[Christian Pappas][1] gave a Morse-theoretic method for constructing all extensions of a codimension 1 immersion $f:\partial N\to W$ to an immersion $F:N\to W$ with $F|_{\partial N}=f$.


  [1]: http://www.ams.org/journals/tran/1996-348-08/S0002-9947-96-01572-3/