https://neckpinch.com/topics/minimal-surfaces/New arXiv papers on minimal & cmc surfaces, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.25833arxiv:2610.25833Mon, 19 Oct 2026 00:00:00 GMTGregŕio Silva Neto — We describe the self-expanders of the mean curvature flow in ℝ^(p+q+2) invariant under O(p+1)× O(q+1). Apart from the Clifford minimal cone, every such hypersurface is complete, proper and noncompact, and belongs to one of three families: two one-parameter families of embedded hypersurfaces diffeomorphic to ℝ^(p+1)×S^(q) or to ℝ^(q+1)×S^(p), whose profile curves leave an axis, and a two-parameter family diffeomorphic to ℝ×S^(p)×S^(q), parametrized by the point of the profile curve closest to the origin, which is the generic family: its set of initial conditions is open and of full measure in the phase space. Every end is asymptotically conical, with explicit decay rates, and every such hypersurface has Euclidean volume growth. A profile curve of the third family has at most one loop, and the point of the curve closest to the origin lies on that loop.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfaces https://arxiv.org/abs/2610.25831arxiv:2610.25831Mon, 19 Oct 2026 00:00:00 GMTGregório Silva Neto — We describe the global structure of the self-shrinkers of the mean curvature flow in ℝ^(p+q+2) invariant under O(p+1)× O(q+1), without assuming properness or embeddedness. The main tool is a pair of piecewise monotone quantities along the profile curves, in the spirit of Hsiang's analysis of constant mean curvature hypersurfaces. We prove that every such self-shrinker other than the two invariant cylinders intersects both of them transversely, and all of them except the Clifford minimal cone are complete. Apart from those of the cylinders, each end is either asymptotic to a cone or meets both cylinders infinitely many times. Consequently, a complete noncompact example is proper if and only if all its ends are asymptotically conical, and this is equivalent to polynomial volume growth, to finite entropy, and to finitely many intersections with the cylinders. The reduced flow preserves a finite measure, and for almost every initial condition the profile curve is either closed or generates a complete nonproper self-shrinker of infinite entropy. We also bound the winding of the profile curve by its number of cylinder crossings and localize its self-intersections.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfaces https://arxiv.org/abs/2610.01713arxiv:2610.01713Thu, 01 Oct 2026 00:00:00 GMTDebora Impera, Michele Rimoldi, Francesco Ruatta — We study complete translating solitons for the mean curvature flow in arbitrary codimension under quadratic pinching of the second fundamental form. We prove a codimension reduction theorem at the Andrews-Baker threshold: if a complete n-dimensional connected translator isometrically immersed in (ℝ^(n+p),⟨·,·⟩) satisfies |B|²≤ (4/3n-ε₀)|H|² for some ε₀>0, then either it is an affine n-plane or |H|>0 everywhere and the translator is contained in an (n+1)-dimensional affine subspace. In particular, this improves in low dimensions the pinching range previously obtained from codimension reduction results for ancient mean curvature flows. Our proof is purely elliptic: viewing translators as weighted minimal submanifolds, we combine Simons-type identities for the drift Laplacian with refined gradient estimates and an Omori-Yau maximum principle. As a consequence, under the stronger pinching condition |B|²≤ (cₙ−ε₀)|H|², cₙ=min\4/3n,1/(n-2)\ for n≥3, with cₙ=2/3 when n=2, we obtain a rigidity theorem: the translator is either an affine plane or a bowl soliton contained in an (n+1)-dimensional affine subspace. No entropy assumption is required.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsCurvature pinchingStability & uniquenessMinimal & CMC surfaces https://arxiv.org/abs/2608.11011arxiv:2608.11011Tue, 11 Aug 2026 00:00:00 GMTJunyoung Park — In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an O(n)× O(m) symmetric minimal quadratic cone for n +m ≥ 10, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a singular minimal cone as the asymptotic model.Mean curvature flowAncient solutionsStability & uniquenessMinimal & CMC surfaces https://arxiv.org/abs/2606.31641arxiv:2606.31641Tue, 30 Jun 2026 00:00:00 GMTAntonio Bueno — We study oriented surfaces in the Heisenberg space Nil₃ whose mean curvature H at each point is H=⟨ N,∂_z⟩+λ, where N is the unit normal, ∂_z is the vertical Killing vector field and λ∈ℝ. These surfaces are known as λ-translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow. The objective in this paper is to classify λ-translators invariant by the following one-parameter groups of isometries of Nil₃: left-translations, rotations and helicoidal motions.Mean curvature flowSelf-shrinkers & translatorsMinimal & CMC surfaces https://arxiv.org/abs/2605.22347arxiv:2605.22347Thu, 21 May 2026 00:00:00 GMTHaoyuan Sun — We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.Ricci flowGeneralized & G₂ flows(Non)collapsingKähler geometryLimit spaces & RCDMinimal & CMC surfacesAnalytic minimal model program https://arxiv.org/abs/2605.10589arxiv:2605.10589Mon, 11 May 2026 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.Mean curvature flowHarnack inequalitiesMinimal & CMC surfaces https://arxiv.org/abs/2602.08850arxiv:2602.08850Mon, 09 Feb 2026 00:00:00 GMTTheodora Bourni, Giada Franz — We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.Mean curvature flowAncient solutionsStability & uniquenessFree boundary & capillaryLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2601.03092arxiv:2601.03092Tue, 06 Jan 2026 00:00:00 GMTKuan-Hui Lee — In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperkähler 4-manifold. In, Wang and Tsai proved a uniqueness theorem and C¹ dynamic stability theorem of the mean curvature flow for minimal surface. We extend their results and obtain a similar dynamic stability theorem of the hyperkähler flow.Mean curvature flowRicci-flat & Calabi–Yau4-manifoldsKähler geometryStability & uniquenessMinimal & CMC surfaces https://arxiv.org/abs/2511.21545arxiv:2511.21545Wed, 26 Nov 2025 00:00:00 GMTTarcios Andrey Ferreira, João Paulo dos Santos — We consider the hyperbolic three-space in the half-space model endowed with a metric Lie group structure. In this setting, translation surfaces are defined as products of two curves α and β with respect to the Lie group operation. We investigate minimal surfaces and solitons to the mean curvature flow arising from specific types of products of these curves. In particular, we provide classification results for minimal surfaces, hyperbolic translators, and conformal solitons to the mean curvature flow.Mean curvature flowSelf-shrinkers & translatorsSurfacesHomogeneous spacesMinimal & CMC surfaces https://arxiv.org/abs/2511.00435arxiv:2511.00435Sat, 01 Nov 2025 00:00:00 GMTYaoting Gui, Yuqiao Li, Jun Sun — In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.Mean curvature flowVolume-preserving & constrained flowsStability & uniquenessBoundaries & non-compactLong-time existence & convergenceGeometric inequalitiesMinimal & CMC surfaces https://arxiv.org/abs/2509.00197arxiv:2509.00197Fri, 29 Aug 2025 00:00:00 GMTRuojing Jiang, Franco Vargas Pallete — This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic 3-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by −1 and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by −6 and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly C⁰-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.Ricci flow3-manifoldsSphere theoremsScalar curvatureMinimal & CMC surfaces https://arxiv.org/abs/2509.00188arxiv:2509.00188Fri, 29 Aug 2025 00:00:00 GMTRuojing Jiang, Franco Vargas Pallete — On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric h₀, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to h₀ in a weighted Hölder norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].Ricci flow3-manifoldsStability & uniquenessLong-time existence & convergenceHarmonic maps & DeTurck trickMinimal & CMC surfaces https://arxiv.org/abs/2508.09806arxiv:2508.09806Wed, 13 Aug 2025 00:00:00 GMTAri J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll, Lisandra Sauer, Rodrigo Soares — We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary differential equation, which allows the construction of explicit boundary barriers and it can also be applied to unbounded domains. In the setting of Hadamard manifolds, this condition relates the geometry of the domain to the admissible boundary data in a direct way. In Euclidean space, the condition leads to solvability under geometric hypotheses of a different nature from those in the classical Jenkins-Serrin theory, and in some configurations it applies where the Jenkins-Serrin method does not. A central point of the present approach is that the geometric restrictions and the boundary data enter independently. The same barrier construction can be used for graphical mean curvature flow. This yields short-time existence with prescribed boundary values even when the boundary of the domain is not mean convex. When mean convexity is present, one recovers the classical graphical setting.Mean curvature flowHarmonic maps & DeTurck trickMinimal & CMC surfaces https://arxiv.org/abs/2507.03837arxiv:2507.03837Fri, 04 Jul 2025 00:00:00 GMTJingwen Chen, Pedro Gaspar — From the perspective of Morse theory, it is natural to investigate gradient flow trajectories between critical points. In this short note, we explore the minimal hypersurface analogue of this phenomenon and present examples that suggest additional topological and variational obstructions to the existence of connecting mean curvature flows.Mean curvature flowMinimal & CMC surfaces https://arxiv.org/abs/2506.23245arxiv:2506.23245Sun, 29 Jun 2025 00:00:00 GMTCaiyan Li, Hengyu Zhou — In this paper, we propose a new assumption (1.2) that involves a small oscillation and C² norms for maps from smooth bounded domains into Euclidean spaces. Furthermore, by assuming that the domain has non-negative Ricci curvature, we establish the Dirichlet problem for the minimal surface system via the mean curvature flow (MCF) with boundary. The long-time existence of such flow is derived using Bernstein-type theorems of higher codimensional self-shrinkers in the whole space and the half-space. Another novel aspect is that our hypothesis imposes no restriction on the diameter of the domains, which implies an existence result for an exterior Dirichlet problem of the minimal surface system.Mean curvature flowSelf-shrinkers & translatorsLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2505.17892arxiv:2505.17892Fri, 23 May 2025 00:00:00 GMTRomina M. Arroyo, Gabriela P. Ovando, Mariel Sáez — In this work we study the existence of solutions to the Mean Curvature Flow for which the initial condition has the structure of a two-dimensional Lie subgroup within a Lie group of dimension three. We consider Lie groups with a fixed left-invariant metric and first observe that if the Lie group is unimodular, then every Lie subgroup is a minimal surface (hence a trivial solution). For this reason we focus on non-unimodular Lie groups, finding the evolution of every Lie subgroup of dimension 2 (within a 3 dimensional Lie group). These evolutions are self-similar for abelian subgroups (i.e. evolve by isometries), but not self-similar in the other cases.Mean curvature flow3-manifoldsSurfacesHomogeneous spacesMinimal & CMC surfaces https://arxiv.org/abs/2505.03499arxiv:2505.03499Tue, 06 May 2025 00:00:00 GMTFranciele Conrado, Detang Zhou — In this paper, we study the drifted Laplacian Δ_f on a hypersurface M in a Ricci shrinker (M,g,f). We prove that the spectrum of Δ_f is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of Δ_f when the hypersurface is an embedded f-minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang's estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers.Ricci solitonsEinstein metricsSelf-shrinkers & translatorsEigenvalues & spectrumMinimal & CMC surfaces https://arxiv.org/abs/2504.03316arxiv:2504.03316Fri, 04 Apr 2025 00:00:00 GMTYukai Sun, Guangrui Zhu — In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in a shrinking gradient Ricci soliton (Mⁿ,g,f) with scalar curvature R≥(n-1)λ must have vanished second fundamental form and vanished normal Ricci curvature. For shrinking gradient Ricci solitons with scalar curvature R≥(n-1)λ, the existence of an area-minimizing hypersurface would imply that M is splitting.Ricci solitonsScalar curvatureMinimal & CMC surfaces https://arxiv.org/abs/2502.18455arxiv:2502.18455Tue, 25 Feb 2025 00:00:00 GMTLiam Mazurowski, Jintian Zhu — We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let (M³,g) be an asymptotically flat manifold with scalar curvature R≥ 0. Then, for each constant c>0, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. Likewise, let (M³,g) be an asymptotically hyperbolic manifold with scalar curvature R≥ −6. Then, for each constant c>2, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any T the inverse mean curvature flow emerging out of a point p far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times t∈ (-∞,T].Mean curvature flowOther curvature flowsInverse mean curvature flowScalar curvatureBoundaries & non-compactFree boundary & capillaryMinimal & CMC surfaces https://arxiv.org/abs/2501.13091arxiv:2501.13091Wed, 22 Jan 2025 00:00:00 GMTCarlo Sinestrari, Jacopo Tenan — We study the volume preserving mean curvature flow of a surface immersed in an asymptotically flat 3-manifold modeling an isolated gravitating system in General Relativity. We show that, if the ambient manifold has positive ADM mass and the initial surface is round in a suitable sense, then the flow exists for all times and converges smoothly to a stable CMC surface. This extends to the asymptotically flat setting a classical result by Huisken-Yau (Invent. Math. 1996) and allows to construct a CMC foliation of the outer part of the manifold by an alternative approach to the ones by Nerz (Calc. Var. PDE, 2015) or by Eichmair-Koerber (J. Diff. Geometry, 2024).Mean curvature flowVolume-preserving & constrained flowsBoundaries & non-compactLong-time existence & convergenceMinimal & CMC surfacesPhysics https://arxiv.org/abs/2501.12759arxiv:2501.12759Wed, 22 Jan 2025 00:00:00 GMTJohn Lott — We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.Ricci flow4-manifoldsLong-time existence & convergenceMinimal & CMC surfacesAnalytic minimal model program https://arxiv.org/abs/2501.06951arxiv:2501.06951Sun, 12 Jan 2025 00:00:00 GMTYipeng Wang — We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of T^(rtimes)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the T^(rtimes)-stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.Ricci flowScalar curvatureStability & uniquenessFree boundary & capillaryMinimal & CMC surfaces https://arxiv.org/abs/2412.03475arxiv:2412.03475Wed, 04 Dec 2024 00:00:00 GMTAlexander Mramor, Ao Sun — In this article we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3-manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument then we construct piecewise almost regular flows which either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in 3-manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods.Mean curvature flowSingularities3-manifoldsMinimal & CMC surfaces https://arxiv.org/abs/2409.12810arxiv:2409.12810Thu, 19 Sep 2024 00:00:00 GMTYaoting Gui, Yuqiao Li, Jun Sun — We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is sufficiently small. Then we show that from sufficiently large initial coordinate sphere, the area preserving mean curvature flow exists for all time and converges exponentially fast to a constant mean curvature surface in 3-dimensional asymptotically Schwarzschild spaces. This provides a new approach to the existence of foliation established by Huisken and Yau. And also a uniqueness result followsMean curvature flowVolume-preserving & constrained flows3-manifoldsSphere theoremsStability & uniquenessLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2409.09315arxiv:2409.09315Sat, 14 Sep 2024 00:00:00 GMTAdrian Chun-Pong Chu, Yangyang Li — In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5 embedded minimal tori. We confirm this conjecture for 3-spheres of positive Ricci curvature. While our proof uses min-max theory, the underlying heuristics are largely inspired by mean curvature flow.Mean curvature flowMinimal & CMC surfaces https://arxiv.org/abs/2408.10751arxiv:2408.10751Tue, 20 Aug 2024 00:00:00 GMTLouis Yudowitz — We prove lower and upper semi-continuity of the Morse index for sequences of gradient Ricci shrinkers which bubble tree converge in the sense of past work by the author and Buzano. Our proofs rely on adapting recent arguments of Workman which were used to study certain sequences of CMC hypersurfaces and were in turn adapted from work on Da Lio-Gianocca-Riviere. Moreover, we are able to refine Workman's methods by using techniques related to polynomially weighted Sobolev spaces. This all also requires us to extend the analysis to handle when the shrinkers we study are non-compact, which we can do due to the availability of a suitable notion of finite weighted volume. Finally, we identify a technical condition which ensures the Morse index of an asymptotically conical shrinker is bounded below by the f-index of its asymptotic cone.Ricci solitonsBoundaries & non-compactMinimal & CMC surfacesDiscrete & graph Ricci https://arxiv.org/abs/2405.18283arxiv:2405.18283Tue, 28 May 2024 00:00:00 GMTNikolaos Kapouleas, Jiahua Zou — For each large enough m∈ℕ we construct by PDE gluing methods a closed embedded smooth minimal hypersurface breveMₘ doubling the equatorial three-sphere Sₑq³ in S⁴(1), with breveMₘ containing m² bridges modelled after the three-dimensional catenoid and centered at the points of a square m× m lattice L contained in the Clifford torus T²⊂ Sₑq³. This answers a long-standing question of Yau in the case of S⁴(1) and long-standing questions of Hsiang. Similarly we construct a self-shrinker breveM_(shr,m) of the Mean Curvature Flow in ℝ⁴ doubling the three-dimensional spherical self-shrinker Sₛₕᵣ³⊂ ℝ⁴ with the bridges centered at the points of a square m× m lattice L contained in a Clifford torus T²⊂ Sₛₕᵣ³. Both constructions respect the symmetries of the lattice L as a subset of S⁴(1) or ℝ⁴ and are based on the Linearized Doubling (LD) methodology which was first introduced in the construction of minimal surface doublings of Sₑq² in S³(1). Furthermore breveMₘ converges as m →∞ in the varifold sense to 2Sₑq³, and its volume |breveMₘ| < 2|Sₑq³|.Mean curvature flowSelf-shrinkers & translators3-manifoldsWeak & singular flowsMinimal & CMC surfaces https://arxiv.org/abs/2405.01062arxiv:2405.01062Thu, 02 May 2024 00:00:00 GMTKyeongsu Choi, Jiuzhou Huang, Taehun Lee — We construct an I-family of ancient graphical mean curvature flows over a minimal hypersurface in ℝⁿ⁺¹ of finite total curvature with the Morse index I by establishing exponentially fast convergence in terms of |x|²−t. As a corollary, we show that these ancient flows have finite total curvature and finite mass drop. Moreover, one family of these flows is mean convex by a pointwise estimate.Mean curvature flowAncient solutionsLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2403.14782arxiv:2403.14782Thu, 21 Mar 2024 00:00:00 GMTAlancoc dos Santos Alencar, Keti Tenenblat — We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by the distance function of the parallel hypersurfaces. The solutions to the flow are obtained explicitly when the distinct principal curvatures have the same multiplicity. This is an additional assumption only for isoparametric hypersurfaces of the hyperbolic space or of the sphere with two or four distinct principal curvatures. The boundaries of the maximal interval of definition, when finite, are determined in terms of the number g of distinct principal curvatures, their multiplicities m and the mean curvature H of the initial hypersurface. We describe the collapsing submanifolds of the flow at the boundaries of the interval. In particular, we show in the Euclidean space the solutions are eternal, while in the hyperbolic space there are eternal and immortal solutions. Starting with a connected isoparametric submanifold of the sphere, we show that the flow is an ancient solution, that collapses into a minimal hypersurface whose square length of its second fundamental form and its scalar curvature are constants given in terms of g and n. The minimal hypersurface is totally geodesic when g=1, it is a Clifford minimal hypersurface of the sphere when g=2 and it is a Cartan type minimal submanifold when g∈\3,4,6\.Mean curvature flowOther curvature flowsInverse mean curvature flowAncient solutions(Non)collapsingSphere theoremsScalar curvatureLong-time existence & convergenceMinimal & CMC surfaces