TAILIEUCHUNG - Đề tài " Global existence and convergence for a higher order flow in conformal geometry"

An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, . a constant. In two dimensions, the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover, J. Moser [20] proved that for every positive function f on S 2 satisfying f (x) = f (−x) for all x ∈ S 2 there exists a conformal metric on S 2 whose Gauss curvature is equal to f . A natural conformal invariant in dimension four is 1 Q = − (∆R −. | Annals of Mathematics Global existence and convergence for a higher order flow in conformal geometry By Simon Brendle Annals of Mathematics 158 2003 323 343 Global existence and convergence for a higher order flow in conformal geometry By Simon Brendle 1. Introduction An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function . a constant. In two dimensions the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover J. Moser 20 proved that for every positive function f on S2 satisfying f x f x for all x E S2 there exists a conformal metric on S2 whose Gauss curvature is equal to f . A natural conformal invariant in dimension four is Q AR R 3 Ric 2 6 where R denotes the scalar curvature and Ric the Ricci tensor. This formula can also be written in the form Q T Ar 6Ơ2 A 6 where A Ric - Rg is the Schouten tensor of M and 2 A 1 tr A 2 2 A 2 is the second elementary symmetric polynomial in its eigenvalues. Under a conformal change of the metric g e2w go the quantity Q transforms according to Q e-4w Qo Pow where P0 denotes the Paneitz operator with respect to g0. The Gauss-Bonnet-Chern theorem asserts that y QdV Ị 1 W 2dV 8n2x M . 324 SIMON BRENDLE Since the Weyl tensor W is conformally invariant it follows that the expression i QdV JM is conformally invariant too. The quantity Q plays an important role in fourdimensional conformal geometry see 2 3 5 16 note that our notation differs slightly from that in 2 3 . Moreover the Paneitz operator plays a similar role as the Laplace operator in dimension two compare 2 3 5 11 12 . We also note that the Paneitz operator is of considerable interest in mathematical physics see 10 . T. Branson . A. Chang and P. Yang 2 studied metrics for which the curvature quantity Q is constant. Since I QdV M is conformally invariant these metrics minimize the functional i Q2 dV M among all .

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