[Paper Review] Positive mass theorem for the Yamabe problem on spin manifolds

Bernd Ammann, Emmanuel Humbert|arXiv (Cornell University)|Apr 3, 2003

Advanced Mathematical Modeling in Engineering7 references4 citations

TL;DR

This paper provides a concise, spinor-based proof of the positivity of the constant term in the Green's function expansion of the conformal Laplacian on compact spin manifolds with positive Yamabe invariant, under local conformal flatness or dimensions 3–5. The proof leverages the Green's function of the Dirac operator to construct a test spinor, simplifying Witten's original argument by avoiding asymptotically flat analysis and relying only on compact manifold analysis.

ABSTRACT

Let $(M,g)$ be a compact connected spin manifold of dimension $n\geq 3$ whose Yamabe invariant is positive. We assume that $(M,g)$ is locally conformally flat or that $n \in \{3,4,5\}$. According to a positive mass theorem of Witten, the constant term in the asymptotic development of the Green's function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. In the present article, we will give a proof for this fact which is considerably shorter than previous proofs. Our proof is a modification of Witten's argument, but no analysis on asymtotically flat spaces is needed.

Motivation & Objective

• To establish the positivity of the constant term in the Green's function expansion of the conformal Laplacian on compact spin manifolds with positive Yamabe invariant.
• To provide a simplified proof of the positive mass theorem in the context of the Yamabe problem, avoiding asymptotically flat geometry.
• To extend Witten's spinor method to compact spin manifolds by constructing a harmonic test spinor via the Dirac operator's Green's function.
• To handle dimensions 3, 4, and 5 uniformly, with additional estimates in higher dimensions.

Proposed method

• Construct a spinor field on $M \setminus \{P\}$ using the Green's function of the Dirac operator, tailored to mimic asymptotically constant behavior.
• Define a test spinor $\psi$ on a punctured neighborhood of $P$ via $\psi = \frac{x}{r^n} \cdot \psi_0 + \theta(x)$, with $\theta$ smooth, ensuring $D\psi = -\omega_{n-1} \delta_P \psi_0$ in distributional sense.
• Use the Green's function of the Dirac operator to correct the spinor and construct a harmonic spinor $\varphi$ satisfying $\overline{D}\varphi \in C^{0,1}(M)$.
• Apply regularity theory and Sobolev embedding to ensure $\varphi$ is Hölder continuous and its gradient decays appropriately via $r^{1+\varepsilon}|\nabla \varphi| \in C^0$.
• Derive dimension-specific estimates: for $n=3$, $\gamma + \gamma' \in L^q$ for $q>1$; for $n=4$, $\gamma + \gamma' = O(1/r)$; for $n=5$, decompose $\gamma$ as homogeneous order $-1$ and $\gamma'$ as $O(1/r)$.
• Construct a correction spinor $\Theta$ via the fundamental solution of the Dirac operator to cancel error terms, ensuring $\overline{D}\Theta = \gamma + \gamma'$ near $P$.

Experimental results

Research questions

• RQ1 Can the positivity of the constant term in the Green's function of the conformal Laplacian be established without asymptotically flat geometry?
• RQ2 How can Witten's spinor method be adapted to compact spin manifolds using only intrinsic analysis?
• RQ3 What modifications are needed in the spinor construction for dimensions 4 and 5 to control the Green's function error terms?
• RQ4 Is the constant term $A$ in the Green's function expansion positive when $M$ is spin, compact, and has positive Yamabe invariant?

Key findings

• The constant term $A$ in the asymptotic expansion of the Green's function of the conformal Laplacian is positive if $(M,g)$ is a compact spin manifold with positive Yamabe invariant and is locally conformally flat or of dimension $n \in \{3,4,5\}$.
• The proof avoids asymptotically flat manifolds and relies solely on compact manifold analysis, significantly shortening Witten's original argument.
• For $n=3$, the error spinor $\gamma + \gamma'$ lies in $L^q$ for all $q>1$, ensuring existence of a correction spinor in $H^1_q$ and Hölder continuity.
• For $n=4$, the correction spinor $\Theta$ satisfies $r^\varepsilon \Theta \in C^0$ and $r^{1+\varepsilon}|\nabla \Theta| \in C^0$, ensuring sufficient decay.
• For $n=5$, the error is split into a homogeneous part $\gamma$ of order $-1$ and a remainder $\gamma'$, each corrected via the Dirac Green's function to yield a $C^{1,a}$ spinor.
• The final spinor $\Psi(\psi_0) = \varphi - \varphi'$ is smooth on $M \setminus \{P\}$ and of class $C^{1,a}(M)$, proving the existence of a harmonic test spinor with controlled decay.