A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
Abstract
On asymptotically Schwarzschild smooth metric measure spaces modeled on the Euclidean half-space, we define a new mass-type invariant analogous to the ADM mass of an asymptotically flat manifold. We prove that this invariant is independent of the choice of admissible asymptotic coordinates. We further show that it fits into a very general framework for mass-type invariants and we also isolate static equations for smooth metric measure spaces under natural boundary conditions. For smooth metric measure spaces obtained through a generalized stereographic projection using the Green's function associated with the fractional Yamabe problem, we identify the mass with the constant term in the asymptotic expansion of the Green's function. This identification gives a geometric interpretation of the constant appearing in the fractional Yamabe problem and recasts its conjectured positivity as a positive-mass-type question.
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BibTeX
@misc{almaraz2026masstype,
title = {{A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem}},
author = {Sergio Almaraz and Levi Lopes de Lima and Shaodong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.23248},
eprint = {2606.23248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23248},
}
RIS
TY - UNPB
TI - A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
AU - Sergio Almaraz
AU - Levi Lopes de Lima
AU - Shaodong Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2606.23248
UR - https://arxiv.org/abs/2606.23248
AB - On asymptotically Schwarzschild smooth metric measure spaces modeled on the Euclidean half-space, we define a new mass-type invariant analogous to the ADM mass of an asymptotically flat manifold. We prove that this invariant is independent of the choice of admissible asymptotic coordinates. We further show that it fits into a very general framework for mass-type invariants and we also isolate static equations for smooth metric measure spaces under natural boundary conditions. For smooth metric measure spaces obtained through a generalized stereographic projection using the Green's function associated with the fractional Yamabe problem, we identify the mass with the constant term in the asymptotic expansion of the Green's function. This identification gives a geometric interpretation of the constant appearing in the fractional Yamabe problem and recasts its conjectured positivity as a positive-mass-type question.
ER -