[Paper Review] Un th\\'eor\\`eme de la masse positive pour le probl\\`eme de Yamabe en dimension paire

Pierre Jammes|arXiv (Cornell University)|Jul 21, 2008

Geometric Analysis and Curvature Flows8 references4 citations

TL;DR

This paper presents an elementary proof of the positive mass theorem for the Yamabe problem on even-dimensional, compact, conformally flat manifolds with positive scalar curvature, using differential forms instead of spinors. It establishes that the mass term in the Green's function expansion is strictly positive unless the manifold is conformally equivalent to the standard sphere, removing the need for spin or orientability assumptions in even dimensions.

ABSTRACT

Let $(M,g)$ be a compact conformally flat manifold of dimension $n\\geq4$ with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. On spin manifolds, there is an elementary proof of this fact by Ammann and Humbert, based on a proof of Witten. Using differential forms instead of spinors, we give an elementary proof on even dimensional manifolds, without any other topological assumption.

Motivation & Objective

• To provide an elementary proof of the positive mass theorem for the Yamabe problem on even-dimensional compact conformally flat manifolds with positive scalar curvature.
• To eliminate the topological assumptions—particularly the spin condition—previously required in Witten-type proofs.
• To replace spinor-based methods with differential forms, leveraging the Weitzenböck formula for middle-degree forms in even dimensions.
• To show that the mass term $ A_P $ in the Green's function expansion is positive unless the manifold is conformally equivalent to the standard sphere.
• To establish the result without requiring orientability or other topological constraints, using only Riemannian geometry and analysis of differential forms.

Proposed method

• Use the conformal transformation induced by the Green's function of the conformal Laplacian to construct an asymptotically Euclidean metric on $ M \setminus \{P\} $.
• Work with $ \frac{n}{2} $-forms on even-dimensional manifolds, where the Hodge star operator $ * $ is conformally invariant, preserving harmonicity under conformal changes.
• Define a $ \frac{n}{2} $-form $ \varphi $ on the asymptotically Euclidean end using a pullback of a constant form from $ \mathbb{R}^n $, ensuring it is smooth and decays appropriately.
• Apply the Bochner formula and Weitzenböck identity for $ \frac{n}{2} $-forms to relate the $ L^2 $-norm of the gradient to curvature terms.
• Use integration by parts and boundary estimates on small spheres $ S_P(r) $ to relate the $ L^2 $-norm of $ \nabla \varphi $ to the mass term $ A_P $, showing $ \int |\nabla \varphi|_{\tilde{g}}^2 \, dv_{\tilde{g}} = 4n(n-1)\omega_{n-1}^2 A + o(1) $.
• Derive the inequality $ A \geq 0 $ from non-negativity of the $ L^2 $-norm of the gradient, and show $ A = 0 $ implies parallelism of all $ \frac{n}{2} $-forms, leading to flatness and hence conformal equivalence to the sphere.

Experimental results

Research questions

• RQ1 Can the positive mass theorem for the Yamabe problem be proven in even dimensions without assuming the manifold is spin?
• RQ2 Is it possible to replace spinor-based methods with differential forms in the positive mass theorem proof?
• RQ3 What role does the conformal invariance of the Hodge star on $ \frac{n}{2} $-forms play in simplifying the proof?
• RQ4 Under what conditions does the mass term $ A_P $ vanish, and what does this imply about the underlying geometry?
• RQ5 Can the proof be made elementary and topologically general, without requiring orientability or other topological restrictions?

Key findings

• The mass term $ A_P $ in the Green's function expansion of the conformal Laplacian is strictly positive for any compact, conformally flat, even-dimensional manifold of dimension $ n \geq 4 $ with positive scalar curvature, provided it is not conformally equivalent to the standard sphere.
• The proof establishes $ A_P \geq 0 $ via integration of the $ L^2 $-norm of the gradient of a carefully constructed $ \frac{n}{2} $-form on the asymptotically Euclidean end.
• Equality $ A_P = 0 $ holds if and only if the manifold is conformally equivalent to the standard $ n $-sphere, as this implies the asymptotically Euclidean metric is flat.
• The method relies on the Weitzenböck formula for $ \frac{n}{2} $-forms in even dimensions, where the Hodge star is conformally invariant, allowing harmonicity to be preserved under conformal changes.
• The result holds without any topological assumptions—neither orientability nor spin structure is required—making the proof applicable to a broader class of manifolds.
• The key estimate $ \int_{M \setminus B_P(r)} |\nabla \varphi|_{\tilde{g}}^2 \, dv_{\tilde{g}} = 4n(n-1)\omega_{n-1}^2 A + o(1) $ leads to the conclusion $ A \geq 0 $, with equality only in the flat case.