TY - JOUR

T1 - On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

AU - Trudinger, Neil S.

AU - Wang, Xu Jia

PY - 2009/7

Y1 - 2009/7

N2 - In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g 0) of dimension n ≥3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g 0) is not conformally equivalent to the unit sphere S n and that the set of all such metrics is compact. When (M, g 0) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2.

AB - In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g 0) of dimension n ≥3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g 0) is not conformally equivalent to the unit sphere S n and that the set of all such metrics is compact. When (M, g 0) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2.

UR - http://www.scopus.com/inward/record.url?scp=61349084181&partnerID=8YFLogxK

U2 - 10.1007/s00526-008-0207-0

DO - 10.1007/s00526-008-0207-0

M3 - Article

SN - 0944-2669

VL - 35

SP - 317

EP - 338

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 3

ER -