On the conformal deformation of Riemannian structures
Yoon-Tae Jung · Bulletin of the Australian Mathematical Society · 1990
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) 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) = ∞.