Uniqueness of spherical self-shrinkers in higher dimensions

Determine whether, for dimensions n ≥ 3, the only compact embedded self-shrinker in Euclidean space R^{n+1} (i.e., a hypersurface satisfying H + <X, ν> = 0) that is diffeomorphic to the sphere S^n is the round sphere.

Background

A λ-hypersurface satisfies H + <X, ν> = λ; the case λ = 0 corresponds to self-shrinkers for the mean curvature flow. The round sphere is a canonical example of a compact embedded self-shrinker.

There is a well-known conjecture that the round sphere is the only embedded self-shrinker diffeomorphic to a sphere. Brendle proved this in dimension 2, but the general higher-dimensional case remains unresolved.

References

For \lambda=0 (that is, self-shrinkers), there is a well-known conjecture that asserts the round sphere should be the only embedded self-shrinker which is diffeomorphic to a sphere. Brendle proved the above conjecture for $2$-dimensional self-shrinker. For the higher dimensional self-shrinkers, the conjecture is still open.

— Examples of compact embedded mean convex $λ$-hypersurfaces  (2603.29371 - Cheng et al., 31 Mar 2026) in Introduction, Section 1