The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:
We have two recent features:
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.
Abstract: Consider a compact Riemannian manifold M of dimension n whose boundary ∂M is totally geodesic and is isometric to the standard sphere Sn-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 Sn+ 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.
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.
Abstract: Given a Hamiltonian H on a compact manifold, the Mane conjecture in Ck topology states that, for a generic potential V (generic w.r.t. the Ck topology), the Aubry set associated to H + V is either a fixed point or a periodic orbit. In this talk we will see how, given a Hamiltonian which possesses a sufficiently smooth viscosity (sub)solution to the Hamilton-Jacobi equation, for any ε > 0 there exists a potential Vε, whose C2-norm is bounded by ε such that the Aubry set associated to H + Vε is either a fixed point or a periodic orbit. This represents a first step through the solution of the Mane Conjecture in C2 topology. Moreover, we will see how these techniques allow to solve the Mane conjecture in C1 topology. This is a joint work with Ludovic Rifford.