Panagiota Daskalopoulos — Neckpinch

Formation and structural stability of nondegenerate neckpinches

Tue, 08 Sep 2026 00:00:00 GMT

Kyeongsu Choi, Panagiota Daskalopoulos, Natasa Sesum — In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.

Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow

Fri, 03 Jul 2026 00:00:00 GMT

Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum, Ziyi Zhao — We obtain the unique asymptotics of SO(k)× SO(n-k+1)-invariant, compact, simply-connected, factorwisely non-self-similar n-dimensional κ-solutions of the Ricci flow (Mⁿ, g(t)), where n≥ 4 and 2≤ k≤ n-2. More precisely, these κ-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere Sⁿ, having a positive curvature operator metric g(t) and a cylindrical tangent flow at -∞, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric g(t) of every SO(k)× SO(n-k+1)-invariant ancient oval is represented in the form g(t)=dz⊗ dz + F²(z,t) g_(S^(k-1)) + G²(z,t)g_(S^(n-k)) (up to flipping k-1 and n-k). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function G(z, t), and prove that the uniqueness of G(z, t) implies the uniqueness of F(z, t). In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional κ-solutions of the Ricci flow.

Mean curvature flow near a peanut solution

Thu, 04 Dec 2025 00:00:00 GMT

Sigurd Angenent, Panagiota Daskalopoulos, Natasa Sesum — It was shown by Angenent, Altschuler and Giga, and by Angenent and Velazquez that there exist closed mean curvature flow solutions that extinct to a point in finite time, without ever becoming convex prior to their extinction. These solutions develop a degenerate neckpinch singularity, meaning that the tangent flow at a singularity is a round cylinder, but at the same time for each of these solutions there exists a sequence of points in space and time, so that the pointed blow up limit around this sequence is the Bowl soliton. These solutions are called peanut solutions and they were first conjectured to exist by Richard Hamilton, while the existence of those solutions was shown by Angenent, Altschuler and Giga. In this paper we show that this type of solutions are highly unstable, in the sense that in every small neighborhood of any such peanut solution we can find a perturbation so that the mean curvature flow starting at that perturbation develops spherical singularity, and at the same time we can find a perturbation so that the mean curvature flow starting at that perturbation develops a nondegenerate neckpinch singularity. We also show that appropriately rescaled subsequence of any sequence of solutions whose initial data converge to the peanut solution, and all of which develop spherical singularities, converges to the Ancient oval solution.

Dynamics of Convex Mean Curvature Flow

Fri, 26 May 2023 00:00:00 GMT

Sigurd Angenent, Panagiota Daskalopoulos, Natasa Sesum — There is an extensive and growing body of work analyzing convex ancient solutions to Mean Curvature Flow (MCF), or equivalently of Rescaled Mean Curvature Flow (RMCF). The goal of this paper is to complement the existing literature, which analyzes ancient solutions one at a time, by considering the space X of all convex hypersurfaces M, regard RMCF as a semiflow on this space, and study the dynamics of this semiflow. To this end, we first extend the well known existence and uniqueness of solutions to MCF with smooth compact convex initial data to include the case of arbitrary non compact and non smooth initial convex hypersurfaces. We identify a suitable weak topology with good compactness properties on the space X of convex hypersurfaces and show that RMCF defines a continuous local semiflow on X whose fixed points are the shrinking cylinder solitons, and for which the Huisken energy is a Lyapunov function. Ancient solutions to MCF are then complete orbits of the RMCF semiflow on X. We consider the set of all hypersurfaces that lie on an ancient solution that in backward time is asymptotic to one of the shrinking cylinder solitons and prove various topological properties of this set. We show that this space is a path connected, compact subset of X, and, considering only point symmetric hypersurfaces, that it is topologically trivial in the sense of Cech cohomology. We also give a strong evidence in support of the conjecture that the space of all convex ancient solutions with a point symmetry is homeomorphic to an n-1 dimensional simplex.

Classification of bubble-sheet ovals in ℝ⁴

Sun, 11 Sep 2022 00:00:00 GMT

Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du, Robert Haslhofer, Natasa Sesum — In this paper, we prove that any bubble-sheet oval for the mean curvature flow in ℝ⁴, up to scaling and rigid motion, either is the O(2)× O(2)-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of ℤ₂²× O(2)-symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.

Uniqueness of entire graphs evolving by Mean Curvature flow

Fri, 22 Oct 2021 00:00:00 GMT

Panagiota Daskalopoulos, Mariel Saez — In this paper we study the uniqueness of graphical mean curvature flow. We consider as initial conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique without any further assumptions. This result is then generalized for rotationally symmetric solutions. In the general n- dimensional case, we prove uniqueness under additional conditions: we require a { \em uniform lower bound } on the second fundamental form and the height function of the initial condition. The latter result extends to initial conditions that are proper graphs over subdomains of ℝⁿ.

Type II smoothing in mean curvature flow

Thu, 19 Aug 2021 00:00:00 GMT

Sigurd Angenent, Panagiota Daskalopoulos, Natasa Sesum — In 1994 Velazquez constructed a smooth O(4)× O(4) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velazquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Here we prove short time existence of Velazquez formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velazquez-Stolarski we therefore show that there exists a solution Mₜ⁷⊂R⁸ mid −t₀