Path connectedness of the full asymptotically flat vacuum constraint space
Fix $0 < \alpha < 1$ and the standard coordinates on $\mathbb{R}^3$, with Euclidean metric $\delta$. For tensors on $\mathbb{R}^3$, let $C^{m,\alpha}_{-q}$ be the weighted Hölder space with norm, applied componentwise, $$\sum_{j=0}^{m} \sup_x \rho(x)^{j+q} |D^j u(x)| + \sup_{x \ne y} \min\{\rho(x), \rho(y)\}^{m+\alpha+q} \frac{|D^m u(x) - D^m u(y)|}{|x - y|^\alpha}, \qquad \rho(x) = (1 + |x|^2)^{1/2}.$$ Let $\mathcal{C}_\alpha$ consist of the pairs $(g, k)$, with $g$ a Riemannian metric and $k$ a symmetric covariant two-tensor, such that $g - \delta \in C^{2,\alpha}_{-1}$ and $k \in C^{1,\alpha}_{-2}$, which satisfy classically the vacuum Einstein constraint equations $$R_g + (\operatorname{tr}_g k)^2 - |k|_g^2 = 0, \qquad \operatorname{div}_g \bigl(k - (\operatorname{tr}_g k) g\bigr) = 0.$$ Give $\mathcal{C}_\alpha$ the topology induced by $C^{2,\alpha}_{-1} \times C^{1,\alpha}_{-2}$, without quotienting by coordinate changes. Marques proved that the subset of $\mathcal{C}_\alpha$ where $\operatorname{tr}_g k = 0$ is path connected in this topology.
Problem. Is $\mathcal{C}_\alpha$ path connected for every fixed $\alpha \in (0, 1)$? Equivalently, can every pair in $\mathcal{C}_\alpha$ be joined to $(\delta, 0)$ by a path in $\mathcal{C}_\alpha$?
Source: Smith and Weinstein (2000), who ask whether the space of asymptotically flat initial data on a 3-manifold is connected, even on R^3; Carlotto (2021), Open Problem 4.18, its path-connectedness question, following the remark of Marques (2012) on the full set of vacuum solutions. Stated by AIM — Open Applied Problems in terms of the second fundamental form.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, asymptotically flat initial data, Einstein constraint equations
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{smithweinstein2000,
author = {Smith, Brian and Weinstein, Gilbert},
title = {On the connectedness of the space of initial data for the {E}instein equations},
journal = {Electron. Res. Announc. Amer. Math. Soc.},
volume = {6},
number = {8},
year = {2000},
pages = {52--63},
doi = {10.1090/S1079-6762-00-00081-0},
}
@article{carlotto2021,
author = {Carlotto, Alessandro},
title = {The general relativistic constraint equations},
journal = {Living Rev. Relativ.},
volume = {24},
number = {1},
year = {2021},
pages = {2},
doi = {10.1007/s41114-020-00030-z},
}
@article{marques2012,
author = {Marques, Fernando C.},
title = {Deforming three-manifolds with positive scalar curvature},
journal = {Ann. of Math. (2)},
volume = {176},
number = {2},
year = {2012},
pages = {815--863},
doi = {10.4007/annals.2012.176.2.3},
eprint = {0907.2444},
archivePrefix = {arXiv},
}
@misc{aim-path-connectedness-of-the-full-asymptotically-flat-vacuum-constraint-space,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Path connectedness of the full asymptotically flat vacuum constraint space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/875-path-connectedness-of-the-full-asymptotically-flat-vacuum-constraint-space.md},
note = {Entry 875 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01120, "Path connectedness of the full asymptotically flat vacuum constraint space", clopen.solutions.