Morse index of a closed minimal surface with a small disc removed

Consider the following observation:

Let $$c_1$$ be a geodesic on the unit round sphere $$S^2$$ with length $$2\pi-\epsilon$$, where $$\epsilon$$ is sufficiently small. Then $$c_1$$ has Morse index one because it contains a pair of conjugate points. If we close the gap of $$c_1$$ and consider the great circle $$c_2$$, it can be shown that $$c_2$$ also has index one.

Is this a mere coincidence? Is there any situation that generalizes this phenomenon, say, to the case of minimal surfaces? e.g. I think that if you consider the totally geodesic $$S^2$$ in $$S^3$$, when you remove a small disc from it, its Morse index remains one. Is there any theorem that tells us when we can cut out a ball from an embedded closed minimal submanifold and have the Morse index unchanged?