# SGS XIX: Monday, December 12, 2011 University of Alabama at Birmingham

The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:

The organizers are: Vladimir Oliker (Emory), Mohammad Ghomi and John McCuan (Georgia Tech), Fernando Schwartz (UTK), Junfang Li (UAB), and Gilbert Weinstein (UAB).

Thanks to an NSF grant, we have funds to support participants, particularly students and recent Ph.D. recipients. We encourage women and minorities to apply. To apply please write to us.

### 9:00 AM - Morning Session - Hill University Center (HUC 411)

9:00 AM Alexander Koldobsky (Univ. of Missouri-Columbia)
Stability in volume comparison problems

Abstract: We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of volume. To prove this generalization, we establish stability in the affirmative part of Zvavitch's extension of the Busemann-Petty problem to arbitrary measures. Then we discuss different stability estimates in similar volume comparison problems.

10:00 AM Yanyan Li (Rutgers Univ.)
Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities

Abstract: We present some results on the behavior of positive solutions in a punctured ball of general second order fully nonlinear conformally invariant elliptic equations. We prove that such a solution, near the puncture, is asymptotic to some radial solution of the same equation in the punctured Euclidean space. This is a joint work with Z.C. Han and E. Teixeira.

11:00 AM Joel Spruck (Johns Hopkins Univ.)
Complete strictly locally convex spacelike hypersurfaces in De Sitter space

### 2:00 PM - Afternoon Session - Hill University Center (HUC 411)

2:00 PM Jeff Viaclovsky ( University of Wisconsin)
Obstruction-Flat Asymptotically Locally Euclidean Metrics

Abstract: Given an even dimensional Riemannian manifold (M^{n},g) with n\geq 4, it was shown in the work of Charles Fefferman and Robin Graham on conformal invariants the existence of a non-trivial 2-tensor which involves n derivatives of the metric, and arises as the first variation of a conformally invariant functional and vanishes for metrics that are conformally Einstein. This tensor is called the Ambient Obstruction tensor and is a higher dimensional generalization of the Bach tensor in dimension 4. We show that any asymptotically locally Euclidean (ALE) metric which is obstruction flat and scalar-flat must be ALE of a certain optimal order using a technique developed by Cheeger and Tian for Ricci-flat metrics. We also prove a singularity removal theorem for obstruction-flat metrics with isolated C^{0}-orbifold singularities. In addition, we show that our methods apply to more general systems. This is joint work with Antonio Ache.

3:00 PM Jiaping Wang (Univ. of Minnesota)
Geometry and topology of Ricci solitons

Abstract: The talk concerns some recent joint work with Ovidiu Munteanu on the geometry and topology of the soliton solutions to the Ricci flows. The issues to be discussed include the rate of volume growth and the connectedness at infinity.

### 4:00 PM - 5:30 PM Poster Session - TBA

Authored by:

Junfang Li and Gilbert Weinstein