https://neckpinch.com/authors/robert-haslhofer/New arXiv papers by Robert Haslhofer, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.01026arxiv:2609.01026Tue, 01 Sep 2026 00:00:00 GMTYueheng Bao, Robert Haslhofer — In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.Mean curvature flowSingularitiesWeak & singular flowsStability & uniquenessFree boundary & capillary https://arxiv.org/abs/2608.30787arxiv:2608.30787Mon, 31 Aug 2026 00:00:00 GMTKyeongsu Choi, Robert Haslhofer — In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is – up to splitting Euclidean factors – either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).Mean curvature flowAncient solutionsSingularities https://arxiv.org/abs/2606.17353arxiv:2606.17353Mon, 15 Jun 2026 00:00:00 GMTYueheng Bao, Robert Haslhofer — We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types ℝ^k× S^(n-k), ℝ^k_+× S^(n-k) or ℝ^k× S^(n-k)_+. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.Mean curvature flowAncient solutionsSingularitiesWeak & singular flowsFree boundary & capillary https://arxiv.org/abs/2604.01981arxiv:2604.01981Thu, 02 Apr 2026 00:00:00 GMTRobert Haslhofer — The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to illustrate many important concepts from nonlinear partial differential equations, including maximum principle estimates, monotonicity formulas, Harnack inequalities and blowup analysis. All these techniques will be combined to give an exposition of Huisken's proof of Grayson's beautiful theorem that the curve shortening flow shrinks any closed embedded curve in the plane to a round point.Curve shorteningSingularitiesHarnack inequalities https://arxiv.org/abs/2510.01355arxiv:2510.01355Wed, 01 Oct 2025 00:00:00 GMTRobert Haslhofer — We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in ℝ³. With the aim of developing a satisfying theory in higher dimensions, we then describe our recent classification of all noncollapsed singularities in ℝ⁴. Finally, we provide a detailed discussion of open problems and conjectures.Mean curvature flowSingularities(Non)collapsingWeak & singular flows https://arxiv.org/abs/2412.10581arxiv:2412.10581Fri, 13 Dec 2024 00:00:00 GMTKyeongsu Choi, Robert Haslhofer — In this paper, we classify all noncollapsed singularities of the mean curvature flow in ℝ⁴. Specifically, we prove that any ancient noncollapsed solution either is one of the classical historical examples (namely ℝ^j× S^(3-j), ℝ×2d-bowl, ℝ×2d-oval, the rotationally symmetric 3d-bowl, or a cohomogeneity-one 3d-oval), or belongs to the 1-parameter family of ℤ₂× O₂-symmetric 3d-translators constructed by Hoffman-Ilmanen-Martin-White, or belongs to the 1-parameter family of ℤ₂²× O₂-symmetric ancient 3d-ovals constructed by Du-Haslhofer. In light of the five prior papers on the classification program in ℝ⁴ from our collaborations with Du, Hershkovits, and Choi-Daskalopoulos-Sesum, the major remaining challenge is the case of mixed behaviour, where the convergence to the round bubble-sheet is fast in x₁-direction, but logarithmically slow in x₂-direction. To address this, we prove a differential neck theorem, which allows us to capture the (dauntingly small) slope in x₁-direction. To establish the differential neck theorem, we introduce a slew of new ideas of independent interest, including switch and differential Merle-Zaag dynamics, anisotropic barriers, and propagation of smallness estimates. Applying our differential neck theorem, we show that every noncompact strictly convex solution is selfsimilarly translating, and also rule out exotic ovals.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsing3-manifoldsHomogeneous spacesBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2308.01448arxiv:2308.01448Wed, 02 Aug 2023 00:00:00 GMTRobert Haslhofer — We introduce a classification conjecture for κ-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of ℤ₂²× O₃-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.Ricci flowMean curvature flowAncient solutionsSingularities3-manifolds4-manifoldsWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2307.01828arxiv:2307.01828Tue, 04 Jul 2023 00:00:00 GMTRobert Haslhofer, Daniel Ketover — In this paper, we prove that every strictly convex 3-ball with nonnegative Ricci-curvature contains at least 3 embedded free-boundary minimal 2-disks for any generic metric, and at least 2 solutions even without genericity assumption. Our approach combines ideas from mean curvature flow, min-max theory and degree theory. We also establish the existence of smooth free-boundary mean-convex foliations. In stark contrast to our prior work in the closed setting, the present result is sharp for generic metrics.Mean curvature flowFree boundary & capillaryMinimal & CMC surfaces https://arxiv.org/abs/2306.07714arxiv:2306.07714Tue, 13 Jun 2023 00:00:00 GMTRobert Haslhofer — In this paper, we prove the existence of mean curvature flow with surgery for mean-convex surfaces with free boundary. To do so, we implement our recent new approach for constructing flows with surgery without a prior estimates in the free boundary setting. The flow either becomes extinct in finite time or for t→∞ converges smoothly in the one or two sheeted sense to a finite collection of stable connected minimal surfaces with empty or free boundary (in particular, there are no surgeries for t sufficiently large). Our free boundary flow with surgery will be applied in forthcoming work with Ketover, where we will address the existence problem for 3 free boundary minimal disks in convex balls.Mean curvature flowSingularitiesSurgeryFree boundary & capillaryLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2305.19137arxiv:2305.19137Tue, 30 May 2023 00:00:00 GMTKyeongsu Choi, Robert Haslhofer, Or Hershkovits — 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 C⁰-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 C⁰-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 ℝ⁴, we derive these estimates in the context of ancient ovals in ℝ³ and noncollapsed translators in ℝ⁴, though our methods seem to apply in other settings as well.Ricci flowMean curvature flowAncient solutionsSelf-shrinkers & translators(Non)collapsing https://arxiv.org/abs/2209.04931arxiv:2209.04931Sun, 11 Sep 2022 00:00:00 GMTBeomjun 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.Mean curvature flowAncient solutionsHomogeneous spaces https://arxiv.org/abs/2208.14280arxiv:2208.14280Tue, 30 Aug 2022 00:00:00 GMTWenkui Du, Robert Haslhofer — Some worrisome potential singularity models for the mean curvature flow are rotating ancient flows, i.e. ancient flows whose tangent flow at -∞ is a cylinder ℝ^k× S^(n-k) and that are rotating within the ℝ^k-factor. We note that while the ℝ^k-factor, i.e. the axis of the cylinder, is unique by the fundamental work of Colding-Minicozzi, the uniqueness of tangent flows by itself does not provide any information about rotations within the ℝ^k-factor. In the present paper, we rule out rotating ancient flows among all ancient noncollapsed flows in ℝ⁴.Mean curvature flowAncient solutionsSingularities(Non)collapsingStability & uniqueness https://arxiv.org/abs/2110.03412arxiv:2110.03412Thu, 07 Oct 2021 00:00:00 GMTRobert Haslhofer — In this short note, we observe that the Bamler-Kleiner proof of uniqueness and stability for 3-dimensional Ricci flow through singularities generalizes to singular Ricci flows in higher dimensions that satisfy an analogous canonical neighborhood property. In particular, this gives a canonical evolution through singularities for manifolds with positive isotropic curvature. The new ingredients we use are the recent classification of higher dimensional κ-solutions by Brendle, Daskalopoulos, Naff and Sesum, and the maximum principle for the linearized Ricci-DeTurck flow on locally conformally flat manifolds due to Chen and Wu.Ricci flowSingularities3-manifoldsSphere theoremsWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2109.13875arxiv:2109.13875Tue, 28 Sep 2021 00:00:00 GMTBeomjun Choi, Robert Haslhofer — We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.Ricci flowAncient solutionsSingularities3-manifoldsWeak & singular flows https://arxiv.org/abs/2108.02944arxiv:2108.02944Fri, 06 Aug 2021 00:00:00 GMTBeomjun Choi, Robert Haslhofer — In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner-Lott, Haslhofer-Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler's precompactness theorem, as well as every singular Ricci flow from Kleiner-Lott, is a weak solution in the sense of Haslhofer-Naber. We also generalize all path-space estimates from Haslhofer-Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.Ricci flowSingularities(Non)collapsingHeat kernelWeak & singular flowsLimit spaces & RCD https://arxiv.org/abs/2107.04443arxiv:2107.04443Fri, 09 Jul 2021 00:00:00 GMTWenkui Du, Robert Haslhofer — We consider ancient noncollapsed mean curvature flows in ℝ⁴ whose tangent flow at -∞ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function u that measures the deviation of the renormalized flow from the round cylinder ℝ² × S¹(√2) and prove that for τ→ -∞ we have the fine asymptotics u(y,θ,τ)= (y^top Qy −2tr(Q))/|τ| + o(|τ|⁻¹), where Q=Q(τ) is a symmetric 2× 2-matrix whose eigenvalues are quantized to be either 0 or −1/√8. This naturally breaks up the classification problem for general ancient noncollapsed flows in ℝ⁴ into three cases depending on the rank of Q. In the case rk(Q)=0, generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or ℝ×2d-bowl. In the case rk(Q)=1, under the additional assumption that the flow either splits off a line or is selfsimilarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be ℝ×2d-oval or belongs to the one-parameter family of 3d oval-bowls constructed by Hoffman-Ilmanen-Martin-White, respectively. Finally, in the case rk(Q)=2 we show that the flow is compact and SO(2)-symmetric and for τ→-∞ has the same sharp asymptotics as the O(2)×O(2)-symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper.Mean curvature flowAncient solutionsSingularities(Non)collapsing3-manifoldsEigenvalues & spectrum