Bounds on area and charge for marginally trapped surfaces with cosmological constant

We sharpen the known inequalities $A\mathrm{\Lambda }\le 4\pi (1-g)$ and $A\ge 4\pi Q^2$betweenthearea$A$andtheelectriccharge$Q$ of a stable marginally outer trapped surface (MOTS) of genus g in the presence of a cosmological constant $\mathrm{\Lambda }$. In particular, instead of requiring stability we include the principal eigenvalue $\lambda$ of the stability operator. For $\Lambda^{} = \Lambda + \lambda > 0$weobtainalowerandanupperboundfor$ \Lambda^{} A$intermsof$ \Lambda^{} Q^2$aswellastheupperbound$ Q \le 1/(2\sqrt{\Lambda^{}})$forthecharge,whichreducesto$ Q \le 1/(2\sqrt{\Lambda})$inthestablecase$\lambda \ge 0$\mathrm{.}For$\Lambda^{*} < 0$thereremainsonlyalowerboundon$A$. In the spherically symmetric, static, stable case one of the area inequalities is saturated iff the surface gravity vanishes. We also discuss implications of our inequalities for “jumps” and mergers of charged MOTS.