A differential geometry study generalises a method developed by Li, Yau, Simon and Topping for closed surfaces immersed in flat Euclidean space, extending it to surfaces immersed in a complete Riemannian three-manifold. The result is a global estimate for the Willmore energy that incorporates an explicit cut-locus contribution and does not require the surface to be embedded: self-intersections are allowed. The bound also applies to ambient spaces whose sectional curvature is bounded above and whose injectivity radius may be finite.

The setting is the study of the Willmore energy, a functional measuring how far a surface is from being a minimal sphere, arising in problems ranging from membrane elasticity to minimal surface theory. Classical inequalities by Li, Yau, Simon and Topping controlled this energy in Euclidean space, but extending them to curved spaces faced two obstacles: the presence of the cut locus, where the geometry ceases to be injective, and the possibility that the surface self-intersects. The new work addresses both issues at once.

The strategy combines weighted global estimates with an explicit analysis of the cut-locus contribution. When the upper bound on sectional curvature is positive, the global weighted estimates additionally require an explicit radius restriction. The Willmore energy is bounded below by a functional depending on the self-intersection number of the surface and the injectivity radius of the ambient space. By investigating in what sense these inequalities are optimal — and are realised by geodesic spheres in constant curvature spaces — the authors introduce a weighted version of the Willmore functional, adapted to geodesic spheres in the model spaces and taking the sectional curvature of the ambient space into account.

As an immediate corollary, the work obtains sufficient conditions for a surface immersed in a curved space to be embedded, that is, free of self-intersections. The result thus connects analytic properties of the Willmore functional with the topology of the immersion, and opens the door to applications in the study of surfaces in manifolds of bounded curvature, where Euclidean methods were not enough.