Let $X$ be a compact 2-dimensional Alexandrov space with curvature at least $\kappa$. My question is somewhat vague.

What is known about the boundary of $X$?

For example:

1) Is the boundary homeomorphic to a finite union of circles?

2) Is it possible to say that in certain generalized sense the boundary has non-negative second fundamental form (in other words, geodesic curvature)?

3) Assume that the diameter of $X$ is at most $D$. It is true that the number of connected components of the boundary is at most a constant depending on $\kappa, D$ only? Does the 1-dimensional Hausdorff measure of the boundary have such an upper bound?

Any other information and references will be helpful.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.