Blow-up phenomena for the Yamabe equation

Abstract

Let (M, g) be compact Riemannian manifold of dimension n ≥ 3. A well-known conjecture states that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M, g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n ≥ 52.

Bifurcating Solutions of the Lichnerowicz Equation

“…We note interesting dimension-dependent phenomena arising in the Yamabe problem [4,16], with similar behaviour expected to occur for the higher-dimensional Lichnerowicz equation.…”

Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium

“…. , p, where S g is the scalar curvature of g. By combining results in [Brendle 2008a;Brendle and Marques 2009], where noncompactness of the Yamabe equation in the nonconformally flat case is investigated, and those in [Druet and Hebey 2005a;Hebey and Vaugon 2001], where unstability of Yamabe type equations in the conformally flat case is investigated, we obtain the following theorem, in view of the remark above.…”

“…The case of low-dimensional manifolds was recently addressed in [Druet 2004;Marques 2005;Li and Zhu 1999;Li and Zhang 2004;2005], and compactness up to dimension 24 was finally proved recently [Khuri et al 2009]. On the other hand, Brendle [2008a] and Brendle and Marques [2009] exhibited counterexamples to the conjecture in dimensions n ≥ 25. For any n ≥ 25 they constructed examples of nonconformally flat n-manifolds with the striking property that their associated Yamabe equations possess sequences of solutions with minimal type energy and unbounded L ∞ -norms.…”

A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture

“…See [18], [33], [35], [36] and [39]. In [10], S. Brendle constructed a metric g in dimension N ≥ 52, with the following properties: (i) g ij = δ ij for |x| ≥ 1 2 ; (ii) g is not conformally flat. Then, for this metric, there exists a sequence of positive smooth solutions u n to (1.5) such that sup |x|≤1 u n (x) → +∞, and u n develops exactly one singularity.…”