# Embeddedness and homology of a limit of minimal surfaces

Consider the following theorem, proved in this paper:

Theorem (Theorem 6.1). Suppose we have a sequence $$(\Sigma_j, \partial \Sigma_j) \subset (M, \partial M)$$ of immersed free boundary minimal $$k$$-dimensional submanifolds, where $$1 \leq k \leq n$$, with uniformly bounded area and second fundamental form. Then, after passing to a subsequence, $$(\Sigma_j, \partial \Sigma_j)$$ converges smoothly and locally uniformly to $$(\Sigma, \partial \Sigma) \subset (M, \partial M)$$, which is a smooth immersed free boundary minimal $$k$$-dimensional submanifold.

My questions are:

Question 1: If we assume additionally that all the submanifolds in the sequence are embedded, is it true that the limit surface is also embedded?

Question 2: If $$M$$ has dimension $$3$$ and $$(\Sigma_j, \partial \Sigma_j)$$ is a sequence of compact, connected, oriented and properly embedded free boundary minimal surfaces that converges as in the theorem to $$(\Sigma, \partial \Sigma)$$, is it true that there exists $$N \geq 1$$ such that $$[\Sigma_j] = [\Sigma] \in H_2(M, \partial M; \mathbb{Z})$$ for all $$j \geq N$$?

Question 3: Let $$M$$ be compact, connected and oriented of dimension $$3$$ and $$(\Sigma_j, \partial \Sigma_j)$$ be a sequence of compact, connected, oriented and properly embedded free boundary minimal surfaces that converges as in the theorem to $$(\Sigma, \partial \Sigma)$$. If $$[\Sigma_j] \neq 0 \in H_2(M, \partial M; \mathbb{Z})$$ for every $$j \geq 1$$, is it true that $$[\Sigma] \neq 0$$?

• What kind of limits do you mean? For instance, on a Moebius band with flat metric you can have a sequence of embeded closed geodesics converging (as maps) to a non-embedded one. – Misha Feb 28 '20 at 3:12
• I admit this would be my 0th question. The authors just say what I transcribed. – Eduardo Longa Feb 28 '20 at 3:19
• (I think I am partly repeating what Misha said but:) The answer to Question 1. I think is trivially "no". These surfaces are arbitrary codimension so e.g. two disjoint lines can meet in the limit. For the second question I'm not sure. Does smooth convergence of the pair $(\Sigma_j, \partial \Sigma_j)$ mean that the boundaries converge smoothly in $\partial M$? – T_M Feb 28 '20 at 8:13
• Yes, the boundaries converge smoothly. – Eduardo Longa Feb 29 '20 at 10:40