# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Gluing Monopoles

A canonical method is established to glue Seiberg-Witten monopoles over a closed 4-manifold split along a 3-dimensional submanifold. The method is quite generic and only requires mild conditions.

##### On the $\sigma_2$-scalar curvature and its application

In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry. As application, we prove that, if a compact 3-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int \sigma_2\ge 0$, then topologically $M$ is a quotient of sphere.

##### Compactness of conformally compact Einstein manifolds of dimension 3+1

##### The Structure of Shrinking Solitons

We discuss a handful of structure theorems for shrinking solitons with bounded curvature. In particular we prove a priori injectivity radi.

We discuss a handful of structure theorems for shrinking solitons, in particular let $(M,g,X)$ be a complete shrinking soliton with bounded curvature, then there exists $k\gt 0$ and a smooth function $f$ such that $(M,g,X)$ is k-noncollapsed and $(M,g,f)$ is a gradient shrinking soliton, generalizing results from the compact case. If $M$ is a noncompact four dimensional shrinking soliton with nonnegative curvature then up to finite quotient it is isometric to $R^4$,$R\times S^3$ or $S^2\times R^2$. Finally we show that the singularity dilation of a Type I singularity on a Ricci Flow is a shrinking soliton.

##### No mass drop for mean curvature flow of mean convex hypersurfaces

A possible evolution of a compact hypersurface in $R^{n+1}$ by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow. As a further application of the techniques used above we give a new variational formulation for mean curvature flow of mean convex hypersurfaces.

##### A priori estimates for special Lagrangian equations

We discuss recent a priori interior Hessian estimates for solutions of the special Lagrangian equation, when the equation has phase at least a certain value, or when the solution is convex. These equations include the sigma-2 equation in dimension three. The gradient graph of any solution is a minimizing Lagrangian surface. While Heinz showed in the 50's that similar estimates hold for the sigma-2 (Monge-Ampere) equation in dimension two, Pogorelov showed that such estimates cannot hold for the sigma-3 (Monge-Ampere) equation in dimension three. This is joint work with Yu Yuan, partly also with Jingyi Chen.

##### The decomposition of global conformal invariants: On a conjecture of Deser and Schwimmer

Global conformal invariants are integrals of geometric scalars which remain invariant under conformal changes of the underlying metric. I will discuss (parts of) my recent proof of a conjecture of Deser and Schwimmer, which states that any such global invariant can be decomposed into standard "building blocks" of three types. (This lecture will be introductory and self-contained; it will be followed by three other technical lectures in the Wedensday seminars where I will present more ideas from the proof).

##### Hamiltonian formulation of general relativity and quasilocal mass

Isometric embeddings of surfaces into the Minkowski space are used as references to derive a quasilocal mass expression from the Hamiltonian formulation of Einstein's equation. This involves an existence and uniqueness theorem of isometric embeddings and a canonical choice of time gauges. We also prove the quasilocal mass is positive under the dominant energy condition and is zero for surfaces in the Minkowski space. This talk is based on joint work with Shing-Tung Yau at Harvard.

##### Optimal curvature decays on asymptotically locally Euclidean manifolds

We present a method to study curvature decays on asymptotically locally Euclidean manifolds. The method is flexible and can also be applied to elliptic systems of reaction-diffusion type.

##### Knots and Topological Growth Laws in the Faddeev Model

In this talk, I present some joint work with Fanghua Lin on the existence of knotted solitons realized as the energy-minimizing configurations in the Faddeev field-theoretical model and the associated universal topological growth laws which relate the knot energy to knot topological charge defined by the Hopf invariant.