Desingularization of branch points of minimal surfaces in R4\mathbb{R}^4 (II)

We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$ is a topological embedding and thus defines a knot in a sphere/cylinder around the branch point, the data of the double points of the $F_t$'s give us a braid representation of this knot as a product of bands

Milnor numbers for 2-surfaces in 4-manifolds

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)

Desingularization of branch points of minimal disks in R4\mathbb{R}^4

We deform a minimal disk in $\mathbb{R}^4$ with a branch point into symplectic minimally immersed disks with only transverse double points