[Paper Review] On the volume functional of compact manifolds with boundary with harmonic Weyl tensor

H. Baltazar, R. Batista+1 more|arXiv (Cornell University)|Oct 17, 2017

Geometric Analysis and Curvature Flows10 references4 citations

TL;DR

This paper classifies critical metrics of the volume functional on compact Riemannian manifolds with boundary and harmonic Weyl tensor, proving that such metrics on simply connected manifolds with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form. The key contribution is establishing the equivalence between Bach-flatness and harmonic Weyl tensor for Miao-Tam critical metrics, extending prior results in the locally conformally flat case.

ABSTRACT

One of the main aims of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ and with harmonic Weyl tensor, which improves the corresponding classification for complete locally conformally flat case, due to Miao and Tam [18]. In particular, we prove that a critical metric with harmonic Weyl tensor on a simply connected compact manifold with boundary isometric to a standard sphere $\mathbb{S}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form $\Bbb{R}^n,$ $\Bbb{H}^n$ and $\Bbb{S}^n.$ In order to achieve our goal, firstly we shall conclude the classification of such critical metrics under the Bach-flat assumption and then we will prove that both geometric conditions are indeed equivalent.

Motivation & Objective

• To classify critical metrics of the volume functional on compact manifolds with boundary under the condition of harmonic Weyl tensor.
• To extend previous classification results in the locally conformally flat case by Miao and Tam to the broader setting of harmonic Weyl tensor.
• To prove that Bach-flatness and harmonic Weyl tensor are equivalent conditions for Miao-Tam critical metrics.
• To show that a simply connected, compact Miao-Tam critical metric with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form.
• To establish a complete classification of such critical metrics under the harmonic Weyl tensor condition.

Proposed method

• The authors first classify Miao-Tam critical metrics under the Bach-flat assumption using geometric analysis and divergence identities.
• They prove that a Miao-Tam critical metric with harmonic Weyl tensor must be Bach-flat by analyzing the divergence of a tensor field involving the Weyl tensor and the gradient of the defining function f.
• The proof relies on constructing a neighborhood Vε near the boundary where f vanishes, and showing that the Weyl tensor contraction with Ricci curvature vanishes in this region.
• Using Stokes’ theorem and the divergence identity, they deduce that the norm of the Weyl tensor component Wν vanishes on Vε, and by real analyticity, globally on the manifold.
• They apply a tensor identity involving Tijk and the Weyl tensor to conclude that the tensor T vanishes, which implies the manifold is Bach-flat.
• The equivalence between harmonic Weyl tensor and Bach-flatness is established via the vanishing of the Cotton tensor and the structure of the Weyl tensor under constant scalar curvature.

Experimental results

Research questions

• RQ1 Under what conditions is a Miao-Tam critical metric on a compact manifold with boundary isometric to a geodesic ball in a space form?
• RQ2 Is the condition of harmonic Weyl tensor equivalent to Bach-flatness for Miao-Tam critical metrics?
• RQ3 Can the classification of critical metrics in the locally conformally flat case be extended to the harmonic Weyl tensor setting?
• RQ4 What geometric structure do simply connected compact Miao-Tam critical metrics with boundary isometric to S^{n-1} admit under harmonic Weyl tensor?
• RQ5 Does the vanishing of the Weyl tensor component Wν on a neighborhood of the boundary imply global vanishing of the Weyl tensor?

Key findings

• A Miao-Tam critical metric with harmonic Weyl tensor on a simply connected compact manifold with boundary isometric to S^{n-1} is isometric to a geodesic ball in a simply connected space form R^n, H^n, or S^n.
• The harmonic Weyl tensor condition implies that the manifold is Bach-flat, and vice versa, establishing the equivalence between harmonic Weyl tensor and Bach-flatness for Miao-Tam critical metrics.
• The Weyl tensor component Wν vanishes identically on a neighborhood of the boundary, and by real analyticity, globally on the manifold.
• The divergence identity involving |Wν|² and ∇f leads to the conclusion that |Wν|² ≡ 0 on the manifold, implying the Weyl tensor vanishes in the direction of ∇f.
• The tensor Tijk vanishes identically, which implies that the manifold is Bach-flat, completing the proof of the equivalence.
• The classification result extends previous results in the locally conformally flat case, providing a complete classification under the harmonic Weyl tensor condition.