Paul C. Yang

An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature

We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.

The scalar curvature equation on 2- and 3-spheres

In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.

Extremal metrics of zeta function determinants on 4-manifolds

In conformal geometry, the Sobolev inequality at a critical exponent has received much attention. In particular, the determination of the best constants has played a crucial role in the Yamabe problem. In dimension two the analogous problem deals with the Moser-Trudinger inequality: on a compact Riemann surface M2, there exists a constant c = c(M) so that f e47w2 < c(M) if f IVw12 < 1 and f w = 0. The connection of this inequality with geometry comes through the zeta functional determinant of the Laplacian as defined by Ray-Singer: for a Riemannian metric g, let 0 < A1 < A2 < ... be the spectrum

Estimates and extremals for zeta function determinants on four-manifolds

AbstractLetA be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose thatA is formally self-adjoint and has positive definite leading symbol. For example,A could be the conformal Laplacian (Yamabe operator)L, or the square of the Dirac operator. Within the conformal class

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

\left\{ {g = e^{2w} g_0 |w \in C^\infty (M)} \right\}

A PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS

of an Einstein, locally symmetric “background” metricgo on a compact four-manifoldM, we use an exponential Sobolev inequality of Adams to show that bounds on the functional determinant ofA and the volume ofg imply bounds on theW2,2 norm of the conformal factorw, provided that a certain conformally invariant geometric constantk=k(M, goA) is strictly less than 32π2. We show for the operatorsL and that indeedk < 32π2 except when (M, go) is the standard sphere or a hyperbolic space form. On the sphere, a centering argument allows us to obtain a bound of the same type, despite the fact thatk is exactly equal to 32π2 in this case. Finally, we use an inequality of Beckner to show that in the conformal class of the standard four-sphere, the determinant ofL or of is extremized exactly at the standard metric and its images under the conformal transformation groupO(5,1).

Embeddability for 3-dimensional Cauchy–Riemann manifolds and CR Yamabe invariants

Theorem. Let g= uj go be a sequence of conformal metrics satisfying the following conditions. (i) Vol(M, gj) = ao for some positive constant ao. (ii) f R2(gj) + lp(gj)12 dVj A > 0; i.e., for each q defined on M, we have

A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group

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