On some conformally invariant fully nonlinear equations

On a fully nonlinear Yamabe problem

“…Combining the results of [19] and [29], the σ 2 -Yamabe problem is solvable if n > 8. Like the ordinary Yamabe problem, there is a well-known difficulty -the loss of compactness of Equation (1).…”

“…Let (M, g 0 ) be a compact, oriented Riemannian manifold with g 0 ∈ Γ + 2 and the dimension n > 4. Then there exists a positive constant λ 1 > 0 depending only on classification in [29] or [8], we show that u ε subconverges to a solution u 0 of (2), provided that…”

“…See also [6] and [23]. When the underlying manifold is locally conformally flat, this problem was solved by Guan-Wang [19] and Li-Li [29] independently. See also [4].…”

“…Then we consider the sequence u ε as ε → 0. Using the blow-up analysis developed in [18] and the classification of "bubbles" in [8] or [29], we can show that the sequence u ε subconverges to a solution of (2) under the condition (3). The flow method to attack the existence of fully nonlinear equations was used by many mathematicians, see for instance [9], [41], [37] and [10].…”

Variational properties of the Gauss–Bonnet curvatures

“…Then Sheng, Trudinger and Wang [10] completed the proof of the remaining cases where 2 ≤ k ≤ n/2 after assuming that the relevant equation is variational. Note also that the σ k -Yamabe problem was solved in the conformally flat case by Li-Li [8], and Guan-Wang [3].…”