1999/12/21 by Michael T. Anderson, Anderson, Michael T. · 1 citation
Mathematics · Physics and Astronomy · #58E11 58J60 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.math/9912177
openalex publication_date 1999/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and is a critical point for the L2 norm of the full curvature R among compact perturbations of (N, g). (ii) (N, g) has non-negative scalar curvature, a free isometric S1 action, and is a critical point of the L2 norm of R among compact volume non-increasing perturbations of (N, g) with non-negative scalar curvature. In either case, (N, g) is flat. The Schwarzschild metric (on the space-like hypersurface) has an isometric S1 action and satisfies the other assumptions in (ii), showing that this result is sharp.