https://neckpinch.com/topics/weak-flows/New arXiv papers on weak & singular flows, from Neckpinch Wed, 04 Nov 2026 13:15:59 GMT https://neckpinch.com/papers/2610.38929/arxiv:2610.38929Wed, 28 Oct 2026 00:00:00 GMTGaia Bombardieri, Mattia Fogagnolo, Valentina Franceschi — We study the arrival time equation of the mean-convex horizontal mean curvature flow in the Heisenberg group. We prove existence of viscosity solutions when the initial set is horizontally mean-convex and either non-characteristic, or axisymmetric and with horizontal mean curvature extending continuously across the characteristic set. A key ingredient is an approximation scheme by sets with prescribed Riemannian mean curvature, which preserves the mean-curvature bounds up to an arbitrarily small error. We further show that, when the evolving level sets form a generalized C²-foliation, the horizontal mean curvature blows up around characteristic points. This provides an obstruction to self-similar evolution in the axisymmetric class considered above and, in particular, shows that the Pansu sphere cannot evolve self-similarly.Mean curvature flowWeak & singular flows https://neckpinch.com/papers/2610.21545/arxiv:2610.21545Thu, 15 Oct 2026 00:00:00 GMTFabius Krämer, Tim Laux, Keisuke Takasao — We prove the generic existence of mean curvature flows in spaces with rough weights. Our weak solutions are both unit-density Brakke flows and distributional solutions. Such a generic statement is optimal in the sense that we consider weights that are too rough to have well-defined traces on every hypersurface.Mean curvature flowWeak & singular flows https://neckpinch.com/papers/2610.20453/arxiv:2610.20453Wed, 14 Oct 2026 00:00:00 GMTJingwen Chen — In this short note, using OpenAI's resolution of the Solomon–Yau conjecture, we extend our joint work with Pedro Gaspar to all dimensions n≥2. We construct eternal Allen–Cahn flows on Sⁿ⁺¹ connecting approximations of any prescribed least-area Clifford hypersurface and equatorial sphere. We also construct infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres.Mean curvature flowWeak & singular flowsPhase fields & multiphase flows https://neckpinch.com/papers/2610.20054/arxiv:2610.20054Wed, 14 Oct 2026 00:00:00 GMTA. Cesaroni, M. Novaga, P. Pozzi — The horizontal mean curvature flow is a geometric evolution associated with the horizontal perimeter in sub-Riemannian spaces. In this paper, we introduce a minimizing movement scheme for this flow in the first Heisenberg group ℍ, and prove the existence of a flat flow which is locally 1/2-Hölder continuous in time, with respect to the L¹-distance. The construction applies to arbitrary bounded sets of finite horizontal perimeter and does not require smoothness or symmetry of the initial datum.Mean curvature flowWeak & singular flows https://neckpinch.com/papers/2610.18761/arxiv:2610.18761Tue, 13 Oct 2026 00:00:00 GMTXinxiu Zhang — We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space ℍⁿ⁺¹, n≥ 2, and prove that equality in each inequality holds if and only if the h-convex body is a geodesic ball. A key ingredient is a rigidity theorem characterizing geodesic balls via proportionality between modified support measures of adjacent orders. Its proof combines the C^(1,α)∩ W^(2,k+1) approximation of the outer parallel hypersurfaces by smooth strictly h-convex hypersurfaces with the estimates for the total variation norm of the signed measure associated with the proportionality relation along the shifted curvature flows independent of initial curvature bounds. For 0≤ k≤ n-1, we construct weak flows preserving the k-th modified quermassintegral Wₖ with arbitrary h-convex initial bodies as Hausdorff limits of the smooth flows. The rigidity theorem above characterizes the limit shape of the flow and gives a positive lower bound on the rate of decrease of Wₖ₊₁ for a short time whenever the initial h-convex body is not a geodesic ball, which characterizes the equality cases in the modified Alexandrov-Fenchel inequality. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which characterizes the equality cases in the Alexandrov-Fenchel inequality.Other curvature flowsWeak & singular flowsStability & uniquenessGeometric inequalities https://neckpinch.com/papers/2610.17229/arxiv:2610.17229Mon, 12 Oct 2026 00:00:00 GMTShaosai Huang — Let compact RCD(κ,N) spaces converge in the measured Gromov–Hausdorff sense, and suppose that they have (δ,ρ)-Reifenberg local covers for one fixed ρ>0, with |κ|ρ²≤1 and δ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of O(n) whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with O(n), with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition Aₛ it is a closed aspherical manifold. Three appendices treat smooth manifolds with Ric≥−(n-1) and small local rewinding Reifenberg error: a uniform short-time Ricci flow with |Rm|≤α/t, also for complete non-compact data, where it is unique among complete flows with curvature at most A/t; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov–Hausdorff approximations with small error.Ricci flow(Non)collapsingWeak & singular flowsLimit spaces & RCDHomogeneous spacesBoundaries & non-compact https://neckpinch.com/papers/2610.10461/arxiv:2610.10461Wed, 07 Oct 2026 00:00:00 GMTBeomjun Choi, Wenkui Du, Seung Chul Park, Junseo Youn — In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow Kₜ starting from the boundary of a smooth bounded domain Ω⊂ℝⁿ⁺¹, as well as for the integral Brakke flow \μₜ\ with initial Radon measure μ₀=𝓗ⁿllcorner∂Ω, we establish the corresponding optimal extinction time estimates \beginequation* T_{\rm ext}(Ω),\, T^B_{\rm ext}(Ω) \leq \frac12n \left(\fracP( Ω){|{\mathbbS^n}|}\right)^{\frac2n}, \endequation* where P(Ω) is the perimeter of Ω representing the area of ∂Ω, and the equality holds if and only if Ω is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp L^p-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling X of outward minimizing level set flow starting from the boundary of a smooth bounded domain Ω⊂ ℝⁿ⁺¹: \beginequation* \mathcal H_par^n+2(X) \leq \fracπ {2n(n+2)^2|{\mathbbS^n}|^{{\frac2n}}} P(Ω)^{\fracn+2n}, \endequation* where the equality holds if and only if Ω is a round ball and the flow is standard smoothly shrinking round sphere.Mean curvature flowSelf-shrinkers & translatorsWeak & singular flowsGeometric inequalities https://neckpinch.com/papers/2610.06473/arxiv:2610.06473Mon, 05 Oct 2026 00:00:00 GMTPapa Badiane, Chinh H. Lu, Ahmed Zeriahi — We study regularity of bounded weak solutions to complex Monge-Ampère equations on compact Hermitian manifolds with right-hand side possibly decreasing in the unknown. Our proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Ampère eigenvalue problem. Our main result roughly says that any bounded solution is smooth in the regular locus of the data. When the right-hand side is strictly positive and smooth, we recover known results by Nie, Ko lodziej-Nguyen for the Hermitian case and Székelyhidi-Tosatti for the Kähler case, which rely on the regularizing property of the Kähler-Ricci flow.Ricci flowWeak & singular flowsEigenvalues & spectrum https://neckpinch.com/papers/2610.03129/arxiv:2610.03129Fri, 02 Oct 2026 00:00:00 GMTLaura Flower — We consider the problem of uniqueness of Ricci flows, starting from singular initial data. Even when the initial data is smooth and smoothly obtained, in the complete noncompact case uniqueness is not known in full generality. When starting from nonsmooth initial data, very little is known, even in the compact case. We obtain a uniqueness result for Ricci flows with scaling-invariant curvature bounds, a suitable noncollapsing assumption, and sectional curvature lower bounds, coming out of (possibly singular) Reifenberg spaces. As a consequence of our result, and existence results obtained by Simon–Topping and Lai, there is a canonical way to define a smooth structure on any Reifenberg uniformly volume-noncollapsed PIC2 limit space. Further, we obtain a diffeomorphic version of Perelman's stability theorem for these limit spaces.Ricci flow(Non)collapsingWeak & singular flowsStability & uniquenessBoundaries & non-compact https://neckpinch.com/papers/2610.00732/arxiv:2610.00732Wed, 30 Sep 2026 00:00:00 GMTAndrea Poiatti — The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.Mean curvature flowWeak & singular flowsLong-time existence & convergencePhase fields & multiphase flows https://neckpinch.com/papers/2609.01026/arxiv: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://neckpinch.com/papers/2608.26842/arxiv:2608.26842Thu, 27 Aug 2026 00:00:00 GMTTim Laux, Keisuke Takasao — Since Ilmanen's pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand the sharp-interface limit of systems of coupled Allen-Cahn equations. We prove - for the first time - a global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke.Mean curvature flowWeak & singular flowsPhase fields & multiphase flows https://neckpinch.com/papers/2608.19152/arxiv:2608.19152Wed, 19 Aug 2026 00:00:00 GMTLongteng Chen, Max Hallgren, Lucas Lavoyer — We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometryWeak & singular flowsBoundaries & non-compactAnalytic minimal model program https://neckpinch.com/papers/2608.05760/arxiv:2608.05760Thu, 06 Aug 2026 00:00:00 GMTKiichi Tashiro — In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.Mean curvature flowWeak & singular flowsLimit spaces & RCD https://neckpinch.com/papers/2608.00293/arxiv:2608.00293Fri, 31 Jul 2026 00:00:00 GMTYuchuan Yang, Selim Esedoglu — We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.Other curvature flowsAnisotropic & nonlocal flowsWeak & singular flowsNumerics & visualizationPhase fields & multiphase flows https://neckpinch.com/papers/2607.10426/arxiv:2607.10426Sat, 11 Jul 2026 00:00:00 GMTJavier Hervás — We develop a weak formulation of spacelike mean curvature flow in pseudo-Euclidean space. The framework is based on spacelike integer rectifiable varifolds and a pseudo-Euclidean version of first variation. Particular attention is paid to the presence of a fixed spacelike boundary. We introduce a generalized mean curvature vector and a weak spacelike conormal along the boundary. Under natural uniform spacelikeness and curvature assumptions, we prove a closure theorem for spacelike varifolds with boundary. We then define spacelike Brakke flows by means of the corresponding pseudo-Euclidean Brakke inequality, and prove a compactness theorem for sequences of spacelike Brakke flows with fixed boundary. Basic monotonicity properties are established, reflecting the sign structure of the ambient indefinite metric. Finally, we adapt Ilmanen's elliptic regularization procedure to prove existence of spacelike Brakke flows, and the local regularity theorem of White to the pseudo-Euclidean setting. The results provide a varifold setting for studying weak spacelike mean curvature flow beyond singularities.Mean curvature flowSingularitiesWeak & singular flowsLimit spaces & RCDPhysics https://neckpinch.com/papers/2607.06698/arxiv:2607.06698Tue, 07 Jul 2026 00:00:00 GMTKai Xu — An ∞-harmonic function is a viscosity solution of ∇² u(∇u,∇u)=0, or equivalently, an absolute minimizer of \|∇u\|_(L^∞). We prove a variety of new structural and regularity results in two dimensions, including: 1. ∞-harmonic functions in domains of ℝ² are C^(1,1/3). 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating ∞-harmonic functions to inverse mean curvature flow (IMCF) clusters – which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the p→∞ limit of the classical duality between p-harmonic and q-harmonic functions in ℝ², where 1/p+1/q=1.Mean curvature flowOther curvature flowsInverse mean curvature flowSingularitiesWeak & singular flowsBoundaries & non-compact https://neckpinch.com/papers/2607.03930/arxiv:2607.03930Sat, 04 Jul 2026 00:00:00 GMTTim Laux, Andrea Poiatti — We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate L² distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows https://neckpinch.com/papers/2607.02067/arxiv:2607.02067Thu, 02 Jul 2026 00:00:00 GMTChaoqun Gao, Yong Wei, Rong Zhou — We prove an anisotropic analogue of the higher regularity theorem of Huisken and Ilmanen for inverse mean curvature flow. For an arbitrary smooth Minkowski norm, we first prove a Huisken–Ilmanen type Harnack estimate for smooth closed strictly star-shaped solutions. We then construct global smooth solutions starting from C¹ strictly star-shaped hypersurfaces with bounded nonnegative weak anisotropic mean curvature. Combining this construction with the asymptotic theory for weak inverse anisotropic mean curvature flow, we show that weak solutions starting from bounded smooth initial sets become smooth outside a compact set.Mean curvature flowOther curvature flowsInverse mean curvature flowAnisotropic & nonlocal flowsHarnack inequalitiesWeak & singular flows https://neckpinch.com/papers/2607.00094/arxiv:2607.00094Tue, 30 Jun 2026 00:00:00 GMTTokuhiro Eto, Jiwoong Jang — This paper studies a Chambolle-type minimizing movement scheme for mean curvature flow with prescribed contact angle in a smooth bounded domain. The scheme is based on the capillary functional and the geodesic signed distance relative to the container, and yields a time-discrete level-set approximation. The main result asserts that, for every Lipschitz-continuous boundary function prescribing a strictly nondegenerate contact angle, the approximate solutions converge locally uniformly to the unique viscosity solution of the corresponding level-set mean curvature equation with oblique derivative boundary condition. This improves a previous convergence theorem, where the container was assumed to be convex and a curvature-type condition relating the tangential derivative of the prescribed contact-angle function to the principal curvatures of the container boundary was imposed. The main new ingredient is a uniform Lipschitz estimate for the solutions of the variational problems defining the scheme. This estimate is derived by applying a Bernstein-type argument to a suitable weighted gradient, rather than to the gradient itself, which rules out boundary maxima without relying on the previous curvature-type condition.Mean curvature flowWeak & singular flowsBoundaries & non-compactFree boundary & capillary https://neckpinch.com/papers/2606.22815/arxiv:2606.22815Mon, 22 Jun 2026 00:00:00 GMTFei Fang, Zhong Tan — This paper investigates the dynamical behaviors of solutions to the Yamabe flow via the modified potential well method. We first establish the local existence and regularity of weak solutions for the flow. Several new results concerning global existence and blowup are obtained by classifying initial data into stable and unstable sets. Specifically, solutions with initial data in the stable set exist globally and extinguish in finite time, whereas those originating from unstable initial data blow up in infinite time. For certain high-energy initial data, we show that the solution decays to zero as time tends to infinity and undergoes finite-time blowup. In addition, we analyze Palais-Smale sequences to reveal the intrinsic relationship between the long-time asymptotic behavior of solutions and steady states. Finally, we derive the Pohozaev identity for the equation and prove the corresponding nonexistence theorem.Yamabe flowSingularitiesWeak & singular flowsStability & uniquenessLong-time existence & convergence https://neckpinch.com/papers/2606.17353/arxiv: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://neckpinch.com/papers/2606.13301/arxiv:2606.13301Thu, 11 Jun 2026 00:00:00 GMTYu Li — Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any t₀>0 and any connected component Z_(t₀)' of the time-slice Z_(t₀) of the completion, we prove that (𝓡ₜ',d_(gₜ)) xrightarrow[t↗ t₀]mathrmGromov-Hausdorff (Z_(t₀)',d_(t₀)^Z), where 𝓡ₜ' is the corresponding connected component of the regular part and 𝓡ₜ' denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic 1-rectifiable, and its time image has vanishing 1/2-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most 1. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.Ricci flow(Non)collapsingHeat kernel3-manifoldsWeak & singular flowsLimit spaces & RCD https://neckpinch.com/papers/2606.04586/arxiv:2606.04586Wed, 03 Jun 2026 00:00:00 GMTTatsuya Miura, Fabian Rupp — We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along proper oriented mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. In fact, our result covers a class of singular calibrations and singular initial data. Even in the smooth setting, this provides a new finite variational framework for mean curvature flow beyond the finite-volume regime. As a main application, we establish a general dynamical rigidity theorem for calibrated cones in arbitrary codimension: no singular calibrated cone can be desingularized by a proper oriented mean curvature flow. Our framework further yields novel rigidity theorems for solitons and convergence for two-dimensional immortal flows.Mean curvature flowSurfacesWeak & singular flowsStability & uniquenessLong-time existence & convergence https://neckpinch.com/papers/2604.10007/arxiv:2604.10007Sat, 11 Apr 2026 00:00:00 GMTSajjad Lakzian — We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to the singular setting in a straightforward manner. These formulations are achieved by weakly formulating super Ricci flows and imposing a saturation condition (solely in terms of metric and measure) to ensure the super Ricci flow inequality is an equality.Ricci flowScalar curvatureWeak & singular flows https://neckpinch.com/papers/2603.24091/arxiv:2603.24091Wed, 25 Mar 2026 00:00:00 GMTVedansh Arya, Seongmin Jeon, Vesa Julin — The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.Mean curvature flowVolume-preserving & constrained flowsSingularitiesWeak & singular flowsLong-time existence & convergence https://neckpinch.com/papers/2603.18942/arxiv:2603.18942Thu, 19 Mar 2026 00:00:00 GMTMarco Flaim, Erik Hupp, Karl-Theodor Sturm — We present a synthetic notion of scalar curvature (and its integral) for Riemannian manifolds and metric measure spaces, defined in terms of the initial slope of a Gaussian (double) integral. We explicitly calculate the integral scalar curvature for Lipschitz gluings of smooth Riemannian manifolds and for cones. In dimension 2, the former coincides with the formula derived by Gauss-Bonnet, whereas the latter differs. The extension to the time-dependent case allows us to characterize Ricci flows as super Ricci flows with minimal integral curvature functional.Ricci flowScalar curvatureWeak & singular flowsLimit spaces & RCD https://neckpinch.com/papers/2602.12186/arxiv:2602.12186Thu, 12 Feb 2026 00:00:00 GMTTheodora Bourni, José M. Espinar, Aakash Mishra — We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow, and as a consequence we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become star-shaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend these results to the non-compact setting in two cases. First, assuming the asymptotic boundary of the solution consists of a single point, we show that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere. Second, assuming the asymptotic boundary consists of two points, we prove that the flow eventually becomes a global graph over a hyperbolic cylinder with uniform gradient bounds; this is achieved through an explicit cylindrical barrier construction analogous to the horospherical one.Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsBoundaries & non-compactLong-time existence & convergence https://neckpinch.com/papers/2511.13473/arxiv:2511.13473Mon, 17 Nov 2025 00:00:00 GMTAlix Deruelle, Vincent Guedj, Henri Guenancia, Ahmed Zeriahi — Given a compact Kähler manifold X and a closed, positive (1,1)-current T on X, we find sufficient conditions for T to induce a metric structure (X,d_T) which is the Gromov-Hausdorff limit of compact Kähler manifolds either in a "static" way or at time zero of smooth Kähler-Ricci flows. In dimension 1 we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a Kähler-Ricci flow.Ricci flowKähler–Ricci flowKähler geometryWeak & singular flowsLimit spaces & RCD https://neckpinch.com/papers/2511.11477/arxiv:2511.11477Fri, 14 Nov 2025 00:00:00 GMTMatthias Erbar, Marco Flaim, Eric Hupp, Zhenhao Li, Timo Schultz, Karl-Theodor Sturm — We review different notions of synthetic Ricci flow that apply to time-dependent families of metric measure spaces and which are based on properties of the heat flow, ideas from optimal transport, and the asymptotic behaviour of volumes. Each notion equivalently characterises (weighted) Ricci flow for smooth families of weighted Riemannian manifolds. We discuss the features of the different notions on various examples.Ricci flowWeak & singular flowsLimit spaces & RCD