Resolve the boundary-area conjecture under nonnegative Ricci curvature and uniform boundary convexity

Determine whether every compact Riemannian manifold with Ric_g ≥ 0 and II_g ≥ g|∂M satisfies the sharp boundary-area inequality |∂M|_g ≤ |S^n|, and characterize the equality case as the Euclidean unit ball.

Background

The boundary-area conjecture asserts that nonnegative Ricci curvature together with the lower bound II_g ≥ g|∂M forces the boundary area not to exceed that of the unit round n-sphere, with equality expected only for the Euclidean unit ball. The paper proves a first-order obstruction to conformal counterexamples near the Euclidean ball and verifies the inequality for rotationally symmetric metrics.

These results exclude certain local conformal and rotationally symmetric mechanisms but do not settle the general inequality. The authors explicitly note that a counterexample, if one exists, would need to use a different mechanism, such as anisotropy, higher-order effects, or topology.