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Natasa Sesum

Publications and source records attributed to Natasa Sesum.

At least 19 recordsLinked to original sources
arXiv2026

Formation and structural stability of nondegenerate neckpinches

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.

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arXiv2026

Mean curvature flow near a peanut solution

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.

math.AP↗
arXiv2026

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

We obtain the unique asymptotics of $SO(k)\times SO(n-k+1)$-invariant, compact, simply-connected, {factorwisely non-self-similar} $n$-dimensional $κ$-solutions of the Ricci flow $(M^n, g(t))$, where $n\geq 4$ and $2\leq k\leq n-2$. More precisely, these $κ$-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere $S^n$, having a positive curvature operator metric $g(t)$ and a cylindrical tangent flow at $-\infty$, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric $g(t)$ of every $SO(k)\times SO(n-k+1)$-invariant ancient oval is represented in the form $g(t)=dz\otimes dz + F^2(z,t) g_{S^{k-1}} + G^2(z,t)g_{S^{n-k}}$ (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.

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arXiv2026

Asymptotic Geometry of Four-Dimensional Steady Solitons

In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak $κ$-solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional steady solitons constructed by Lai.

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arXiv2025

Stability of neckpinch singularities

In this paper, we study the stability of neckpinch singularities. We show that if a mean curvature flow $\{M_t\}$ develops only finitely many neckpinch singularities at the first singular time, then the mean curvature flow starting at any sufficiently small perturbation of $M_0$ can also develop only neckpinch type singularities at the first singular time. We also show stability of nondegenerate neckpinch singularities in the above sense, which speaks in favor of stability of Type I singularities.

math.DG↗
arXiv2025

Local singularities of compact multiply warped Ricci flow solutions

We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal B \times {\mathbb S^2}, g)$, where $\mathcal B$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal B$, has to be isometric to the generalized cylinder $\mathbb R^2\times\mathbb S^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products -- but not products -- and provide rigorous examples of the formation of generalized cylinder singularity models $\mathbb R^k\times\mathbb S^\ell$.

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arXiv2024

Classification of bubble-sheet ovals in $\mathbb{R}^{4}$

In this paper, we prove that any bubble-sheet oval for the mean curvature flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-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.

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arXiv2023

Unique Asymptotics of Steady Ricci Solitons with Symmetry

In this paper we study 4d gradient steady Ricci solitons, which are weak $κ$-solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those for 3d compact $κ$-solutions found in [ABDS22]. This is the first step towards the classification of 4d gradient steady Ricci solitons and more general ancient Ricci flows.

math.DG↗
arXiv2023

Dynamics of Convex Mean Curvature Flow

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.

math.AP↗
arXiv2023

On the rate of convergence of the rescaled mean curvature flow

We estimate from above the rate at which a solution to the rescaled mean curvature flow on a closed hypersurface may converge to a limit self-similar solution, i.e. a shrinker. Our main result implies that any solution which converges to a shrinker faster than any fixed exponential rate must itself be shrinker itself.

math.DG↗
arXiv2021

Type II smoothing in mean curvature flow

In 1994 Velazquez constructed a smooth \(O(4)\times O(4)\) 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 \(\{M_t^7\subset\R^8 \mid -t_0 <t<t_0\}\) that has an isolated singularity at the origin \(0\in\R^8\), and at \(t=0\); moreover, the mean curvature is uniformly bounded on this solution, even though the second fundamental form is unbounded near the singularity.

math.AP↗
arXiv2021

Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which we describe. This analysis applies in particular to the ancient solution constructed by Perelman.

math.DG↗
arXiv2021

Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

In this paper, we study the classification of $κ$-noncollapsed ancient solutions to n-dimensional Ricci flow on $S^n$, extending the result in [13] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.

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arXiv2020

Singularity formation of complete Ricci flow solutions

We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cylinders as Ricci flow singularity models.

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arXiv2019

Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as $t\to-\infty$ and we give their precise asymptotic description. This description applies in particular to the solution constructed by G.Perelman

math.DG↗
arXiv2019

Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these solutions is expected to be the "blowdown soliton" discovered in [FIK03]. Our partial results support the conjecture that the blowdown soliton is stable under Ricci flow, as well as the conjectured stability of the subspace of Kahler metrics under Ricci flow.

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