invariant, is central in the analysis of the Yamabe problem. The survey paper under review is an introduction to nonlinear analysis on manifolds and in problems of conformal geometry on surfaces and \(4\)-manifolds.
It is divided into four parts. The first reviews the Trudinger inequality [N. S. Trudinger, J. Math. Mech. 17, 473–483 (1967; Zbl 0163.36402)] and its extension for compact \(C^\infty\)-surfaces without boundary, the Moser inequality for the \(2\)-sphere with its standard metric and an optimal version of it due to E. Onofri [Commun. Math. Phys. 86, 321–326 (1982; Zbl 0506.47031)], as well as J. Moser’s amelioration for the setting of symmetric functions [Indiana Univ. Math. J. 20, 1077–1092 (1971; Zbl 0203.43701)]. The Moser-Trudinger inequality is exploited in order to discuss the problem of prescribed (in particular, constant) Gauss curvature. Here the author presents well-known results of J. L. Kazdan and F. W. Warner [Ann. Math. (2) 101, 317–331 (1975; Zbl 0297.53020)] and Moser [op. cit.] concerning this problem [see also, S.-Y. A. Chang and P. C. Yang, Duke Math. J. 64, No. 1, 27–69 (1991; Zbl 0739.53027); S.-Y. A. Chang, M. J. Gursky and P. C. Yang, Calc. Var. Partial Differ. Equ. 1, No. 2, 205–229 (1993; Zbl 0822.35043); A. Bahri and J.-M. Coron, J. Funct. Anal. 95, No. 1, 106–172 (1991; Zbl 0722.53032); Y. Li, Commun. Pure Appl. Math. 49, No. 6, 541–597 (1996; Zbl 0849.53031)].
The second part is devoted to the isospectrality problem for compact smooth surfaces without boundary. The author gives applications [among them a theorem of B. Osgood, R. Philips and P. Sarnak, J. Funct. Anal. 80, No. 1, 212–234 (1988; Zbl 0653.53021), concerning the family of isospectral metrics on such surfaces] of Polyakov’s formula [Phys. Lett. B 103, 207–210 (1981)], which describes the behavior of the functional \(g\mapsto \log\det(-\Delta_g)\), where \(\Delta_g\) denotes the Laplace-Beltrami operator associated to the metric \(g\) on a compact smooth surface.
In the third one, results of M. J. Gursky [Ann. Math. (2) 148, No. 1, 315–337 (1998; Zbl 0949.53025); Commun. Math. Phys. 207, No. 1, 131–143 (1999; Zbl 0988.58013)], M. J. Gursky and J. A. Viaclovsky [J. Differ. Geom. 63, No. 1, 131–154 (2003; Zbl 1070.53018)] concerning the kernel and the positivity of the conformally covariant differential operator of S. Paneitz [“A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds”, preprint (1983)] on a compact smooth Riemannian \(4\)-manifold, are presented. Also the author discusses the existence of extremal metrics for the log-determinant functional associated to a self-adjoint geometric differential operator with positive principal symbol (in particular, the conformal Laplacian) on such manifolds. Concerning this matter, a result due to S.-Y. A. Chang and P. C. Yang [Ann. Math. (2) 142, No. 1, 171–212 (1995; Zbl 0842.58011)] is given. To obtain such a result, an inequality of D. R. Adams [Ann. Math. (2) 128, No. 2, 385–398 (1988; Zbl 0672.31008)], generalizing Moser’s one, has to be exploited.
The last part contains three sections, and is devoted to applications and to the utilization of the Paneitz operator in conformal geometry in the setting of compact smooth Riemannian \(4\)-manifolds: the existence of metrics of constant Q-curvature, a lower bound (in terms of the Euler-Poincaré characteristic and the signature of the manifold) for a conformally invariant quadratic functional associated to the self-dual component of the Weyl tensor field if the Yamabe invariant is positive, and consequences, and – finally – rigidity results in conformal geometry. The first section covers results of Chang and Yang [loc. cit. (1995; Zbl 0842.58011)], the author and A. Malchiodi [Ann. Math. (2) 168, No. 3, 813–858 (2008; Zbl 1186.53050)], the second one, results of M. J. Gursky [loc. cit. (1998; Zbl 0949.53025); Indiana Univ. Math. J. 43, No. 3, 747–774 (1994; Zbl 0832.53032)], among them the classification of smooth \(4\)-manifolds \(M\) with boundary that are locally conformally flat, with the Yamabe invariant \(Y(M)>0\), and with Euler-Poincaré characteristic \(\chi(M)\geq 0\), and, the third one, results of S.-Y. A. Chang, M. J. Gursky and P. C. Yang [Publ. Math., Inst. Hautes Étud. Sci. 98, 105–143 (2003; Zbl 1066.53079)], and C. Margerin [Commun. Anal. Geom. 6, No. 1, 21–65 (1998; Zbl 0966.53022)].

Reviewer: Mircea Craioveanu (Timişoara)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C24	Rigidity results
58J53	Isospectrality
35J60	Nonlinear elliptic equations
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

Keywords:

Gauss curvature; surface; conformal geometry; conformal metric; isospectrality; conformally covariant differential operator; Riemannian manifold; Laplace-Beltrami operator; scalar curvature; conformal Laplacian; Yamabe invariant; Q-curvature; Euler-Poincaré characteristic; signature; Weyl tensor; log-determinant functional

Citations:

Zbl 0163.36402; Zbl 0506.47031; Zbl 0203.43701; Zbl 0297.53020; Zbl 0739.53027; Zbl 0822.35043; Zbl 0722.53032; Zbl 0849.53031; Zbl 0653.53021; Zbl 0949.53025; Zbl 0988.58013; Zbl 1070.53018; Zbl 0842.58011; Zbl 0672.31008; Zbl 0832.53032; Zbl 1066.53079; Zbl 0966.53022; Zbl 1186.53050