Weakly $$\delta $$-trapped spacelike submanifolds in weighted standard static spacetimes: height estimates, parabolicity, and rigidity

Abstract We investigate complete weakly $$\delta $$ δ -trapped spacelike submanifolds immersed in weighted standard static spacetimes. Our approach is based on the analysis of the height function associated with the static time coordinate and on suitable differential inequalities involving its drift Laplacian. Within this framework, we establish several rigidity and nonexistence results under distinct global geometric and analytic assumptions. In particular, we obtain slice characterization theorems under polynomial weighted volume growth, nonlinear differential inequalities, $$L^p$$ L p -integrability conditions and gradient Ricci soliton structures. We also develop a parabolicity approach, showing that weighted p -parabolicity, combined with an appropriate integrability condition on the gradient of the height function, forces a complete weakly $$\delta $$ δ -trapped submanifold to be $$\delta $$ δ -minimal. Further rigidity results are derived from weighted parabolicity and integrability hypotheses, yielding conditions under which the submanifold must be contained in a spacelike slice.