Tilted spacetime positive mass theorem with arbitrary ends (Q6596114)

The positive mass theorem of \textit{R. Schoen} and \textit{S.-T. Yau} [Surv. Differ. Geom. 24, 441--480 (2022; Zbl 07817751)] is a central result in the study of the geometry of scalar curvature, mathematical relativity and geometric analysis. It says that if an \(n\)-dimensional complete Riemannian manifold with non-negative scalar curvature has an asymptotically flat end \(E\), then the ADM mass of \(E\) is non-negative.\N\NAn initial data set \((M, g, k)\) is a Riemannian manifold \((M, g)\) with a symmetric \((0, 2)\)-tensor \(k\). This satisfies the interior dominant energy condition if\N\[\N\mu -|J| \geq 0 \ ,\N\]\Nwhere\N\[\N\mu = \frac{1}{2}(\mathrm{Scal}_{g}+(\mathrm{trace}_{g}k)^2-|k|_{g}^2)\N\]\Nis the energy density and \(J= \mathrm{div}_g k - d(\mathrm{trace}_g k)\) is the momentum density. Moreover, \((M, g, k)\) satisfies the tilted boundary dominant energy condition if\N\[\NH_{\partial M} \pm \cos(\alpha)\mathrm{trace}_{\partial M}k \geq \sin(\alpha)|k(\eta, \cdot)^{T}|\N\]\Neverywhere along the boundary \(\partial M\), where \(H_{\partial M}\) is the mean curvature of \(\partial M\) computed with respect to the outward-pointing normal \(\eta\), \(\alpha \in [0, \pi/2]\) is a constant angle and \(k(\eta, \cdot)^{T}\) denotes the component of the \(1\)-form \(k(\eta, \cdot)\) tangential to \(\partial M\).\N\NThis paper establishes some general results of the spacetime positive mass theorem for asymptotically flat initial data sets with arbitrary ends and a non-compact boundary under the tilted boundary dominant energy condition.