Sun-Yung A. Chang

Some weighted norm inequalities concerning the Schrödinger operators.

A method and an arrangement for operating a supersonic tuyere having a convergent and a divergent passage portion separated by a neck in such a manner so as to produce in the divergent passage portion a shock wave to atomize a combustible material fed in the region of the neck into the tuyere. The shock wave is produced by more or less restricting the open cross section of the neck, preferably by adjusting the axial position of a solid body in the convergent portion of the tuyere with respect to the neck, to maintain the pressure of a combustion sustaining material fed into the tuyere upstream of the convergene portion, during variation of the fed amount of such material within predetermined limits, at such a magnitude to assure production of a shock wave at the divergent portion.

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

Products of Toeplitz operators

A sufficient condition is found for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator. The condition, which comprehends all previously known sufficient conditions, is shown to be necessary under additional hypotheses. The question whether the condition is necessary in general is left open.

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

Isospectral conformal metrics on 3-manifolds

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

A Class of Variational Functionals in Conformal Geometry

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).