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arxiv: 2604.17599 · v2 · submitted 2026-04-19 · 🧮 math.DG · math-ph· math.MG· math.MP

Geometric Stability of the Schoen-Yau Zero Mass Theorem

Christina Sormani This is my paper

Pith reviewed 2026-05-10 05:09 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MGmath.MP
keywords geometric stabilitypositive mass theoremADM massasymptotically flat manifoldsrigiditynotions of convergencezero mass theoremSchoen-Yau
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The pith

The geometric notion of convergence that best captures stability of the Schoen-Yau zero-mass rigidity theorem remains an open question even in three dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews the stability question that follows from the rigidity part of the Schoen-Yau Positive Mass Theorem: an asymptotically flat three-manifold with nonnegative scalar curvature and zero ADM mass must be Euclidean space. It asks what this implies when the mass is merely small rather than zero, and examines how close such manifolds must be to flat space. The review collects examples of sequences whose mass approaches zero yet whose geometry stays away from Euclidean space under several standard notions of convergence. It compares a range of candidate convergences and concludes that no single notion has yet been shown to be the right one for quantifying the stability.

Core claim

While sequences of asymptotically flat manifolds with nonnegative scalar curvature and ADM mass tending to zero can fail to converge to Euclidean space under some geometric notions, progress has been made under others; nevertheless, the paper states that it is still unknown which notion of convergence best encodes the geometric stability of the zero-mass rigidity theorem, even in dimension three.

What carries the argument

Geometric notions of convergence (such as pointed C0, Gromov-Hausdorff, intrinsic flat, and volume convergence) applied to sequences of asymptotically flat manifolds whose ADM mass approaches zero.

If this is right

  • Sequences whose mass tends to zero but whose geometry does not converge under a given notion demonstrate that the notion is too weak to capture stability.
  • Convergences that do force the manifolds to become close to Euclidean space when mass is small provide quantitative versions of the zero-mass rigidity.
  • The zero-mass case is recovered exactly when the mass reaches zero under any notion that is continuous with respect to mass.
  • Additional assumptions such as uniform bounds on curvature or diameter allow some convergences to succeed where they otherwise fail.
  • Identifying the optimal notion would turn the rigidity theorem into a stable statement useful for approximating physical spacetimes with tiny mass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A convergence that works in three dimensions might extend to higher-dimensional stability questions for the same theorem.
  • The problem suggests testing whether a hybrid convergence that controls both distances and volumes simultaneously could resolve the open issue.
  • Numerical construction of near-zero-mass manifolds on a computer could be used to check which of the reviewed notions actually detect the expected closeness to flat space.
  • Connections to other rigidity results in general relativity, such as stability of the Penrose inequality, may become clearer once the right convergence is fixed.

Load-bearing premise

The original Schoen-Yau theorem is valid and there exist sequences of manifolds with ADM mass approaching zero that fail to converge geometrically under at least some of the candidate notions.

What would settle it

A sequence of three-dimensional asymptotically flat manifolds with nonnegative scalar curvature, ADM mass less than 1/n for each n, whose intrinsic flat distance to Euclidean space stays bounded below by a positive constant would show that intrinsic flat convergence is too weak; a proof that any sequence with mass less than 1/n must have intrinsic flat distance less than epsilon(n) with epsilon(n) to zero would confirm that notion works.

Figures

Figures reproduced from arXiv: 2604.17599 by Christina Sormani.

Figure 1
Figure 1. Figure 1: A simple example of a sequence, M3 j , of spher￾ically symmetric asymptotically flat Riemannian manifolds with nonnegative scalar curvature such that mADM (M3 j ) → 0 which smoothly converge to flat Euclidean space. In Section 3 we present many examples of sequences of Riemannian mani￾folds which satisfy the hypotheses of this conjecture. These include examples [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Riemannian Schwarzschild Spaces with mass con￾verging to zero as in Example 3.1. Example 3.1. A sequence of Riemannian Schwarzschild manifolds, M3 Sch,mj , with mj → 0 as depicted in [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Here we construct Example 3.5. 3.5. Bubbling. It has been well known that sequences of manifolds with positive scalar curvature can form bubbles. See for example work of Miao [103] which builds upon work of Corvino [46] and work of Beig-O Murchadha ´ in [21]. Here is a construction of such a sequence using spherical zones and tun￾nels: Example 3.6. Take N3 j to be the Riemannian manifolds with spherical zo… view at source ↗
Figure 4
Figure 4. Figure 4: Example 3.6. Remark 3.7. Note that in Example 3.6 we see why the class M of Defi￾nition 1.2 cannot include any interior minimal surfaces if we hope to prove [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example 3.12. Example 3.13. Take N3 j to be the Riemannian manifolds with spherical zones constructed in Example 3.5 and attach to each an increasingly thin well of increasing depth Dj as described in Example 3.11. This creates M3 j ∈ M of Definition 1.2 as in [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example 3.13 Example 3.14. Take N3 j to be the Riemannian manifolds with spherical zones constructed in Example 3.5 and attach to each increasingly many in￾creasingly thin wells of uniform depth D0 as described in Example 3.11. This creates M3 j ∈ M of Definition 1.2 as in [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example 3.14. 3.7. Sewing. Here we will see a collection of examples constructed using the tunnels described in Example 3.4. This time rather than running the tunnels between two different manifolds, we will run the tunnel between two balls in the same manifold. We begin with a simple example: Example 3.16. Take N3 j to be the Riemannian manifolds with spherical zones constructed in Example 3.5 and remove … view at source ↗
Figure 8
Figure 8. Figure 8: Example 3.16. In [18], Basilio-Dodziuk-Sormani described a method called ”sewing along a curve” in which they edit in increasingly small increasingly short tun￾nels along a curve in a three-sphere to obtain a sequence of manifolds with positive scalar curvature that converge to a sphere with a pulled string. Such pulled string spaces were first described by Burago while working with Ivanov and Sormani on e… view at source ↗
Figure 9
Figure 9. Figure 9: Example 3.17 constructed by sewing along a curve as in work of Basilio-Dodziuk-Sormani. In [17], Basilio-Sormani also described a method of ”sewing” a region to a point in which they edit in an increasingly small increasingly dense network of tunnels across a region in a three-sphere to obtain a sequence of manifolds with positive scalar curvature that converge to a three sphere with the entire region iden… view at source ↗
Figure 10
Figure 10. Figure 10: A sequence of manifolds with ADM mass con￾verging to 0 that have sets (depicted as blue curves) which “scrunch” to a point so that the limit space is Euclidean space with a set identified to a point. Remark 3.23. If one can construct a sequence of M3 j as in [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Here we estimate the GH distance between Ωj (R) ⊂ Mj on the left with a well and the GH limit, Ω∞(R) ⊂ M∞, of Example 3.12 which is a Euclidean ball with a line segment attached on the right. They are embed￾ded together in Z in the center where the red vertical and horizontal segments are estimating the Hausdorff distance between them in Z. Notice how the segment in M∞ must be close in length to the depth… view at source ↗
Figure 12
Figure 12. Figure 12: Here we estimate the Wasserstein distance be￾tween Ωj (R) ⊂ Mj on the left with a well and the met￾ric measure limit, Ω∞(R) ⊂ M∞, of Example 3.12 which is a Euclidean ball, B0(R) ⊂ E 3 . Their renormalized volume measures as in (68) are pushed forward into a common met￾ric space, Z, at the center. Notice how the image of the well in Ωj (R) has very small volume and so it costs very little to transport it … view at source ↗
Figure 13
Figure 13. Figure 13: Here we see Ωj (R) ⊂ Mj on the left with a well and on the right the F limit, B0(R) ⊂ E 3 , of Example 3.12. They are embedded together in Z in the center where the blue region is the filling volume between them with volume, MAK(B), and the green cylinder has volume, MAK(A). No￾tice how the depth of the well does not contribute signifi￾cantly to this F distance between the spaces as long as it is very thi… view at source ↗
read the original abstract

In 1979, Schoen and Yau proved their famous Positive Mass Theorem which is a combination of a comparison theorem: {\em a three dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature has nonnegative ADM mass}, and a rigidity theorem: {\em if such a manifold has zero ADM mass then it is isometric to Euclidean space}. Here we review results and open questions on the geometric stability of their zero mass rigidity theorem: {\em if such a manifold has almost zero mass, how close is its geometry to that of Euclidean space}? We review the geometry of these spaces, examples of sequences of such spaces with mass approaching zero, and a variety of geometric notions of convergence. Although there has been much progress, it is still an open question (even in dimension three): exactly which geometric notion of convergence works best to capture the geometric stability of this famous rigidity theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript surveys results on the geometric stability of the rigidity statement in the Schoen-Yau Positive Mass Theorem. It recalls the 1979 theorem (nonnegative ADM mass for asymptotically flat 3-manifolds with nonnegative scalar curvature, with rigidity to Euclidean space when mass vanishes), reviews examples of sequences with ADM mass approaching zero that fail to converge geometrically under various notions, discusses several notions of geometric convergence, and concludes that determining the optimal notion remains open even in dimension three.

Significance. If the review is accurate, it provides a useful consolidation of the literature on stability for this foundational result in geometric analysis and mathematical relativity. By organizing examples of non-converging sequences and comparing convergence notions, the survey clarifies the current state of the open question and may help direct subsequent work on precise formulations of stability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for the recommendation to accept. We are pleased that the survey is viewed as a useful consolidation of the literature on this topic.

Circularity Check

0 steps flagged

No significant circularity: survey of prior results with no derivations

full rationale

This manuscript is a review paper that summarizes the 1979 Schoen-Yau Positive Mass Theorem, known examples of sequences with ADM mass approaching zero, and various notions of geometric convergence from the existing literature. It advances no new theorems, equations, predictions, or fitted parameters. The central statement is that the optimal notion of convergence remains an open question even in dimension three. All references are to prior independent work; no load-bearing claim reduces by construction to a self-definition, self-citation chain, or fitted input. The paper is self-contained as a survey.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper with no new mathematical derivations. It relies entirely on the established Schoen-Yau theorem and prior literature on stability questions, introducing no free parameters, axioms, or invented entities of its own.

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