Lectures on Stability and Constant Scalar Curvature

K-stability of constant scalar curvature Kähler manifolds

We show that a polarised manifold with a constant scalar curvature Kähler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.

Constant scalar curvature metrics on toric surfaces

Hypersurfaces with Constant Scalar Curvature

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Har tman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of com...

Lectures on Mean Curvature Flow and Stability (MAT 1063 HS)

The mean curvature flow (MCF) arises material science and condensed matter physics and has been recently successfully applied to topological classification of surfaces and submanifolds. It is closely related to the Ricci and inverse mean curvature flow. The most interesting aspect of the mean curvature flow is formation of singularities, which is the main theme of these lectures. In dealing wit...

Constant scalar curvature metrics with isolated singularities

We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by...