In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact Riemannian manifold (M,g) is compact. Established in the locally conformally flat case [41,42] and for n\leq 24 [23], it has revealed to be generally false for n\geq 25 [8,9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1)Scal_g, Scal_g being the Scalar curvature of (M,g). Even tough the Yamabe equation is compact in some cases, surprisingly we show that a-priori L^\infty-bounds fail on all manifolds with n\geq 4 as well as H_1^2-bounds do in the locally conformally flat case when n\geq 7. In several situations, the results are optimal.
Esposito, P., Pistoia, A., Vétois, J. (2014). The effect of linear perturbations on the Yamabe problem. MATHEMATISCHE ANNALEN, 358(1-2), 511-560 [10.1007/s00208-013-0971-9].
The effect of linear perturbations on the Yamabe problem
ESPOSITO, PIERPAOLO;
2014-01-01
Abstract
In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact Riemannian manifold (M,g) is compact. Established in the locally conformally flat case [41,42] and for n\leq 24 [23], it has revealed to be generally false for n\geq 25 [8,9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1)Scal_g, Scal_g being the Scalar curvature of (M,g). Even tough the Yamabe equation is compact in some cases, surprisingly we show that a-priori L^\infty-bounds fail on all manifolds with n\geq 4 as well as H_1^2-bounds do in the locally conformally flat case when n\geq 7. In several situations, the results are optimal.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
