UBC Mathematics Department
Abstract: One can associate two topological numbers to any surface Sigma in a 4-manifold M, namely, the euler number and the self-intersection number of the surface. If M is complex and Sigma is holomorphic, the sum a_M(Sigma) of those two topological numbers is the same as c_1(M) (Sigma), where c_1(M) denotes the first Chern class of M. In this talk, we discuss recent results on the topological number a_M(Sigma) for any minimal surface Sigma in a general Riemannian 4-manifold. In particular, we will show that a_M is always negative under certain pinched curvature conditions. Finally, we will discuss a theorem for minimal surfaces in symplectic 4-manifolds and its consequences.