## Topological Methods in Nonlinear Analysis

### The Yamabe problem on subdomains of even-dimensional spheres

Frank Pacard

#### Abstract

We prove the existence of complete conformally flat metrics of constant positive scalar curvature on the complement in ${\mathbb S}^n$ of a finite number of $(n-2)/2$-dimensional smooth submanifolds, provided $n \geq 4$ is even.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 6, Number 1 (1995), 137-150.

Dates
First available in Project Euclid: 16 November 2016

https://projecteuclid.org/euclid.tmna/1479265353

Mathematical Reviews number (MathSciNet)
MR1391949

Zentralblatt MATH identifier
0854.53037

#### Citation

Pacard, Frank. The Yamabe problem on subdomains of even-dimensional spheres. Topol. Methods Nonlinear Anal. 6 (1995), no. 1, 137--150. https://projecteuclid.org/euclid.tmna/1479265353

#### References

• \ref\key1 T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampère equations, Springer-Verlag, New York, 252
• \ref\key2 P. Aviles, On isolated singularities in some nonlinear partial differential equations , Indiana Univ. Math. J., 35 (1983), 773–791 \ref\key3 ––––, Local behavior of solutions of some elliptic equations , Comm. Math. Phys., 108 (1987), 177–192
• \ref\key4 W.-Y. Ding, On a conformally invariant elliptic equation on ${\Bbb R}^{n}$ , Comm. Math. Phys., 107 (1986), 331–335
• \ref\key5R. Mazzeo, D. Pollack and K. Uhlenbeck, Moduli spaces of singular Yamabe metrics, preprint \ref\key6––––, Dipole solutions of the singular Yamabe problem, in preparation
• \ref\key7R. Mazzeo and N. Smale, Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere , J. Differential Geom., 34 (1991), 581–621
• \ref\key8F. Pacard, Solutions with high dimensional singular set, to a conformally invariant elliptic equation in ${{\Bbb R}^4}$ and in ${{\Bbb R}^6}$ , Comm. Math. Phys., to appear \ref\key9––––, Le problème de Yamabe sur des sous domaines de ${\Bbb S}^6$ , C. R. Acad. Sci. Paris, to appear
• \ref\key10R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation , Comm. Pure Appl. Math., 41 (1988), 317–392
• \ref\key11R. Schoen and S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature , Invent. Math., 92 (1988), 47–72