Rigidity and the interpretation of mass with a positive cosmological constant

We provide an explicit counterexample to the rigidity properties underlying the interpretation of mass in the presence of a positive cosmological constant $\Lambda$. Specifically, we construct a family of regular static stellar configurations satisfying the dominant and strong energy conditions, and containing no surface layers, for which the mass parameter of the outer Schwarzschild-de Sitter region can be positive, zero, or negative. The zero-mass configuration is precisely the one for which the outer vacuum region becomes exactly de Sitter, yielding a spacetime in which a de Sitter domain coexists with regular perfect-fluid matter. This contrasts sharply with the $\Lambda=0$ case, where a Minkowski domain cannot coexist with perfect-fluid regions satisfying the energy conditions. These results show that the static, spherically symmetric realization of the rigidity principle associated with the positive mass theorem for $\Lambda=0$ does not carry over to $\Lambda>0$.