arXiv subjects

Or Hershkovits

Publications and source records attributed to Or Hershkovits.

At least 19 recordsLinked to original sources

arXiv2026

A gradient estimate for the linearized translator equation

In this paper, we develop some analytic foundations for the linearized translator equation in $\mathbb{R}^4$, i.e. in the first dimension where the Bernstein property fails. This equation governs how the (noncompact) singularity models of the mean curvature flow in $\mathbb{R}^4$ fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for $W_v$, namely for the derivative of the variation field $W$ in the tip region. This serves as a substitute for the fundamental quadratic concavity estimate from Angenent-Daskalopoulos-Sesum, which has been crucial for controlling $Y_v$, namely the derivative of the profile function $Y$ in the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for $W_τ$ as well. Hence, our gradient estimate also serves as substitute Hamilton's Harnack inequality, which has played an important role for controlling $Y_τ$ in the tip region.

math.AP↗

arXiv2026

Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

In this paper we demonstrate that if two mean curvature flows of compact hypersurfaces $M^1_t$ and $M^2_t$ encounter only isolated, multiplicity one, asymptotically conical singularities at the first singular time $T$, and if $M^1_T=M^2_T$ then $M^1_t=M^2_t$ for every $t\in [0,T]$. This is seemingly the first backwards uniqueness result for any geometric flow with singularities, that assumes neither self-shrinking nor global asymptotically conical behaviour. This necessitates the development of new global tools to deal with both the core of the singularity, its asymptotic structure, and the smooth part of the flows simultaneously. As an immediate application, we show that low entropy flows in $\mathbb{R}^4$ are backwards unique

math.DG↗

arXiv2025

The linearized translator equation and applications

In this paper, we consider the linearized translator equation $L_ϕu=f$, around entire convex translators $M=\textrm{graph}(ϕ)\subset\mathbb{R}^4$, i.e. in the first dimension where the Bernstein property fails. Here, $L_ϕu=\mathrm{div} (a_ϕD u)+ b_ϕ\cdot Du$ is a mean curvature type elliptic operator, whose coefficients degenerate as the slope tends to infinity. We derive two fundamental barrier estimates, specifically an upper-lower estimate and an inner-outer estimate, which allow to propagate $L^\infty$-control between different regions. Packaging these and further estimates together we then develop a Fredholm theory for $L_ϕ$ between carefully designed weighted function spaces. Combined with Lyapunov-Schmidt reduction we infer that the space $\mathcal{S}$ of noncollapsed translators in $\mathbb{R}^4$ is a finite dimensional analytic variety and that the tip-curvature map $κ:\mathcal{S}\to\mathbb{R}$ is analytic. Together with the main result from our prior paper (Camb. J. Math. '23) this allows us to complete the classification of noncollapsed translators in $\mathbb{R}^4$. In particular, we conclude that the one-parameter family of translators constructed by Hoffman-Ilmanen-Martin-White is uniquely determined by the smallest principal curvature at the tip.

math.DG↗

arXiv2025

How close is too close for singular mean curvature flows?

Suppose $(M^i_t)_{t\in [0,T)}$, $i=1,2$, are two mean curvature flows in $\mathbb{R}^{n+1}$ encountering a multiplicity one compact singularity at time $T$, in such a manner that for every $k$, the Hausdorff distance between the two flows, $d_H$, satisfies $d_{H}(M^1_t,M^2_t)/(T-t)^k \rightarrow 0$. We demonstrate that $M^1_t=M^2_t$ for every $t$. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where $M^1_t$ is itself a self-similarly shrinking flow.

math.DG↗

arXiv2023

Avoidance for Set-Theoretic Solutions of Mean-Curvature-Type Flows

We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint as long as one of the subsolutions is compact; previously, this was only known for Euclidean space (with no ambient vectorfield). The new version (Sept 15, 2023) corrects a minor typo.

math.DG↗

arXiv2023

Mean Curvature Flow in de Sitter space

We study mean convex mean curvature flow $M_s$ of local spacelike graphs in the flat slicing of de Sitter space. We show that if the initial slice is of non-negative time and is graphical over a large enough ball, and if $M_s$ is of bounded mean curvature, then as $s$ goes to infinity, $M_s$ becomes graphical in expanding balls, over which the gradient function converges to $1$. In particular, if $p_s$ is the point lying over the center of the domain ball in $M_s$, then $(M_s,p_s)$ converges smoothly to the flat slicing of de Sitter space. This has some relation to the mean curvature flow approach to the cosmic no hair conjecture.

math.DG↗

arXiv2023

Moving plane method for varifolds and applications

In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compact varifolds with smooth boundary and in the setting of complete varifolds. A key ingredient is a Hopf lemma for stationary and CMC-varifolds. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compact CMC-varifolds in hyperbolic space.

math.DG↗

arXiv2023

Classification of noncollapsed translators in $\mathbb{R}^4$

In this paper, we classify all noncollapsed singularity models for the mean curvature flow of 3-dimensional hypersurfaces in $\mathbb{R}^4$ or more generally in $4$-manifolds. Specifically, we prove that every noncollapsed translating hypersurface in $\mathbb{R}^4$ is either $\mathbb{R}\times$2d-bowl, or a 3d round bowl, or belongs to the one-parameter family of 3d oval bowls constructed by Hoffman-Ilmanen-Martin-White.

math.DG↗

arXiv2023

A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$

Some of the most worrisome potential singularity models for the mean curvature flow of $3$-dimensional hypersurfaces in $\mathbb{R}^4$ are noncollapsed wing-like flows, i.e. noncollapsed flows that are asymptotic to a wedge. In this paper, we rule out this potential scenario, not just among self-similarly translating singularity models, but in fact among all ancient noncollapsed flows in $\mathbb{R}^4$. Specifically, we prove that for any ancient noncollapsed mean curvature flow $M_t=\partial K_t$ in $\mathbb{R}^4$ the blowdown $\lim_{λ\to 0} λ\cdot {K_{t_0}}$ is always a point, halfline, line, halfplane, plane or hyperplane, but never a wedge. In our proof we introduce a fine bubble-sheet analysis, which generalizes the fine neck analysis that has played a major role in many recent papers. Our result is also a key first step towards the classification of ancient noncollapsed flows in $\mathbb{R}^4$, which we will address in a series of subsequent papers.

math.DG↗

arXiv2023

Enhanced profile estimates for ovals and translators

We consider the profile function of ancient ovals and of noncollapsed translators. Recall that pioneering work of Angenent-Daskalopoulos-Sesum (JDG '19, Annals '20) gives a sharp $C^0$-estimate and a quadratic concavity estimate for the profile function of two-convex ancient ovals, which are crucial in their papers as well as a slew of subsequent papers on ancient solutions of mean curvature flow and Ricci flow. In this paper, we derive a sharp gradient estimate, which enhances their $C^0$-estimate, and a sharp Hessian estimate, which can be viewed as converse of their quadratic concavity estimate. Motivated by our forthcoming work on ancient noncollapsed flows in $\mathbb{R}^4$, we derive these estimates in the context of ancient ovals in $\mathbb{R}^3$ and noncollapsed translators in $\mathbb{R}^4$, though our methods seem to apply in other settings as well.

math.AP↗

arXiv2022

Ancient asymptotically cylindrical flows and applications

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon=\varepsilon(x_0,t_0)>0$ such that $M_t\cap B(x_0,\varepsilon)$ is mean-convex for all $t\in(t_0-\varepsilon^2,t_0+\varepsilon^2)$. As in the case $n=2$, which was resolved by the first three authors in arXiv:1810.08467, the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors, we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when $n\geq 3$, smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call "moving plane method without assuming smoothness" - where smoothness and symmetry are established in tandem.

math.DG↗

arXiv2021

Ancient low entropy flows, mean convex neighborhoods, and uniqueness

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in $\mathbb{R}^3$ with entropy at most $\sqrt{2π/e}+δ$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed.

math.DG↗

arXiv2020

A de Sitter no-hair theorem for 3+1d Cosmologies with isometry group forming 2-dimensional orbits

We study, using Mean Curvature Flow methods, 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying the dominant and the strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are the closed orbits of a symmetry group. If these surfaces have non-positive Euler characteristic (or in the case of 2-spheres, if the initial 2-spheres are large enough) and also if the initial spatial slice is expanding everywhere, then we prove that asymptotically the spacetime becomes physically indistinguishable from de Sitter space on arbitrarily large regions of spacetime. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations.

hep-th↗

arXiv2019

A note on the selfsimilarity of limit flows

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establish a local version of this equivalence for neck singularities in arbitrary dimension. In particular, we see that the ancient ovals occur as limit flows if and only if there is a sequence of spherical singularities converging to a neck singularity.

math.DG↗

arXiv2018

Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

Let $M\subset {\mathbf R}^{m+1}$ be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial $k^{\rm th}$ homology. We show that the entropy of $M$ is greater than or equal to the entropy of a round $k$-sphere, and that if equality holds, then $M$ is a round $k$-sphere in ${\mathbf R}^{k+1}$.

math.DG↗

arXiv2018

Translators asymptotic to cylinders

We show that the Bowl soliton in $\mathbb{R}^3$ is the unique translating solutions of the mean curvature flow which has the family of shrinking cylinders as an asymptotic shrinker at $-\infty$. As an application, we show that for a generic mean curvature flow, all (non-static) translating limit flows are the bowl soliton. The crucial point is that we do not make any global convexity assumption, while as the same time, the asymptotic requirement is very weak.

math.DG↗

arXiv2018

Non-fattening of mean curvature flow at singularities of mean convex type

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. This generalizes the well known fact that the level set flow of a mean convex initial hypersurface M does not fatten. This also provides the first instance where non-fattening is concluded from local information around the singular set.

math.DG↗

arXiv2017

The moduli space of two-convex embedded spheres

We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds.

math.DG↗