Sun-Yung Alice Chang

On the Chern-Gauss-Bonnet integral for conformal metrics on R4

where χ(M) is the Euler number ofM . Later, Huber [Hu] extended this inequality to metrics with much weaker regularity. More importantly, he proved that such a surface M is conformally equivalent to a closed surface with finitely many punctures. The deficit in formula (1.1) has an interpretation as an isoperimetric constant. On a complete and open surface with Gaussian curvature absolutely integrable, one may represent each end conformally as R2 \K for some compact set K . We consider the isoperimetric ratio

Regularity of a fourth order nonlinear PDE with critical exponent

In this paper we demonstrate the regularity of minimizers for a variational problem, a special case of which arises in spectral theory and conformal geometry. The associated Euler-Lagrange equation is fourth order semilinear; the leading term is the bilaplacian, and lower order terms appear at critical powers.

On a class of locally conformally flat manifolds

An important class of conformally flat structures are those derived from Kleinian groups. As an application of the Yamabe equation, Schoen and Yau (1986) proved that if M n ,g) is a smooth compact locally conformally flat manifold on which the Yamabe operator is positive, then the developing map of the holonomy cover of M is injective and the complement of its image in S n (the limit set) has Hausdorff dimension bounded by (n−2)/2. There are a family of fully nonlinear equations which prescribes the symmetric functions of the eigenvalues of the Weyl Schouten tensor A=(1/(n−2))(Rc−(R/(n−2))g), and a family of conformally covariant operators which prescribes higher-order curvature invariants. The positivity of such operators on a Kleinian group gives further control of the size of the limit set. In this paper, we carry out this analysis for the σ 2 (A) equation as well as for the fourth-order Paneitz equation. In particular, under suitable additional bounds on the Ricci tensor, we show that the limit set has Hausdorff dimension bounded by (n−4)/2 if either of these two equations are positive. As a consequence, we extend the range of indices for which the homotopy groups vanish, that is, π i (M)=0 for 2 ≤ i ≤ [n/2]+1.

Sobolev trace inequalities of order four

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean 4-balls.