University of Tennessee Knoxville

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

**The National Science Foundation****The University of Alabama at Birmingham****The Georga Institute of Technology****Emory University****University of Tennessee Knoxville**

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

We have two recent features:

- A poster session. Please contact us if you would like to present a poster.
- An afternoon public lecture by our plenary speaker, Hugh Bray. Click here for the poster for this event.

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.

8:30 AM -
**Catherine Williams (Columbia U)**

Good and bad asymptotic behavior of marginally trapped tubes

Good and bad asymptotic behavior of marginally trapped tubes

**Abstract:** In general relativity, marginally trapped tubes are hypersurfaces of spacetime that are foliated by apparent horizons. When they exist, these hypersurfaces generally lie inside of black holes and (roughly speaking) form a boundary between the regions of weak and strong gravitational fields there. The expectation in the physics community is that marginally trapped tubes will form during gravitational collapse, in particular, that any physically reasonable black hole will contain one, and that they will asymptotically approach the black hole's event horizon. In this talk, we will discuss the extent to which this expectation has been proven true. We will give an overview of the various settings in which the `good' asymptotic behavior is known to hold but in addition describe a recently constructed example in which it does not.

**Abstract:** Beginning with a geometric motivation for dark matter going back to the axioms of general relativity, we show how scalar field dark matter, which naturally forms dark matter density waves due to its wave nature, may cause the observed barred spiral pattern density waves in many disk galaxies and triaxial shapes with plausible brightness profiles in many elliptical galaxies. If correct, this would provide a unified explanation for spirals and bars in spiral galaxies and for the brightness profiles of elliptical galaxies. We compare the results of preliminary computer simulations with photos of actual galaxies.

10:30 AM -
**Simon Brendle (Stanford U)**

Counterexamples to Min-Oo's Conjecture

Counterexamples to Min-Oo's Conjecture

**Abstract:** Consider a compact Riemannian manifold *M* of dimension *n* whose boundary ∂*M* is totally geodesic and is isometric to the standard sphere *S*^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of *M* is at least *n*(*n*-1), then *M* is isometric to the hemisphere *S*^{n}_{+} equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases.

I will present joint work with F.C. Marques and A. Neves which shows that Min-Oo's conjecture fails in dimension *n* ≥ 3.

2:00 PM -
**Spyros Alexakis (U of Toronto)**

Renormalized area and bubbling phenomena for complete minimal surfaces in hyperbolic space.

Renormalized area and bubbling phenomena for complete minimal surfaces in hyperbolic space.

**Abstract:** We consider the space of complete minimal surfaces in hyperbolic space with an asymptotic boundary at infinity. We consider the renormalized area functional on this space (introduced by Graham and Witten) and show it to be analogous to the Willmore energy for these complete surfaces. We then study the compactness of the space of such minimal surfaces with bounded energy. We show that loss of compactness can occur due to bubbling near infinity. This seems to be the first study of bubbling phenomena in the context of surfaces with a free boundary. Joint work with R. Mazzeo.

3:00 PM -
**Alessio Figalli (U of Texas at Austin)**

Closing Aubry sets

Closing Aubry sets

**Abstract:** Given a Hamiltonian *H* on a compact manifold, the Mane conjecture in *C ^{k}* topology states that, for a generic potential