For every closed Riemannian n-manifold (M,g) and every real number lambda, there exists a constant C(M,g,lambda) such that, for every compact Riemannian (n+1)-manifold X with boundary (partial X,g_partial X) isometric to (M,g) and scalar curvature satisfying operatornameScal_X≥lambda, one has int_partial XH_X,dA≤ C(M,g,lambda), where H_X is the mean curvature of partial X with respect to the outward unit normal.

Gromov’s mean-of-the-mean-curvature conjecture

Gromov's conjecture asks whether, for a fixed closed boundary $M$ and a lower scalar-curvature bound, the total mean curvature of every compact filling has a uniform upper bound. Gromov formulated related conjectures in 2020; the unrestricted statement is not established.

Known results

Gromov (2020) proved intrinsic rigidity theorems motivating these conjectures, including a boundary length-decreasing-map obstruction.
A separate preprint claims, in the spin case and assuming $H\ge -\eta$ , the estimate $\frac{1}{\operatorname{vol}(M)}\int_M H\le C(M)+\sqrt{\frac{n}{n+1}}\lambda+\eta$ ; versions without the spin assumption are reported for boundary dimension $n\le 6$ when $\eta=0$ .

September 1, 2026 dimension-two advance

A report dated September 1, 2026 describes a preprint confirming the conjecture in dimension two under intrinsic boundary-length data and a lower Gauss-curvature bound. It also gives higher-dimensional examples relevant to removing the mean-curvature lower bound, without disproving Gromov's original bounded-setting formulation; the claims are unverified.

Current status (as of September 2026): the conditional dimension-two result and several bounded-mean-curvature variants are claimed, while the unrestricted higher-dimensional conjecture remains open and unverified.

Gromov’s mean-of-the-mean-curvature conjecture

Primary source

Additional references

Progress summary

Solutions 0