Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem in Riemannian Geometry

Abstract: We generalize to the setting of compact Riemannian manifolds a recent result of Barkatou on quadrature surfaces. Following the geometric and variational framework recently developed by Djité and Seck, we formulate the quadrature surface free boundary problem as a shape optimization problem on a compact Riemannian manifold. Using the Riemannian $RC$-GNP condition introduced in \cite{DjiteSeck2026} and the stability results established therein, we prove that the quadrature surface problem $QS(f,k)$ admits a solution strictly containing the totally convex hull $C$ of the support of $f$ if and only if the following integral condition holds: [ \int_C f(x)\,dv(g) > k |\partial C|_g, ] where $dv(g)$ is the Riemannian volume element and $|\partial C|_g$ is the perimeter of $C$ with respect to the metric $g$. This work extends the results of Barkatou et al. (2005) by replacing the Euclidean space with a Riemannian manifold. We also provide explicit examples on the round sphere where the condition can be verified explicitly. This paper complements the recent works \cite{DjiteSeck2026} and \cite{DjiteSeck2026b} by establishing a sharp necessary and sufficient condition in the spirit of the Euclidean result of Barkatou.