Global existence and convergence for a higher order flow in conformal geometry

By S. Brendle. 2003

109 citations recorded by the source corpus.

An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, e.g. 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 1 2 Q (AR R2+ 3 IRicI2), where R denotes the scalar curvature and Ric the Ricci tensor. This formula can also be written in the form

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4a1bfd2cb513d5573123a393d7f3d66b6271a734

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Morse theory and a scalar field equation on compact surfaces (2008, background): “A positive answer was given in [7, 15] under the assumptions Pg ≥ 0 and qP < 8π, which are naively the counterpart of ρ < 8π for (1) (also in this case there is a variant of the Moser-Trudinger inequality which makes the problem coercive).”
Compactness of solutions to some geometric fourth-order equations (2004, methodology): “In the case of the sphere a similar result has been shown in [9], and our proof basically goes along the same line.”
The Nirenberg problem and its generalizations: a unified approach (2014, methodology): “On S4, Moser’s type result was obtained by Brendle [9] via a flow method (see also Baird et al.”
Existence of conformal metrics with constant Q-curvature (2004, methodology): “The result in [16] has also been extended in [9] to higher-dimensional manifolds (regarding higher-order operators and curvatures) using a geometric flow.”
Constant Q-curvature metrics in arbitrary dimension (2007, methodology): “Our assumptions include those made in [ 11 ] and (one) of the following two possibilities (or both)”