Second variation of superminimal surfaces into self-dual Einstein four-manifolds

Second Variation of Superminimal Surfaces into Self-dual Einstein Four-manifolds

The index of a compact orientable superminimal surface of a selfdual Einstein four-manifold M with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of its nullity is obtained. Applications to the cases M = S4 and M = CP2 are given, characterizing the standard Veronese immersions and their twistor deformations as those with lowest index.

Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

Let Σ be a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold (M,ω). We consider the evolution of Σ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms ω and ω that determine different orientations and Σ is symplectic with...

Four - Manifolds without Einstein

Second Variation of Energy for Minimal Surfaces in Riemannian Manifolds

This article describes a formula for second variation of energy for twodimensional parametrized minimal surfaces, in which the conformal structure on the two-dimensional domain is allowed to vary. Moreover, it shows that minimal surfaces with branch points have tangential Jacobi fields which are not visible via the standard formulae for second variation of area which is commonly used to study s...

Four-manifolds without Einstein Metrics

It is shown that there are infinitely many compact simply connected smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2χ > 3|τ |. The examples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory.