Q-curvature in Conformal Geometry

In this talk , I will survey some analytic results concerned with the top order Q-curvature equation in conformal geometry. Q-curvature is the natural generalization of the Gauss curvature to even dimensional manifolds. Its close relation to the Pfaffian, the integrand in the Gauss-Bonnet formula, provides a direct relation between curvature and topology.

The notion of Q-curvature arises naturally in conformal geometry in the context of conformally covariant operators. In 1983, Paneitz gave the first construction of the fourth order conformally covariant Paneitz operator in the context of Lorentzian geometry in dimension four. The ambient metric construction, introduced by Fefferman and Graham , provides a systematic construction in general of conformally covariant operators. Each such operator gives rise to a semi-linear elliptic equation analogous to the Yamabe equations which we shall call the Q-curvature equation. These equations share a number of common features. Among these we mention the following: (i) the lack of compactness: the nonlinearity always occur at the critical exponent, for which the Sobolev imbedding is not compact; (ii) the lack of maximum principle: for example, it is not known whether the solution of the fourth order $Q$-curvature equation on manifolds of dimensions greater than four may touch zero. In spite of these difficulty, there has been significant progress on questions of existence, regularity and classification of entire solutions for these equations in the recent literature.

In the talk, I will give a brief survey of the subject with emphasize on applications to problems in conformal geometry and the connection between the a class of conformal covariant operators to that of the fractional Laplacian opeators.

Sun-Yung Alice Chang, Princeton University

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