Rigidity results with curvature conditions from Lichnerowicz Laplacian and applications

TL;DR

This paper extends Bochner-type rigidity theorems to non-compact Riemannian manifolds using the Lichnerowicz Laplacian, establishing new vanishing and rigidity results for curvature tensors under integral bounds and decay conditions.

Contribution

It generalizes classical rigidity results to non-compact settings via $L^Q$-harmonic tensors and applies these to various geometric contexts including hypersurfaces and ALE spaces.

Findings

Vanishing of curvature tensors under integral curvature bounds.

Rigidity results for Ricci-flat and Einstein manifolds.

Obstructions to nontrivial solutions on ALE 4-manifolds.

Abstract

The Bochner technique is a classical tool in global differential geometry for proving vanishing and rigidity results by exploiting curvature conditions. Building on recent extensions of this method to complete non-compact settings by Petersen and Wink, we investigate $L^Q$-harmonic tensors with $Q>1$ governed by the Lichnerowicz Laplacian on complete Riemannian manifolds. Our results generalize Bochner-type theorems to the non-compact realm, revealing new geometric rigidity phenomena not visible in compact cases. We establish vanishing theorems under integral curvature bounds and weighted Poincar\'e inequalities, and derive conditions under which harmonic tensors must vanish. In particular, we show that on Ricci-flat or Einstein manifolds, curvature tensors such as $\mathrm{Rm}$ or the Weyl tensor $W$ vanish identically under natural $L^Q$-integrability and positivity assumptions on the curvature operator. These results imply strong rigidity: flatness in the Ricci-flat case and constant sectional curvature in the Einstein case. We further apply our framework to closed hypersurfaces in space forms and derive vanishing results for intermediate Betti numbers under positivity conditions on the second fundamental form. Finally, we extend our theory to asymptotically locally Euclidean (ALE) spaces, proving that harmonic Weyl tensors and Codazzi tensors must vanish under curvature positivity and decay conditions. Our analysis also links these results to ADM mass rigidity, establishing new obstructions to nontrivial decaying solutions on ALE 4-manifolds.