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A gradient estimate for the linearized translator equation Open
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 linearized translator equation and applications Open
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_…
Backwards uniqueness for Mean curvature flow with asymptotically conical singularities Open
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 $M^1_T=M^2_T$…
How close is too close for singular mean curvature flows? Open
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,…
Mean Curvature Flow in de Sitter space Open
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 bounde…
Enhanced profile estimates for ovals and translators Open
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 pr…
Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness Open
The equation of course also makes sense in higher dimension and co-dimension and in other ambient manifolds.In this paper, however, we focus on evolving surfaces in R 3 .k. choi, r. haslhofer and o. hershkovits Classification of ancient lo…
The moduli space of two-convex embedded spheres Open
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-connec…
Classification of noncollapsed translators in $\mathbb{R}^4$ Open
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 hype…
A nonexistence result for wing-like mean curvature flows in\n $\\mathbb{R}^4$ Open
Some of the most worrisome potential singularity models for the mean\ncurvature flow of $3$-dimensional hypersurfaces in $\\mathbb{R}^4$ are\nnoncollapsed wing-like flows, i.e. noncollapsed flows that are asymptotic to a\nwedge. In this pa…
A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$ Open
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,…
A note on the selfsimilarity of limit flows Open
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 conv…
A de Sitter no-hair theorem for 3+1d Cosmologies with isometry group forming 2-dimensional orbits Open
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 …
Moving plane method for varifolds and applications Open
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…
Ancient asymptotically cylindrical flows and applications Open
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…
Translators asymptotic to cylinders Open
We show that the Bowl soliton in ℝ 3 {\mathbb{R}^{3}} is the unique translating solution of the mean curvature flow whose tangent flow at - ∞ {-\infty} is the shrinking cylinder. As an application, we show that for a generic mean curvature…
Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type Open
We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝ n + 1 remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface …
Ancient low entropy flows, mean convex neighborhoods, and uniqueness Open
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 …
Avoidance for Set-Theoretic Solutions of Mean-Curvature-Type Flows Open
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 import…
The Moduli Space of Two-Convex Embedded Tori Open
EPSRC [EP/M011224/1 to R.B.]; NSERC [RGPIN-2016-04331], NSF [DMS-1406394], and a Connaught New Researcher Award (to R.H.); AMS-Simons travel grant (to O.H.).
Non-fattening of mean curvature flow at singularities of mean convex\n type Open
We show that a mean curvature flow starting from a compact, smoothly embedded\nhypersurface M remains unique past singularities, provided the singularities\nare of mean convex type, i.e., if around each singular point, the surface moves\ni…
Mean curvature flow of Reifenberg sets Open
In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in [math] starting from any [math] –dimensional [math] –Reifenberg flat set with [math] sufficiently small. More precisely, we show that the …
Isoperimetric properties of the mean curvature flow Open
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimension…
Ancient solutions of the mean curvature flow Open
In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near the center they have asymptotic shrinkers model…