Rigidit\'e conforme des h\'emisph\`eres S^4_+ et S^6_+

TL;DR

This paper proves a rigidity result for four and six-dimensional locally conformally flat manifolds with boundary, showing they are conformally equivalent to standard hemispheres under certain topological and geometric conditions.

Contribution

It establishes a new conformal rigidity theorem for hemispheres in dimensions 4 and 6 with specific topological and curvature assumptions.

Findings

Manifolds with Euler characteristic 1 and positive Yamabe invariant are conformally isometric to standard hemispheres.

The result confirms a special case of the Min-Oo conjecture for these manifolds.

Provides a new rigidity criterion based on topological and conformal invariants.

Abstract

Let (M,g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its Yamabe invariant is positive, then (M,g) is conformally isometric to the standard hemisphere. As an application and using a result of Hang-Wang, we prove a rigidity result for these hemispheres regarding the Min-Oo conjecture.