Conformal geometry, contact geometry, and the calculus of variations

σk-Scalar curvature and eigenvalues of the Dirac operator

“…For a symmetric n × n matrix A, set σ k (A) = σ k (Λ), where Λ = (λ 1 , λ 2 , · · · , λ n ) is the set of eigenvalues of A. Following Viaclovsky [22], a σ k -scalar curvature or k-scalar curvature of g is defined by…”

“…Note that when n = 2m > 4 and the underlying manifold (M 2m , g) is locally conformally flat, it was proved in [22] that…”

“…However, the proof given here relies on the conformal invariance of F n/2 , which was only proved to be true in the case that n = 4 or (M, g) is locally conformally flat ( [22]). The average Q-curvature might be a good quantity to bound the eigenvalues under certain positivity assumptions on the Paneitz-Branson operator on a higher dimensional manifold.…”

“…The notion of k-scalar curvature σ k (g) was introduced by Viaclovsky in [22]. See [8] for another interesting application.…”

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

“…Moreover, this operator appear as an object of investigation by many remarkable geometers. For example, Viaclovsky (see [17], [18]) observed that the k−Hessian operator is an important class of fully nonlinear operators which is closely related to a geometric problem of the type (1.1), where we cite the work of Bao-Ji-Li [2] for a more detailed discussion. Moreover, equation (1.1) arises via the study of the quasilinear parabolic problem ( see for example the introduction of MollPetitta [15]).…”