I am a bit confused about the statement of Theorem 1.1 in this paper by Brian White. For convenience, I will restate it here.
Theorem: Let $\Omega$ be an open subset of a Riemannian $3$-manifold. Let $g_i$ be a sequence of smooth Riemannian metrics on $\Omega$ converging smoothly to a Riemannian metric $g$. Let $M_i \subseteq \Omega$ be a sequence of properly embedded surfaces such that $M_i$ is minimal with respect to $g_i$. Suppose also that the area and the genus of $M_i$ are uniformly bounded on compact subsets of $\Omega$. Then (after passing to a subsequence) the $M_i$ converge to a smooth, properly embedded $g$-minimal surface $M$. For each connected component $\Sigma$ of $M$, either
- the convergence to $\Sigma$ is smooth with multiplicity one, or
- the convergence is smooth (with some multiplicity $> 1$) away from a discrete set $S$.
In the second case, if $\Sigma$ is two-sided, then it must be stable.
My question concerns the last sentence of the statement above. It seems very surprising to me the conclusion about the stability. What if $g$ is a metric with strictly positive Ricci curvature?
I was thinking about the following 2-dimensional situation. Imagine to have a sequence of metrics $g_i$ on the 2-sphere such that the limit metric $g$ has positive curvature. Imagine that each $g_i$ carries two closed geodesics which are converging to an unstable geodesic with multiplicity 2. See this poor quality picture:
The compactness theorem above says that one cannot construct such an example on the $3$ sphere and using minimal surfaces instead of geodesics. Can anyone give me an intuition of what is going on here?