Rigidity Conditions for the Boundaries of Submanifolds in a Riemannian Manifold
A. P. Kopylov, Михаил Вячеславович Коробков · arXiv (Cornell University) · 2014
Developing A.D. Aleksandrov's ideas, the first-named author of this article proposed the following approach to study of rigidity problems for the boundary of a $C^0$-submanifold in a smooth Riemannian manifold: Let $Y_1$ be a 2-dimensional compact connected $C^0$-submanifold with nonempty boundary in a 2-dimensional smooth connected Riemannian manifold $(X,g)$ without boundary satisfying the condition $$ρ_{Y_1}(x,y) = \liminf_{x' \to x, y' \to y, x',y' \in \mathop{\rm Int} Y_1} \{[l(γ_{x',y',\mathop{\rm Int} Y_1})]\} 2$, is also considered.