Jeffrey S. Case

THE NONEXISTENCE OF QUASI-EINSTEIN METRICS

m d f d fD 0 by studying the associated PDE 1 f fC m exp.2 f=m/D 0 for 0. By developing a gradient estimate for f , we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ricci flat warped products with fibers that have nonpositive Einstein constant. We also show that for nontrivial steady gradient Ricci solitons, the quantity RCjr fj 2 is a positive constant.

Singularity theorems and the Lorentzian splitting theorem for the Bakry–Emery–Ricci tensor

Abstract We consider the Hawking–Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry–Emery–Ricci curvature Ric f m in timelike directions. We prove that they still hold when m is finite, and when m is infinite, they hold under the additional assumption that f is bounded from above.

A PANEITZ-TYPE OPERATOR FOR CR PLURIHARMONIC FUNCTIONS

We introduce a fourth order CR invariant operator on pluriharmonic functions on a three-dimensional CR manifold, generalizing to the abstract setting the operator discovered by Branson, Fontana and Morpurgo. For a distinguished class of contact forms, all of which have vanishing Hirachi-Q curvature, these operators determine a new scalar invariant with properties analogous to the usual Q-curvature. We discuss how these are similar to the (conformal) Paneitz operator and Q-curvature of a four-manifold, and describe its relation to some problems for three-dimensional CR manifolds.

Some energy inequalities involving fractional GJMS operators

Under a spectral assumption on the Laplacian of a Poincare--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order

Smooth metric measure spaces, quasi-Einstein metrics, and tractors

2\gamma\in(0,2)

The P'-operator, the Q'-curvature, and the CR tractor calculus

or

Extremal metrics for the Q′-curvature in three dimensions

2\gamma\in(2,4)

A notion of the weighted σk-curvature for manifolds with density☆

and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption is necessary and sufficient for such an inequality to hold. We prove the energy inequalities by introducing conformally covariant boundary operators associated to the weighted conformal Laplacian and weighted Paneitz operator which generalize the Robin operator. As an application, we establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.

On the Lichnerowicz conjecture for CR manifolds with mixed signature

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.

Rigidity of quasi-Einstein metrics ☆

We establish an algorithm which computes formulae for the CR GJMS operators, the P0-operator, and the Q0-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the P0-operator, and shows that the Q0-curvature is constant, with the constant explicitly given in terms of the Webster scalar curvature. We also use our algorithm to derive local formulae for the P0-operator and Q0-curvature of a five-dimensional pseudo-Einstein manifold. Comparison with Marugame’s formulation of the Burns-Epstein invariant as the integral of a pseudohermitian invariant yields new insights into the class of local pseudohermitian invariants for which the total integral is independent of the choice of pseudo-Einstein contact form.