Rigidity of bordered polyhedral surfaces

TL;DR

This paper studies the rigidity properties of bordered polyhedral surfaces, demonstrating they are uniquely determined by boundary data and interior curvatures, and reestablishes classical congruence results for cyclic polygons.

Contribution

It introduces a variational approach to prove rigidity of bordered polyhedral surfaces and provides a new proof of classical cyclic polygon congruence results.

Findings

Bordered polyhedral surfaces are determined by boundary values and interior curvatures.

Reproves classical congruence of Euclidean and hyperbolic cyclic polygons with equal side lengths.

Uses variational principles to establish rigidity results.

Abstract

This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary, we reprove the classical result that two Euclidean cyclic polygons (or hyperbolic cyclic polygons) are congruent if the lengths of their sides are equal.