Closed Hypersurfaces Driven by their Mean Curvature and Inner Pressure
We introduce a hyperbolic equation that describes the motion of closed hypersurfaces in some Riemannian manifold. In the case of spherical surfaces this equation can be considered as an idealised mathematical model of a moving soap bubble. The equation is derived from an action integral as an Euler-Lagrange equation. In addition to the kinetic energy this action integral contains terms for the surface tension and the inner pressure, which depends on the enclosed volume. The resulting Euler-Lagrange equation is a quasilinear degenerate hyperbolic partial differential equation of second order, which describes the motion of the surface extrinsically. We first explain basic properties of this equation and then focus on the proof of the short time existence result. We also present a continuation criterion and a stability estimate.
 

The surgery and level-set approaches to mean curvature flow
Huisken and Sinestrari have recently developed a surgery approach to mean curvature flow according to which one removes, by hand, any regions of large curvature before a singularity forms. This does not constitute a weak solution of the flow in the traditional sense since it involves a non-canonical modification of the surface at each surgery time. In this talk we discuss the estimates and parameters controlling the surgery construction, and explain the relationship between this solution and the well-known weak solution of the level set flow.

 
 

Conformal Type of Minimal Surfaces with Finite Topology 
The conformal type and asymptotic geometry of minimal ends with finite total curvature has been well known for some time. The elusive case remained (understanding an infinite total curvature end) as the typical harmonic analysis results could not be applied. Fundamental work by Colding and Minicozzi on the structure of embedded minimal surfaces with one end paved the way for many new results, including the Meeks-Rosenberg proof of the uniqueness of the helicoid. In this talk, we demonstrate how these structural results can be used to determine the conformal type and asymptotic geometry of one ended minimal surfaces with finite topology.

Area minimizing projective planes in RP³
I will present joint work with H. Bray, S. Brendle, and A. Neves. We characterize round RP³ among all metrics on this manifold with scalar curvature is bounded below by 6 by the size of the least area projective plane that they contain. The proof uses elementary differential geometry and a basic application of the Ricci flow.

 
 

Curvature contraction of convex hypersurfaces
Consider compact, convex hypersurfaces without boundary contracting by functions of their curvature. Much is known in this setting, however, I will describe some behaviour which is not well known, for certain fully nonlinear flows. Inspiration comes from initial hypersurfaces which are nonsmooth and uniformly convex, or smooth and weakly convex, but some results also carry over for smooth uniformly convex initial hypersurfaces as well. This is joint work with Ben Andrews and Zheng Yu. We show in particular that under certain flows, convexity may be lost, smoothness may be lost, flat sides or cylindrical regions may persist and hypersurfaces may even fail to contract to a point, instead collapsing to a line segment or lower dimensional disk.
 

 
 

 
 

On the distribution of eigenvalues of an invariant elliptic operator
We study the spectrum of an invariant elliptic operator on a closed G- manifold M, where G is a compact, connected Lie group acting effectively and isometrically on M. Using resolution of singularities, we determine the asymptotic distribution of the eigenvalues along the isotypic components, and relate it with the reduction of the corresponding Hamiltonian flow, proving that the equivariant spectral counting function satisfies Weyl's law, together with an estimate for the remainder.