The Yamabe problem on subdomains of even-dimensional spheres

Abstract

Dedicated to Louis Nirenberg on the occasion of his 70th birthday We prove the existence of complete conformally flat metrics of constant positive scalar curvature on the complement in S n of a finite number of (n − 2)/2dimensional smooth submanifolds, provided n ≥ 4 is even.

Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities

“…The second proof regards (47) as a perturbation of (14), and makes a continuous dependence argument, with the help of (48), (49), and (51) to prove that, when h < 0, either 1 − γ 2 t (t) → 0 as t → ∞, which contradicts (49), or 1 − γ 2 t (t) → −∞ as t → ∞ in the case 1 − γ 2 t (t) < 0 and k is odd, which contradicts (46). The case h = 0 can also be ruled out along similar lines by a more careful argument.…”

Weak Solutions of Nonlinear Elliptic Equations with Prescribed Singular Set

“…All these solutions are obtained perturbation-theoretically starting with exact or approximate solutions. F. Pacard, in his paper [14], has studied the Eq. (2) and he proved the existence of a weak solution of the Eq.…”

“…The method used to this end is a mixture of arguments used by both R. Mazzeo and N. Smale in their paper [10] and by F. Pacard in his paper [14].…”

Incomplete Yamabe flows and removable singularities

“…Moreover, Schoen [19] studied the case where N ⊂ S m is a finite set of at least two points. In the setting of Theorem 3 (ii), Pacard [16] considered the borderline case n = m−2 2 for even m first and then solved the singular Yamabe Problem for n ≤ m−2 2 in joint work with Mazzeo [14,15].…”