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

By Neil S. Trudinger, Xu-jia Wang. 2005

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In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g0) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g0) is not conformally equivalent to the unit sphere Sn and that the set of all such metrics is compact. When (M, g0) 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/b14c22df90d86f969be3582ac839e7277a81e210

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In Applied Mathematics

Citations in the corpus

Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary (2006, background): “Li-Li [26, 29] also established a general result that (1.8) is still solvable if (M, g) is locally conformally flat and if (f,Γ) satisfies (1.1)–(1.6) with f |∂Γ = 0.”
A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry* (2006, methodology): “0 ≤ δ < 1 n−2 . Then there exists a solution to problem (1.1). Remark 2.3. Theorem 2.4 was ﬁrst proved by Gursky and Viaclovsky [GV1] for the conformal k-Hessian equation for k > n 2 , see also [TW2] for a diﬀerent proof. By the 5 very recent a priori estimates in [Ch], Gursky and Viaclovsky e”