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

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.

• The paper constructs global, exponentially convergent STCMC foliations on asymptotically hyperboloidal data, addressing challenges from non-time-symmetric settings.
• It employs a volume-preserving spacetime mean curvature flow with sharp decay estimates to ensure uniqueness and stability of each foliation leaf.
• The paper defines a gauge-invariant geometric center of mass via barycenters of the STCMC leaves, extending methods from asymptotically Euclidean manifolds.

Foliations by Constant Spacetime Mean Curvature Surfaces in Asymptotically Hyperboloidal Initial Data

Background and Motivation

The study of geometric foliations of initial data sets in general relativity underpins several crucial aspects of gravitational physics, notably the definition of conserved quantities such as mass and center of mass. In the asymptotically Euclidean (AE) setting, the seminal work of Huisken-Yau constructed constant mean curvature (CMC) foliations of asymptotically Schwarzschildean manifolds and used these structures to define a geometric center of mass. These ideas have been extended and refined by numerous authors in both AE and asymptotically hyperbolic (AH) contexts. The present work addresses analogous constructions for asymptotically hyperboloidal initial data sets, focusing on the existence, properties, and applications of foliations by surfaces of constant spacetime mean curvature (STCMC).

The STCMC condition is natural when the initial data set is not time-symmetric; the spacetime mean curvature $\mathcal{H} = \sqrt{H^2 - P^2}$, where $H$ is the mean curvature and $P = \mathrm{tr}_g\,\overline{K}$ is the trace of the second fundamental form, interpolates standard CMC ($\overline{K} = 0$) with more general kinematic situations.

Main Results

Construction of STCMC Foliations

The core achievement of the paper is the construction of an exhaustive family of STCMC surfaces for initial data sets asymptotic to the anti-de Sitter–Schwarzschild (AdS-Schwarzschild) hyperboloid, i.e., triples $(M,\overline{g},\overline{K})$ with specified decay to the AdS-Schwarzschild metric and second fundamental form. This is accomplished via an analytic study of the volume-preserving spacetime mean curvature flow (VPSTMCF):

$\partial_t F = -(\mathcal{H} - \hbar) \nu$

where $F$ denotes the immersion, $\nu$ the normal, and $\hbar$ the average of $\mathcal{H}$ on the evolving surface. Notably, the flow is not area-decreasing, complicating the convergence analysis relative to the standard (Euclidean-space) VPMCF.

By initializing the VPSTMCF from the CMC-foliation constructed by Neves-Tian for (Riemannian) AH manifolds, and leveraging sharp decay estimates, the author demonstrates:

• Global existence and convergence: For sufficiently "round" initial data (essentially, large CMC leaves with controlled higher-order geometry), the flow exists for all time and converges exponentially in $H$0 to a surface of constant spacetime mean curvature.
• Exhaustive foliation: As the parameter $H$1 (area-radius) ranges over large values, the limiting leaves exhaust the asymptotic region, yielding a foliation $H$2 with each leaf uniquely determined in its roundness class.
• Volume-preserving stability: Each leaf is stable under normal volume-preserving variations, with spectral gap estimates quantified in terms of the mass $H$3 of the background AdS-Schwarzschild metric.

Sharp quantitative results include exponential decay rates for the $H$4-norms of $H$5 under the flow, with explicit dependencies on the geometric radius parameter and mass.

Geometric Center of Mass

Building on the STCMC foliation, a geometric definition of center of mass is formulated for asymptotically hyperboloidal manifolds, generalizing the construction in the AE setting by Cederbaum-Sakovich. The center of mass is defined as a limit of barycenters, computed via the canonical immersion of $H$6 into Minkowski space, of the STCMC leaves as their area tends to infinity. The paper proves that, for the AdS-Schwarzschild background, these barycenters converge to the origin of hyperbolic space, mirroring the centering property of the original Huisken-Yau foliation.

Technical Approach

The analysis relies on several conceptual and technical innovations relative to prior work:

• VPSTMCF Analysis: The VPSTMCF incorporates the presence of $H$7 (i.e., non-time-symmetric data) and requires handling the persistence of non-decaying terms at infinity (specifically, $H$8 approaches a nonzero limit). This affects both evolution equations and stability properties.
• Graphical Regularity and Schauder Theory: Detailed control of the leaves is derived by representing them as graphs over large coordinate spheres in hyperbolic space, with $H$9-norm estimates propagated via bootstrapping and Sobolev-type inequalities adapted to the AH setting.
• Stability Operator Spectral Analysis: The spectrum of the linearized stability operator around large spheres is crucial for deducing both convergence rates and uniqueness in foliation. The paper establishes lower bounds on the first nontrivial eigenvalue, uniformly for large $P = \mathrm{tr}_g\,\overline{K}$0, in terms of the mass $P = \mathrm{tr}_g\,\overline{K}$1.
• Volume Correction Terms: Careful tracking of volume corrections and decay orders in the asymptotically hyperboloidal regime is essential, in contrast to Euclidean or time-symmetric cases.

Implications and Future Directions

This work generalizes the geometric analysis of center of mass and foliations to a broader gravitational context (non-time-symmetric, asymptotically AdS-like spacetimes), strengthening the robustness of these constructions in geometric relativity. Important theoretical implications include:

• Gauge-invariant Center of Mass: The barycenter prescription provides a geometric, coordinate-invariant notion of "center" for solutions to Einstein's constraint equations with AdS-type infinity.
• Framework for Quasi-local Quantities: The geometric foliations offer a coherent background for further study of quasilocal mass, angular momentum, and their fluxes in non-isolated gravitating systems.

Potential future developments include:

• Extension to general positive mass aspect functions (removing the diagonal restriction), building on the techniques outlined here and in related works (especially [nevestian2], [cederbaumcortiersakovich]).
• Relaxation or removal of coordinate "balancedness" conditions, enabling a broader class of initial data sets.
• Connections with the AdS/CFT correspondence and the geometry of conserved charges in AdS asymptotics.
• Stability and rigidity theorems exploiting the VPSTMCF as a geometric flow in the initial data landscape.

Conclusion

The paper establishes, via refined geometric analysis and flow techniques, the global existence, uniqueness, and stability of STCMC foliations for asymptotically hyperboloidal initial data sets near AdS-Schwarzschild, and utilizes these foliations to define a geometric center of mass in this setting. The results provide a crucial building block for quasi-local mass and global structure theory in general relativity with AdS-type infinity, and set the stage for further generalization and application to geometric analysis in mathematical general relativity.

Reference: "Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets" (2607.02244)