Blow-up phenomena for the Yamabe equation

By S. Brendle. 2007

191 citations recorded by the source corpus.

It is well known that the PDE (1) has at least one positive solution for any choice of (M, g). If n > 6 and (M, g) is not locally conformally flat, this follows from results of T. Aubin [3]. The remaining cases were solved by R. Schoen using the positive mass theorem [16]. Solutions to (1) are not usually unique. As an example, consider the product metric on 51(L) x 5n_1(l). If L is sufficiently small, then the Yamabe PDE has a unique solution. On the other hand, there are many non-minimizing solutions if L is large. D. Pollack [14] has used gluing techniques to construct high energy solutions on more general background manifolds: given any conformai class with positive Yamabe constant and any positive integer iV, there exists a new conformai class which is close to the original one in the C?-norm and contains at least N metrics of constant scalar curvature (see [14], Theorem 0.1). It is an interesting question whether the set of all solutions to the Yamabe PDE is compact (in the C2-topology, say). A well-known conjecture states that this should be true unless (M, p) is conformally equivalent to the round sphere (see [17], [18], [19]). This conjecture has been verified in low dimensions and in the locally conformally flat case: if (M, g) is locally conformally flat, compactness follows from work of R. Schoen [17], [18]. Moreover, Schoen proposed a strategy for proving the conjecture in the non-locally conformally flat case based on the Pohozaev identity.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2ca678664c61ebe0121f6a83e6cb8e04e6cc312a

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Citations in the corpus

A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture (2010, background): “In [10], S. Brendle constructed a metric g in dimension N ≥ 52, with the following properties: (i) g ij = δ ij for | x | ≥ ; (ii) g is not conformally ﬂat.”
An existence theorem for the Yamabe problem on manifolds with boundary (2009, methodology): “In order to extend the testfunction to all of M , we glue the function uε +w to the Greens function of the conformal Laplacian with pole at p.”
Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds (2009, methodology): “3) with ε = 0 in high dimensions have been proved by Brendle [4] in case n ≥ 52, and by Brendle–Marques [5] in case 25 ≤ n ≤ 51.”
Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres (2011, background): “l (2.2). A natural question YAMABE PROBLEM ON HOMOGENEOUS SPHERES 25 is then to study the nature of the space of solutions to the Yamabe problem in a given conformal class. In this direction, Brendle [12] and Brendle and Marques [14] obtained remarkable results on compactness and non compactness of ”
Blow-up phenomena for the Yamabe equation II (2009, background): “Moreover, the blowingup sequences obtained in [4] form exactly one bubble.”
A Compactness Theorem for the Yamabe Problem (2009, background): “In a surprising recent paper Simon Brendle [4] has constructed examples of C∞ metrics on spheres of dimension at least 52 for which the compactness statement fails.”