Paper · Jan 14, 2026

Classification of ancient ovals in higher dimensional mean curvature flow

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Abstract

We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in $R^{n + 1}$ , called ancient ovals. Our main result is the classification of $k$ -ovals: any $k$ -oval (characterized by having cylindrical blow down $R^{k} \times S^{n - k}$ and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open $(k - 1)$ -simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of $k$ -ovals in arbitrary dimensions based on a different spectral parametrization.

Mean curvature flow Ancient solutions (Non)collapsing Eigenvalues & spectrum

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

12B.4
Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.
9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

Classification of ancient ovals in higher dimensional mean curvature flow

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