×

zbMATH — the first resource for mathematics

An a priori estimate for a fully nonlinear equation on four-manifolds. (English) Zbl 1067.58028
Let \((M,g)\) be a 4-dimensional Riemannian manifold without boundary and let \(W\), Ric, \(R\) and \(A\) denote, respectively, the Weyl curvature tensor, Ricci curvature, scalar curvature of \(g\) and \(A=A_g=\text{Ric}-\frac{1}{6}Rg\). For a metric \(g_0\), the corresponding quantities defined by \(g_0\) are denoted by attaching a sub- or superscript \(0\). For \(g=e^{2\omega}g_0\), \[ \sigma_k(A_g)=\sigma_k(A_0-2\nabla_0^2 \omega+2d\,\omega\otimes d\,\omega-| d\,\omega| ^2g_0), \] where \(\sigma_k\) denotes the \(k\)th elementary symmetric polynomial, applied to the eigenvalues of \(A_g\). The equation \[ \sigma_k(A_g)=f\tag{*} \] was introduced by J. A. Viaclovsky [Duke Math. J. 101, 283–316 (2000; Zbl 0990.53035)] and the present authors [Ann. Math. 155, 711–789 (2002)] prove that any metric \(g_0\) satisfying \[ \int\sigma_2(A_{g_0})d\text{vol}_{g_0}> 0,\quad Y(g_0) >0, \tag{**} \] where \(Y(g_0)\) is the Yamabe invariant of \(g_0\), is conformal to a metric \(g\) with \(\sigma_2(A_g)>0\). As the main existence result the authors prove that for any positive (smooth) function \(f\) there exist a solution \(g=e^{2\omega}g_0\) of (*) if \(g_0\) satisfies (**).

MSC:
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Citations:
Zbl 0990.53035
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Besse,Einstein Manifolds, Springer-Verlag, Berlin, 1987.
[2] Chang, S. Y. A.; Gursky, M. J.; Yang, P., An equation of Monge-Ampere type in conformal geometry, andfour-manifolds of positive Ricci curvature, Ann. of Math., 155, 711-789, (2002) · Zbl 1031.53062
[3] Evans, L. C., Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35, 333-363, (1982) · Zbl 0469.35022
[4] Fontana, L., Sharp boderline Sobolev inequality on compact Riemannian manifold, Comment. Math. Helv., 68, 415-454, (1993) · Zbl 0844.58082
[5] Gursky, M. J., The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Comm. Math. Phys., 207, 131-143, (1999) · Zbl 0988.58013
[6] Korevaar, N.; Mazzeo, R.; Pacard, F.; Schoen, R., Refined asymptotics for constand scalar curvature metrics with isolated singularities, Invent. Math., 135, 233-272, (1999) · Zbl 0958.53032
[7] Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR, Ser. Mat., 47, 75-108, (1983)
[8] Li, Y., Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14, 1541-1578, (1989) · Zbl 0702.35094
[9] Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Math. J., 20, 1077-1091, (1971) · Zbl 0203.43701
[10] Obata, M., The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom., 6, 247-258, (1971) · Zbl 0236.53042
[11] Viaclovsky, J., Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., 101, 283-316, (2000) · Zbl 0990.53035
[12] J. Viaclovsky,Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds, Comm. Anal. Geom., to appear. · Zbl 1023.58021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.