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On the conformal deformation of Riemannian structures
Published online by Cambridge University Press: 17 April 2009
Abstract
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In this paper, we study a nonlinear partial differential equation on a compact manifold;
where a > 1 is a constant, r is a positive constant, and H is a prescribed smooth function.
Kazdan and Warner showed that if λ1(g) < 0 and 
< 0, where is the mean of H, then there is a constant 0 < r0(H) ≤ ∞ such that one can solve this equation for 0 < r < r0(H), but not for r > r0(H). They also proved that if r0(H) = ∞, then H(x) ≤ 0 (≢0) for all x ∈ M. They conjectured that this necessary condition might be sufficient.
I show that this conjecture is right; that is, if H(x) ≤ 0 (≠ 0) for all x ∈ M, then r0(H) = ∞.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 42 , Issue 2 , October 1990 , pp. 307 - 313
- Copyright
- Copyright © Australian Mathematical Society 1990
References
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