2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127815We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that is uniquely determined by their curvatures.28 pages, 7 figures Minor changes in the introduction, Subsection 4.3 added, references addedDifferential GeometryMetric Geometry52B10, 53C45; 52C25text