How curved spacetime shapes the size and survival of topological solitons

Scalar field theories with nontrivial vacuum structure typically cannot form stable localized defects in higher dimensions — a collapse driven by simple scaling arguments. By building a Bogomol'nyi (first-order) framework for rotationally symmetric curved backgrounds, the authors show that geometry itself, encoded in a single radial function, can counteract this instability. A key result is that the orbital paths traced in field-space are the same across different spacetimes, even though the actual soliton profiles differ — geometry sculpts the defect's size and existence without altering its target-space topology. Exact solutions are worked out for Minkowski, Schwarzschild, de Sitter, Schwarzschild–de Sitter, and conformally flat backgrounds.