Yamabe-type Equations on Complete, Noncompact Manifolds

Lichnerowicz-type equations on complete manifolds

“…We note that the existence of solutions for equation (1.3) can be easily obtained under the hypoteses of Theorem 3.9 by direct application of the monotone iteration scheme of H. Amann (see for instance [27] or [19]). Indeed in this case it is relatively easy to find an ordered pair of global sub and supersolutions.…”

Gradient estimates for a nonlinear parabolic equation and Liouville theorems

“…Indeed, for any n-dimensional (n ≥ 3) complete manifold (M, g), consider a pointwise conformal metric g = u 4 n−2 g for some 0 < u ∈ C ∞ (M). Then the scalar curvatureS of metricg related to the scalar curvature S of metric g is given by (see [33]) (1.4) ∆u − n − 2 4(n − 1) S u + n − 2 4(n − 1)S u n+2 n−2 = 0, which is a special form of equation (1.1). If M is compact andS is constant, the existence of a positive solution u is the well-known Yamabe problem and it has been solved in the affirmative by the combined efforts of Yamabe [45], Trudinger [39], Aubin [2] and Schoen [36]; see the survey [25] for more details.…”

A note on Killing fields and CMC hypersurfaces

“…The search for f matching the first requirement of (1.9) can be made via the Laplacian comparison theorem (see for instance [15,20]) by assigning a lower bound on the Ricci tensor of M , and since the behavior at +∞ of a carefully chosen f can be easily detected under this assumption, the second requirement in (1.9) turns out simple to check (we refer the reader to [4] for details). In view of (1.9), we can define the critical curve…”