A priori estimates for the Yamabe problem in the non-locally conformally flat case

By F. C. Marques. 2004

144 citations recorded by the source corpus.

Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show that, when the dimension is greater than or equal to 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6c55e95cbf8c6ebc9e97f32ee2329c827cb21901

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound (2012, background): “This allows us to prove the existence of solutions when the associated cone Γ satisfies µ+Γ ≤ 1, which includes the σk−Yamabe problem for k not smaller than half of the dimension of the manifold.”
Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres (2011, background): “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.”
A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture (2010, unknown): “See [18], [33], [35], [36] and [39].”
Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds (2009, background): “His conjecture was very recently proved to be true in case n ≤ 24 by Khuri–Marques–Schoen [13].”
A Compactness Theorem for the Yamabe Problem (2009, background): “The details may be found in [17] or [19].”