The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:
Abstract: While most 4-manifolds do not admit Einstein metrics, a direct variational approach to the problem leads to natural decompositions of many 4-manifolds into Einstein and collapsed pieces. I will discuss explicit constructions of minimizing sequences for two natural quadratic curvature functionals, and relate this picture to the geometrization of manifolds via Ricci flow.
Abstract: The classical Liouville theorem says that a positive entire harmonic function must be a constant. We give a fully nonlinear version of it. This extension enables us to establish local gradient estimates of solutions to general conformally invariant fully nonlinear elliptic equations of second order. This talk will start from a proof of the classical Liouville theorem using only the comparison principle and the invariance of harmonicity under Mobius transformations and scalar multiplications. We will then outline the proof of the comparison principle used in establishing the new Liouville theorem. Finally we outline the proof of the gradient estimates via the Liouville theorem.
Abstract: I will discuss some Liouville-type uniqueness theorems for orthogonal complex structures on domains in R4, and give a classification of quadrics in CP3 under the action of the conformal group. These quadrics give rise to some interesting orthogonal complex structures defined on the complement of an unknotted solid torus in R4. I will also discuss some results on the maximal domain of definition for other algebraic orthogonal complex structures. This is joint work with Simon Salamon.
Abstract: The existence and uniqueness of an optimal Sobolev norm for a function is shown to be essentially equivalent to the existence and uniqueness of the solution to the Lp Minkowski problem for even measures. The former is established using the latter. This leads to new affine analytic inequalities as well as a new proof of a previously established affine Lp Sobolev inequality.
Abstract: In this talk we explain how to derive the short time asymptotics of the K-Energy map along certain "algebraic potentials" from the long time asymptotics of a certain line bundle on the Hilbert Scheme. One consequence is that properness of the K-Energy implies the K-Stability (in Donaldson's sense) of the variety. Another is that Donaldson's weight is NOT the only contribution to the asymptotics of the K-Energy. This is a joint work with Gang Tian.
Abstract: I will discuss a proof of nonlinear stability of Lorentz cones over Riemannian negative Einstein spaces M. In spacetime dimensions D ≤ 11, this work allows one to construct families of vacuum spacetimes with quiescent singularity and asymptotically Friedman behavior in the expanding direction. This talk is based on joint work with Vince Moncrief.