Proof of the Penrose Conjecture for Maximally Parameterized Initial Data Sets
Abstract
We prove the spacetime Penrose inequality for asymptotically flat initial data sets (Mn,g,k) in all dimensions n≥ 3, under the dominant energy condition, for data admitting the 'maximally parameterized form' defined in equation (4.7). In this case, the second fundamental form k is locally parameterized by n(n+1)/2 functions, giving the same formal local functional degrees of freedom as an arbitrary symmetric $2$-tensor. The proof is based on a new 1-parameter family of generalized Schoen-Yau identities and their gauge-invariant specialization, inspired by the generalized Jang framework of Bray-Khuri [8, 9]. For maximally parameterized data, these identities produce a Riemannian comparison metric g̅=g+β⊗β with nonnegative scalar curvature Rg̅ ≥ 0. Under a natural nondegeneracy condition (5.7) at the apparent horizon, the comparison metric admits a smooth minimal completion up to ∂ Mn. Together with suitable asymptotic control, this reduces the Penrose inequality to its Riemannian counterpart. The all-dimensional conclusion uses the Riemannian Penrose theorem of Bi-Zhu [6], together with a minimizing enclosure argument that allows singular boundaries. More generally, in the spirit of [8, 9], we note that the Penrose conjecture follows whenever a certain system of PDEs (3.18) involving k can be solved, in which case our new identity (3.15) immediately implies Rg̅ ≥ 0. A natural problem is thus to regularize our degenerate system to prove a general existence theory, thereby proving the Penrose conjecture. Essentially, we believe this is what the OpenAI paper [27] posted last week on October 6, 2026 claims to have done.
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BibTeX
@misc{bray2026proof,
title = {{Proof of the Penrose Conjecture for Maximally Parameterized Initial Data Sets}},
author = {Hubert Bray and Tuan Dolmen and Marcus Khuri and David Schmidt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20559},
eprint = {2610.20559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20559},
}
RIS
TY - UNPB
TI - Proof of the Penrose Conjecture for Maximally Parameterized Initial Data Sets
AU - Hubert Bray
AU - Tuan Dolmen
AU - Marcus Khuri
AU - David Schmidt
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20559
UR - https://arxiv.org/abs/2610.20559
AB - We prove the spacetime Penrose inequality for asymptotically flat initial data sets (M^n,g,k) in all dimensions n≥ 3, under the dominant energy condition, for data admitting the 'maximally parameterized form' defined in equation (4.7). In this case, the second fundamental form k is locally parameterized by n(n+1)/2 functions, giving the same formal local functional degrees of freedom as an arbitrary symmetric $2$-tensor. The proof is based on a new 1-parameter family of generalized Schoen-Yau identities and their gauge-invariant specialization, inspired by the generalized Jang framework of Bray-Khuri [8, 9]. For maximally parameterized data, these identities produce a Riemannian comparison metric g̅=g+β⊗β with nonnegative scalar curvature R_g̅ ≥ 0. Under a natural nondegeneracy condition (5.7) at the apparent horizon, the comparison metric admits a smooth minimal completion up to ∂ M^n. Together with suitable asymptotic control, this reduces the Penrose inequality to its Riemannian counterpart. The all-dimensional conclusion uses the Riemannian Penrose theorem of Bi-Zhu [6], together with a minimizing enclosure argument that allows singular boundaries. More generally, in the spirit of [8, 9], we note that the Penrose conjecture follows whenever a certain system of PDEs (3.18) involving k can be solved, in which case our new identity (3.15) immediately implies R_g̅ ≥ 0. A natural problem is thus to regularize our degenerate system to prove a general existence theory, thereby proving the Penrose conjecture. Essentially, we believe this is what the OpenAI paper [27] posted last week on October 6, 2026 claims to have done.
ER -