Bounding immersed curves

The extending surfaces of immersions into surfaces

“…The extensions related to different normal numberings are inequivalent ( [8,Lemma 5.3.2]). If Σ is a 2-sphere or a closed oriented surface of genus greater than one, [11] showed the maximum image of ψ F has an upper bound if f a filling immersed curve, and solved Question 3 (i) (the existence problem) in these cases.…”

“…The topology type of an immersed surface bounded by the immersed circle is determined by the normal numbering given by it (note that some changes need to be adopted for 3-dimensional cases). This basically follows from [11,Lemma 3]. In the remainder of this paper, we state with fixing a normal numbering of the immersed circle, and we will not discuss the topology types specially.…”

“…In [11], D i (f, ψ) is denoted by S i . The definition of Gaussian circles basically follows from [5, Section 2].…”

“…(i) (H.Hopf and R.Thom) When does it extend to an immersed surface? (see [11]) (ii) Fix a nonnegative integer n, how many inequivalent extension F are there such that min p∈Σ #(F −1 (p)) = n?…”

“…D. Frisch has claimed to solve Question 3 (ii) ( [8]). Question 3 (i) was solved by McIntyre (in the cases of 2-sphere and closed oriented surfaces of genus greater than 1) ( [11]). There are some other works related to these questions, for example, D.Calegari discussed the rationally bounding ( [3]).…”