Seminar

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Given an Alexandrov space with curvature bounded below by $\kappa$, dimension $m$ and radius $r$, one may ask how large the boundary $\partial X$ can be. In the case where $\kappa = 1$ and $r = \pi/2$, this is known as Lytchak’s problem and was answered by Petrunin, who showed the sharp upper bound $\mathcal{H}^{m-1}(\partial X) \leq \mathcal{H}^{m-1}(\mathbb{S}^{m-1})$. Rigidity was later analyzed by Grove-Petersen, who showed that in the case of equality $X$ must be a hemisphere or the intersection of two hemispheres. In this talk, I will address both the bound and rigidity for arbitrary $\kappa$ and $r$. A main technical ingredient is the gradient flow of semiconcave functions developed by Perelman-Petrunin. Time permitting, I will also discuss some related open questions under the setting of Ricci curvature lower bounds. This is joint work with Vitali Kapovitch.