A generalization of the ADM mass for asymptotically Euclidean manifolds of weak regularity

Abstract We propose a new definition of the ADM mass for asymptotically Euclidean manifolds inspired by the definition of mass for weakly regular asymptotically hyperbolic manifolds by Gicquaud and Sakovich. This version of the mass allows one to work with metrics of local Sobolev regularity $$ W^{1,2}_\text {loc} \cap L^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mtext>loc</mml:mtext> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>∩</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:mrow> </mml:math> and we show, under suitable asymptotic assumptions, that the mass is finite, invariant under a change of coordinates at infinity and that it agrees with the classical ADM mass in the smooth setting. We also provide an expression in terms of the Ricci tensor that agrees with the Ricci version of the ADM mass studied by Herzlich.