Differential and Integral Equations

Existence results on prescribing zero scalar curvature

Qi S. Zhang and Z. Zhao

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We study some time independent Schrödinger equations on complete noncompact manifolds. By showing that these equations have global positive solutions bounded away from zero, we prove that a large class of manifolds are conformal to complete manifolds with zero scalar curvature. They include those with Ricci curvature being nonnegative and the scalar curvature satisfying $V(x) \le C/d^{2}(x, x_{0})$ for an arbitrary $C>0$. As we have shown that there are noncompact manifolds with positive scalar curvature, which are not conformal to manifolds with positive constant scalar curvature, the issue of prescribing zero scalar curvature becomes interesting. The current result gives a partial answer to the Yamabe problem on noncompact manifolds with nonnegative Ricci curvatures.

Article information

Differential Integral Equations, Volume 13, Number 4-6 (2000), 779-790.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.)


Zhang, Qi S.; Zhao, Z. Existence results on prescribing zero scalar curvature. Differential Integral Equations 13 (2000), no. 4-6, 779--790. https://projecteuclid.org/euclid.die/1356061249

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