Conformal transformation of metrics on the -sphere

Prescribing Morse scalar curvatures: incompatibility of non existence

“…On the other hand such flatness situations may be viewed as a borderline case to the realm of Morse functions, when for instance the sign of the Laplacian changes at critical points, and indeed do permit conclusions, which otherwise are not readily available, cf. [10], [18].…”

On the Chen-Lin conjecture for the prescribed scalar curvature problem

“…Such a characterization is of course quite difficult and technical. In principle, it relies on the construction of a special pseudogradient W at infinity, as in ([3], [6], [23]). The construction is more difficult when the function K satisfies (f ) β -condition with flatness order β(y) varies in (1, n) for any y ∈ Γ and leads to a new interesting phenomenon drastically different from the previous ones.…”

The scalar curvature problem on four-dimensional manifolds

“…Our aim is to prove global compactness and existence results for the problem when the prescribed function K satisfies the so-called "β-flatness" condition near its critical points. The main novelty of our results is that the flatness order β(y) varies in the entire interval (1, ∞) for any critical point y of K. Let us point out that existence results for the scalar curvature problem under the "β-flatness" condition were discussed for S n , n ≥ 3; in [19] under the assumption that β(y) ∈ (1, n − 2], for any critical point y of K and previously in [26] and [27] under the assumption that β(y) ∈ [n − 2, n) for any critical point y of K.…”

“…The scalar curvature problem has always been one of major subject in differential geometry. Intensive studies were dedicated to this topic, in dimension 3 and 4, see [1,8,10,11,13,18,28,34] as well as in high dimensions, see [3,4,5,12,14,15,16,17,19,22,23,26,27,30,35], and the references therein.…”