Milnor numbers for 2-surfaces in 4-manifolds

TL;DR

This paper investigates the behavior of Milnor numbers associated with 2-surfaces in 4-manifolds, especially their convergence and relation to bubbling phenomena in the Grassmannian, under certain boundedness conditions.

Contribution

It introduces conditions under which tangent and normal Milnor numbers are well-defined and related, and connects these to bubbling-off phenomena in the Grassmannian.

Findings

 and  are well-defined under specific assumptions.

 is always smaller than  when both exist.

The paper relates Milnor numbers to bubbling-off in the Grassmannian.

Abstract

In this paper (S_n) is a sequence of surfaces immersed in a 4-manifold which converges to a branched surface S_0. Up to sign, \mu^T_p (resp. \mu^N_p) will denote the amount of curvature of the tangent bundles TS_n (resp. the normal bundles NS_n) which concentrates around a singular point p of S_0 when n goes to infinity. By a slight abuse of notation, we call \mu_p^T (resp. \mu_p^N) the tangent (resp. normal) Milnor number of S_n at p. These numbers are not always well-defined; we discuss assumptions under which, if \mu^T exists, then \mu^N also exists and is smaller than -\mu^T . When the second fundamental forms of the S_n's have a common L^2 bound, we relate \mu^T and \mu^N to a bubbling-off in the Grassmannian G_2^+(M).