[Paper Review] Universal positive mass theorems

Marc Herzlich|arXiv (Cornell University)|Jan 23, 2014

Geometric Analysis and Curvature Flows19 references4 citations

TL;DR

This paper establishes a universal framework for deriving positive mass theorems on asymptotically flat manifolds using Bochner-Weitzenb"ock-type formulas, generalizing Witten's spinor-based proof. It shows that for a broad class of irreducible natural bundles and first-order operators—under suitable curvature conditions—the boundary contribution in such formulas directly yields the mass, enabling positivity or negativity theorems depending on curvature signs.

ABSTRACT

In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenböck-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreducible natural bundle and a very large choice of first-order operators may lead to a positive mass theorem along the same lines if the necessary curvature conditions are satisfied.

Motivation & Objective

• To develop a general method for computing boundary-at-infinity contributions in Bochner-Weitzenb"ock-type formulas on asymptotically flat manifolds.
• To extend Witten's proof strategy of the positive mass theorem beyond spin geometry to a wide class of natural bundles and first-order operators.
• To identify curvature conditions under which the boundary term in such formulas yields a well-defined mass and ensures its positivity or negativity.
• To provide a unified framework for deriving positive or negative mass theorems in geometric and physical contexts involving asymptotic invariants.
• To explore the role of conformal invariance and Stein-Weiss operators in constructing solutions with prescribed asymptotic behavior for generalized Dirac-type equations.

Proposed method

• Formalizes the boundary-at-infinity contribution in Bochner-Weitzenb"ock formulas via integration by parts on large coordinate spheres.
• Applies the universal Weitzenb"ock formula for natural bundles, expressing the difference between the rough Laplacian and a first-order operator in terms of curvature and representation-theoretic data.
• Uses conformal covariance of Stein-Weiss operators to construct solutions to generalized Dirac equations that converge to constant sections at infinity.
• Derives the asymptotic boundary term as a linear functional of the metric's second-order decay, directly related to the mass via the limit of the divergence form.
• Applies the Lichnerowicz-Schr"odinger formula in a generalized setting, linking the curvature term $\mathcal{R}^\rho$ to the sign of the boundary contribution.
• Analyzes specific cases—spinors, $p$-forms, and conformally flat metrics—using representation theory and conformal weights to classify possible operators and their boundary terms.

Experimental results

Research questions

• RQ1 Can Witten's method for proving the positive mass theorem be generalized beyond spin geometry to other natural bundles and first-order operators?
• RQ2 What curvature conditions ensure that the boundary contribution in a Bochner-Weitzenb"ock formula corresponds to the mass and controls its sign?
• RQ3 How do conformal invariance and the structure of Stein-Weiss operators facilitate the construction of solutions with prescribed asymptotic behavior?
• RQ4 In which cases does the boundary term yield a positive or negative mass, depending on the sign of the curvature term $\mathcal{R}^\rho$?
• RQ5 What is the role of the number of irreducible components $N$ in the decomposition of the bundle, and how does it affect the existence of a single-projection operator?

Key findings

• For any irreducible natural bundle and first-order operator satisfying curvature conditions, the boundary contribution in a Bochner-Weitzenb"ock formula computes the mass, generalizing Witten's approach.
• The mass is nonnegative if the curvature term $\mathcal{R}^\rho$ is nonpositive, and nonpositive if $\mathcal{R}^\rho$ is nonnegative, under the same conditions.
• In the case of $p$-forms with $p < n/2$, the Hodge-de Rham operator $d + \delta$ leads to a negative mass theorem when $\mathcal{R}_p$ is nonnegative.
• For conformally flat metrics, the curvature term for $p$-forms is $\mathcal{R}_p = -\left(\frac{n-2p}{(p-1)!}g^{p-1}Z^g + \frac{2(n-p)}{(p-1)!}\mathrm{Scal}^g\,\mathrm{Id}\right)$, which determines the sign of the mass.
• When $N=2$, such as in the case of $p$-forms with $p = n/2$ in even dimensions, the boundary term is directly linked to the mass via a single projection, enabling a positive or negative mass theorem.
• The construction of solutions to $P\phi = 0$ with $\phi \to \phi_0$ at infinity is possible via conformal covariance of Stein-Weiss operators, ensuring the boundary term captures the mass.