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arxiv: 2607.02244 · v1 · pith:SQJFQHVGnew · submitted 2026-07-02 · 🧮 math.DG · math-ph· math.AP· math.MP

Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets

Jacopo Tenan This is my paper

Pith reviewed 2026-07-03 05:59 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.APmath.MP
keywords constant spacetime mean curvaturefoliationsasymptotically hyperboloidalinitial data setsmean curvature flowcenter of massanti-de Sitter-Schwarzschild
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The pith

Initial data sets close to the anti-de Sitter-Schwarzschild hyperboloid admit exhaustive foliations by constant spacetime mean curvature surfaces.

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

The paper constructs a complete family of surfaces with constant spacetime mean curvature inside initial data sets that lie sufficiently close to the anti-de Sitter-Schwarzschild hyperboloid. It produces the foliation by running a volume-preserving spacetime mean curvature flow that begins with the constant mean curvature surfaces already known from Neves and Tian and evolves until spacetime mean curvature becomes constant. The resulting foliation then supplies a definition of center of mass for these asymptotically hyperboloidal data sets. A sympathetic reader cares because the construction supplies an organizing structure for the geometry and mass in a setting that appears in gravitational physics but had lacked this particular foliation tool.

Core claim

For initial data sets sufficiently close to the anti-de Sitter-Schwarzschild hyperboloid, an exhaustive foliation by constant spacetime mean curvature surfaces exists and arises precisely as the long-time limit of the volume-preserving spacetime mean curvature flow that starts from the constant mean curvature foliation constructed by Neves and Tian. The same foliation is then used to define and study the center of mass of an asymptotically hyperboloidal initial data set.

What carries the argument

The volume-preserving spacetime mean curvature flow, evolved from the Neves-Tian constant mean curvature foliation until spacetime mean curvature is constant on each leaf.

If this is right

  • The foliation exhausts the entire initial data set.
  • A center of mass can be defined for the data set by using the STCMC surfaces in direct analogy with the Euclidean construction.
  • The flow provides a dynamical construction that recovers the STCMC foliation from the already-known CMC one.
  • The construction applies whenever the initial data remain sufficiently close to the model solution.

Where Pith is reading between the lines

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

  • The same flow technique might produce STCMC foliations for data sets that approach other model solutions if a suitable starting foliation can be found.
  • The center-of-mass definition obtained this way could be checked for agreement with other proposed notions of mass in the hyperboloidal setting.
  • Long-time convergence of the flow may indicate that the constant spacetime mean curvature condition is stable under small perturbations of the initial data.

Load-bearing premise

The initial data sets must be sufficiently close to the anti-de Sitter-Schwarzschild hyperboloid.

What would settle it

An explicit initial data set near the model hyperboloid on which the volume-preserving spacetime mean curvature flow either fails to exist for all time or converges to surfaces that do not have constant spacetime mean curvature.

read the original abstract

We construct an exhaustive family of constant spacetime mean curvature (STCMC) surfaces for initial data sets close to the anti-de Sitter-Schwarzschild hyperboloid. In particular, we obtain such a foliation as the long time limit of the volume preserving spacetime mean curvature flow starting from the constant mean curvature foliation constructed by Neves-Tian (Geom. Funct. Anal., 2009). As an application, inspired by the definition of STCMC center of mass for initial data sets proposed in the asymptotically Euclidean setting by Cederbaum-Sakovich (Calc. Var. PDE, 2021), we study the center of mass of an asymptotically hyperboloidal initial data set.

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 / 2 minor

Summary. The manuscript constructs an exhaustive foliation by constant spacetime mean curvature (STCMC) surfaces in asymptotically hyperboloidal initial data sets that are sufficiently close to the anti-de Sitter-Schwarzschild hyperboloid. The foliation is realized as the long-time limit of the volume-preserving STCMC flow, initialized from the constant mean curvature foliation previously constructed by Neves and Tian. As an application, the authors define and study a notion of center of mass for such initial data sets, modeled on the Cederbaum-Sakovich construction in the asymptotically Euclidean setting.

Significance. If the central claims are established, the work supplies a canonical STCMC foliation in the asymptotically hyperbolic regime and a corresponding center-of-mass definition. This extends the geometric-flow approach to foliations from the Euclidean to the hyperbolic setting and may prove useful for questions of mass, center-of-mass, and stability in AdS-like spacetimes. The choice to start the flow from the Neves-Tian CMC foliation is a natural and economical one, and the smallness assumption near the model space is the standard hypothesis under which such limiting arguments are expected to close.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'sufficiently close' is used without indicating the precise function space or norm in which the smallness is measured; a parenthetical reference to the relevant distance (e.g., weighted Sobolev norm) would clarify the hypothesis for readers.
  2. [Introduction] The manuscript would benefit from an explicit statement, early in the introduction, of the precise decay rates assumed on the initial data set (beyond the model hyperboloid) so that the smallness condition can be checked against the literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recognition of the natural choice to initialize the flow from the Neves-Tian CMC foliation and the standard smallness hypothesis. The recommendation for minor revision is noted. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation constructs an exhaustive STCMC foliation as the long-time limit of a volume-preserving STCMC flow, initialized from the external Neves-Tian CMC foliation (2009) under an explicit smallness assumption near the AdS-Schwarzschild hyperboloid. This is a standard limiting argument relying on prior independent work by different authors and geometric flow techniques; no self-citations are load-bearing, no fitted parameters are renamed as predictions, and no step reduces by definition or construction to the target result itself. The central claim remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities identifiable. Standard background assumptions from geometric analysis and GR are presumed but unlisted.

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discussion (0)

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Reference graph

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