Positive mass theorems for manifolds with ALH toroidal ends

In work with P. Chruściel, L. Nguyen and T.-T. Paetz [8], a positive mass theorem was obtained for asymptotically locally hyperbolic manifolds with boundary, having a toroidal end. The proof made use of properties of marginally outer trapped surfaces (MOTS). Here we present some new PMT results for such manifolds, but without boundary, which allow for other more general ends. The proofs, while still MOTS-based, involve a more elaborate technique (related to $μ$-bubbles) introduced in work of D. A. Lee, M. Lesourd, and R. Unger [20] for manifolds with an asymptotically flat end, and further developed in [23] for manifolds with an asymptotically hyperbolic end.

💡 Research Summary

In this paper the authors extend positive mass theorems to a class of asymptotically locally hyperbolic (ALH) manifolds that possess a toroidal end but no boundary, allowing for more general “other” ends. The main result (Theorem 1.1) states that for a complete orientable ALH manifold ((M^{n},g)) of dimension (4\le n\le7) with a conformally compactifiable toroidal end (E) and scalar curvature satisfying (R\ge -n(n-1)), if the complement (N=M\setminus E) is non‑compact and asymptotically retractable onto (\partial E), then the mass associated with the end (E) is non‑negative.

The proof follows the strategy pioneered in earlier work with Chruściel, Nguyen and Paetz, but replaces the relatively simple MOTS (marginally outer trapped surface) barrier argument with a more sophisticated construction that incorporates the (\mu)-bubble technique introduced by Lee, Lesourd and Unger. The authors first assume, for contradiction, that the mass (m) is negative. By examining a small toroidal slice (\Sigma_{1}) at coordinate (x=x_{1}) they compute its mean curvature (H_{1}) and show that, after a suitable rescaling of the extrinsic curvature (K) (taking (K=-\lambda g) with (\lambda>1)), the null expansion (\theta_{+}) on (\Sigma_{1}) becomes positive.

A smooth function (h) is then constructed on an open region (M_{0}\subset M) with the properties required in the (\mu)-bubble framework: (h\ge0), vanishing on an inner region, blowing up near the outer boundary, and satisfying precise gradient bounds. Defining (\hat K=K-hg), the authors verify that the modified initial data set ((M_{0},g,\hat K)) satisfies the dominant energy condition (DEC) provided the distance parameter (D_{0}) is chosen large enough.

With the DEC in hand, the authors consider the region (W) bounded by (\Sigma_{0}) (a hypersurface close to the outer boundary where (\theta_{+}<0)) and (\Sigma_{1}) (where (\theta_{+}>0)). Lemma 2.1 guarantees the existence of an outermost MOTS (\Sigma^{}) in (W) homologous to (\Sigma_{1}). Because (\Sigma_{1}) satisfies a cohomology condition (its first cohomology classes cup‑product to a non‑zero top class), Lemma 2.3 implies that (\Sigma^{}) also satisfies this condition. Consequently, at least one component of (\Sigma^{*}) cannot support a metric of positive scalar curvature. This component must be weakly outermost; otherwise Lemma 2.1 would produce a MOTS further outward, contradicting the outermost property. Lemma 2.2 then forces a foliation of a neighbourhood of this component by MOTS, again contradicting outermostness. Hence the assumption (m<0) is untenable, establishing (m\ge0).

The authors also present a “quantitative shielding” theorem (Theorem 3.1) that removes the completeness hypothesis. By selecting three nested neighborhoods (U_{0}\supset U_{1}\supset U_{2}) of the toroidal end and imposing a strict scalar curvature inequality (R+n(n-1)>4D_{0}D_{1}) on the annular region (U_{1}\setminus U_{2}) (where (D_{0},D_{1}) are the distances between the respective boundaries), the same contradiction argument works with (\lambda=1). This result shows that a sufficiently large curvature “shield” around the end prevents negative mass, even when the manifold is not globally complete.

The paper discusses two illustrative families of metrics. The Birmingham–Kottler family (with flat toroidal horizons) satisfies the hypotheses exactly when the mass parameter (m) is non‑negative; the zero‑mass case corresponds to the locally hyperbolic metric, known to be uniquely mass‑zero. The Horowitz–Myers soliton provides a counterexample with negative mass, but it fails the non‑compactness/retractability condition on (N), illustrating the sharpness of the assumptions.

Dimension restrictions arise because the existence and regularity theory for MOTS is currently established only for (3\le n\le7). An additional technical restriction (n\ge4) appears in the original proof of Theorem 1.3 in