Blow-up phenomena for the Yamabe equation II
By S. Brendle, F. C. Marques. 2009
147 citations recorded by the source corpus.
Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts 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 \geq 52.
Subfields: Computational Mechanics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/2b00a09ee1557fff12697260df2493c9ed22d553
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In Computational Mechanics
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres (2011, methodology): “It follows that the C2,α-a priori estimates proved by Li and Zhang [14,15] and Marques [16] hold for the metrics in [gtk ], which implies that the set { g ∈ [gt ], t ∈ {tk, k ∈ N} ∪ {t∗} : g ∈ Met1 ( S2n+1 ) , scal(g) = const.”
- 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.”
- A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound (2012, unknown): “, 33] for 8 ≤ n≤ 11, and Khuri, Marques and Schoen [24] for 12 ≤ n≤ 24. On the other hand, the answer to Problem 1.2 is negative in dimension n≥ 25; see Brendle [5] for n≥ 52, and Brendle and Marques [6] for 25 ≤ n≤ 51. Fully nonlinear elliptic equations involving f(λ(∇2u)) were investigated in the ”
- A Compactness Theorem for the Yamabe Problem (2009, background): “In a subsequent paper Brendle and Marques [6] extend these examples to the dimensions 25 ≤ n ≤ 51.”