Surgery and total mean curvature

We prove Gromov’s conjecture on the total mean curvature of fill-ins in various cases. Our methods are based on surgery to reduce the statement to fill-ins of spheres, which can be treated by instances of the positive mass theorem. For spin fill-ins, where we permit the mean curvature to take negative values, we build on a classical surgery result of Lawson-Michelsohn and a recent positive mass theorem with creases by Kazaras-Khuri-Lin. For non-spin fill-ins of spin manifolds, where we assume the mean curvature to be non-negative, we develop a novel quantitative surgery process to reduce the general situation to a result of Shi-Wang-Wei. We also treat the case of fill-ins of non-spin manifolds, provided there is a fixed positive lower bound on the mean curvature.

💡 Research Summary

The paper addresses Gromov’s conjecture that, for a fixed closed Riemannian manifold ((M,g_M)) and a scalar curvature lower bound (\sigma), the total mean curvature (\int_{\partial\Omega} H,d\mathrm{vol}) of any fill‑in ((\Omega,g_\Omega)) with (\operatorname{scal}(g_\Omega)\ge\sigma) cannot become arbitrarily large. The authors prove the conjecture in three broad settings, each requiring a different combination of geometric analysis, surgery theory, and positive mass theorems.

Theorem (A) assumes a positive mean‑curvature lower bound (\kappa>0) and dimension (n\le6). The proof proceeds by a topological reduction: using surgeries of codimension at least two, the authors replace (M) by a sphere (S^n) while simultaneously attaching corresponding handles to the fill‑in. By choosing the handles sufficiently thin, the mean curvature of the new boundary changes only by a controlled error. The resulting spherical fill‑in falls under the Shi–Wang–Wei result, which supplies a uniform constant (\Lambda(M,g_M,\sigma,\kappa)). An induction on the number of handles completes the argument.

Theorem (B) treats the case (\kappa=0) (non‑negative mean curvature) but requires that (M) be spin. This is the most delicate part. The authors develop a new quantitative surgery procedure that refines the classical Gromov–Lawson construction. For a submanifold (S\subset M) of codimension (k\ge3) they build a family of “Gromov–Lawson handles” (\Sigma_\varepsilon) inside (M\times