Willmore inequalities beyond the injectivity radius

We generalize a method first introduced by Li, Yau, Simon, and Topping (for closed surfaces immersed in the flat Euclidean space), and we establish a global-in-nature estimate, with an explicit cut-locus contribution, on the Willmore energy of closed surfaces that are immersed in a complete Riemannian three-manifold and need not be embedded. Our estimate applies to ambient spaces whose sectional curvature is bounded above and whose injectivity radius may be finite. For positive upper curvature bounds, the global weighted estimates require an explicit radius restriction. The Willmore energy is bounded below by a functional involving the self-intersection number of the surface and the injectivity radius of the ambient space. Furthermore, by investigating in what sense our inequalities are optimal and realized by geodesic spheres in constant curvature spaces, we are led to introduce a weighted version of the Willmore functional, which is adapted to geodesic spheres in the model spaces and takes the sectional curvature of the ambient space into account. As a corollary, we obtain sufficient conditions for an immersed surface in a curved space to be embedded.