3
$\begingroup$

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?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.