https://neckpinch.com/topics/numerics/New arXiv papers on numerics & visualization, from Neckpinch Wed, 04 Nov 2026 13:15:59 GMT https://neckpinch.com/papers/2610.31672/arxiv:2610.31672Thu, 22 Oct 2026 00:00:00 GMTG. Bonelli, S. Lacroix, T. Pedroni, A. Tanzini — We construct a family of integrable 2d sigma models through 4d Chern-Simons theory and show that their classical point-particle reduction corresponds to a Hitchin system. The integrable Hitchin quantisation then provides a natural candidate Hamiltonian for the one-loop reduced dynamics of the sigma models, which coincides with the Perelman operator in all cases examined. This allows us to study the corresponding spectral problems using 4d 𝓝=2 gauge theory and 2d conformal field theory methods. For the sausage model, we derive exact quantisation conditions and compute exponentially small corrections essential for the agreement with numerical finite-volume results.Ricci flow4-manifoldsEigenvalues & spectrumPhysicsNumerics & visualization https://neckpinch.com/papers/2610.27340/arxiv:2610.27340Mon, 19 Oct 2026 00:00:00 GMTFan Zhang, Lee Lindblom — This study develops numerical methods for computing homogeneous and isotropic cosmological models on manifolds with non-trivial spatial topologies. The use of Ricci flow is introduced here to aid in the construction of globally homogeneous initial data on these manifolds. The initial data produced in this way are evolved by solving Einstein's equation numerically to construct homogeneous and isotropic cosmological models on three exemplar manifolds in this study: Seifert-Weber space, the lens space L(5,2), and Sixth-Turn space.Ricci flowHomogeneous spacesPhysicsNumerics & visualization https://neckpinch.com/papers/2608.07985/arxiv:2608.07985Sat, 08 Aug 2026 00:00:00 GMTYeong-Bok Bae, Young-Hwan Hyun, Gungwon Kang — We present a numerical method that constructs geometrically defined coordinates on black-hole horizons, from which the multipole moments are computed without assuming axisymmetry. This method, which we denote the conformal-mapping method (CMM), provides a numerical realization of the conformal construction proposed by Ashtekar et al. in 2022, combining discrete Ricci flow, spectral embedding onto the unit sphere, and Möbius gauge fixing by the vanishing-area-dipole condition. We first test the CMM against analytic Kerr benchmarks, and then apply it to an equal-mass, non-spinning binary black-hole merger. We also compare it with an approximate-symmetry-based method. The CMM allows the multipole moments to be expressed in a fixed reference frame, whereas the symmetry-adapted frame can reorient abruptly when the preferred approximate axis changes. In a frame aligned with the orbital angular momentum, the amplitude of the quadrupole mode grows during inspiral and decays after merger, displaying a qualitative ringdown behavior. These results show that the CMM is a useful tool for studying horizon geometry in dynamical situations where no stable symmetry axis is available.Ricci flowEigenvalues & spectrumPhysicsDiscrete & graph RicciNumerics & visualization https://neckpinch.com/papers/2608.02840/arxiv:2608.02840Mon, 03 Aug 2026 00:00:00 GMTEric Cochran, Arseny Mingajev, Lawrence Mouillé, Nazia Valiyakath — We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.Ricci flowHomogeneous spacesNumerics & visualization 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/2607.06100/arxiv:2607.06100Tue, 07 Jul 2026 00:00:00 GMTJihui Wang, Matteo Lulli, Antonino Marciano — We investigate the stochastic Ricci flow of spherically symmetric perturbations of the Schwarzschild–Anti de Sitter black-hole metric. Elaborating on the Ricci-flow analysis of Headrick and Wiseman, we include a negative cosmological constant through a Ricci-target term and study how the flow is correlated with the thermodynamic heat capacity of the black hole. Numerical simulations show that, in the positive heat-capacity regime, perturbations of the angular sector of the metric relax toward the Schwarzschild–Anti de Sitter fixed point, while in the negative heat-capacity regime they grow under the deterministic Ricci flow. We then introduce a multiplicative stochastic noise and find that sufficiently strong stochasticity can suppress the growth of these perturbations, effectively stabilizing configurations that would otherwise be thermodynamically unstable. Finally, we reformulate the dynamics in terms of an entropy variable evolving on a thermodynamic free-energy landscape, and support the metric-flow results through Monte Carlo simulations and the associated Fokker–Planck equation. These results suggest that stochastic fluctuations can modify the relation between geometric stability under Ricci flow and thermodynamic stability in asymptotically Anti de Sitter black-hole spacetimes.Ricci flowStability & uniquenessRotational symmetry & warped productsPhysicsNumerics & visualization https://neckpinch.com/papers/2605.21445/arxiv:2605.21445Wed, 20 May 2026 00:00:00 GMTKlaus Deckelnick, Vanessa Styles — The paper is concerned with the error analysis of a numerical scheme for the approximation of parametric mean curvature flow. The scheme we study is based on a reparametrization using the DeTurck trick and was proposed by Elliott and Fritz in [15]. In the semidiscrete case, for a spatial discretization by finite elements of order k ≥ 2 we prove an optimal H¹-error estimate for the position vector. We present numerical experiments that confirm this error bound and demonstrate that the scheme has good properties with respect to the distribution of mesh points as already observed in [15].Mean curvature flowHarmonic maps & DeTurck trickNumerics & visualization https://neckpinch.com/papers/2509.19989/arxiv:2509.19989Wed, 24 Sep 2025 00:00:00 GMTShuliang Bai, Rui Li, Shuang Liu, Xin Lai — Ricci curvature and Ricci flow have proven to be powerful tools for analyzing the geometry of discrete structures, particularly on undirected graphs, where they have been applied to tasks ranging from community detection to graph representation learning. However, their development on directed graphs remains limited, with Ricci flow being especially underexplored. In this work, we introduce a rigorous formulation of Ricci flow on directed weighted graphs, which evolves edge weights while preserving distances, and establish both the existence and uniqueness of its solutions. To capture the essence of asymmetry in directed networks and to enhance the capability of modeling more flexible structures, we incorporate a node-wise balancing factor that regulates between outflow and inflow. Building on the continuous Ricci flow evolution framework, we propose a discrete Ricci flow algorithm that is applicable to numerical computing. Numerical studies on various directed graph examples demonstrate the capacity of the proposed flow to reveal structural asymmetry and dynamic evolutions.Ricci flowStability & uniquenessDiscrete & graph RicciNumerics & visualization https://neckpinch.com/papers/2507.12088/arxiv:2507.12088Wed, 16 Jul 2025 00:00:00 GMTShuhui He, Ben Whale, Glen Wheeler, Valentina-Mira Wheeler — In this paper we propose and study a curvature-based mathematical model for dorsal closure in embryonic drosophila. Using an analysis that mixes maximum-principle and integral-estimates, we establish global existence and convergence for data that mimics the initial geometry of a dorsal closure event. Further, we present a numerical approximation scheme for the flow, establishing stability, consistence, and convergence. We also give sample simulations of the flow with initial configurations that include experimentally observed data.Other curvature flowsStability & uniquenessLong-time existence & convergenceNumerics & visualization https://neckpinch.com/papers/2505.12775/arxiv:2505.12775Mon, 19 May 2025 00:00:00 GMTMiroslav Kolar, Daniel Sevcovic — We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of nonlinear parabolic equations describing the motion of curves belonging to a given two-dimensional manifold. Using the abstract theory of analytic semiflows, we prove the local existence, uniqueness of Hölder smooth solutions to the governing system of nonlinear parabolic equations for the position vector parametrization of evolving curves. We apply the method of flowing finite volumes in combination with the methods of lines for numerical approximation of the governing equations. Qualitative analytical results are illustrated by various numerical experiments.Mean curvature flow3-manifoldsSurfacesStability & uniquenessNumerics & visualization https://neckpinch.com/papers/2503.15033/arxiv:2503.15033Wed, 19 Mar 2025 00:00:00 GMTPatrick Donovan — Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is SU(2) (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected SU(2)-invariant expanding and shrinking cohomogeneity one gradient Ricci solitons. The first result is the construction of new 3-parameter families of complete SU(2)-invariant asymptotically conical expanding gradient Ricci solitons. New shrinking Kähler U(2)-invariant gradient Ricci solitons in dimension 4 with orbifold singularities are also constructed, leading to a classification of such metrics when the base space of the orbifold is a simply-connected smooth manifold. Finally, we highlight numerical evidence that all the compact cohomogeneity one shrinking gradient Ricci solitons are known.Ricci solitonsSingularities4-manifoldsKähler geometryHomogeneous spacesBoundaries & non-compactNumerics & visualization https://neckpinch.com/papers/2411.15814/arxiv:2411.15814Sun, 24 Nov 2024 00:00:00 GMTGiovanna Citti, Nicolas Dirr, Federica Dragoni, Raffaele Grande — We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multiscale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the models of the visual cortex proposed by Petitot and Citti-Sarti, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field equation to curvature flows which are used in image processing. From a pure mathematical point of view, it provides a new approximation and regularization of Heisenberg mean curvature flow. Using the local structure of the rototranslational group, we extend the result to cover the model by Citti and Sarti. Numerically, the parameters in our algorithm interpolate between solving an Ementrout-Cowan type of equation and a Bence-Merriman-Osher algorithm type algorithm for sub-Riemannian mean curvature. We also reproduce some known exact solutions in the Heisenberg case.Mean curvature flowOther curvature flowsNumerics & visualization https://neckpinch.com/papers/2407.01240/arxiv:2407.01240Mon, 01 Jul 2024 00:00:00 GMTDaniel Ketover — For each positive integer g we use variational methods to construct a genus g self-shrinker Σ_g in ℝ³ with entropy less than 2 and prismatic symmetry group D_(g+1)×ℤ₂. For g sufficiently large, the self-shrinker Σ_g has two graphical asymptotically conical ends and the sequence Σ_g converges on compact subsets to a plane with multiplicity two as g→∞. Angenent-Chopp-Ilmanen conjectured the existence of such self-shrinkers in 1995 based on numerical experiments. Using these surfaces as initial conditions for large g, we obtain examples of mean curvature flows in ℝ³ with smooth initial non-compact data that evolve non-uniquely after their first singular time.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactNumerics & visualization https://neckpinch.com/papers/2406.16920/arxiv:2406.16920Mon, 10 Jun 2024 00:00:00 GMTRoman Bahadursingh — In this paper, we investigate the stochastic mean curvature flow (SMCF) on networks, a niche area within stochastic processes and geometric analysis. By applying Ito calculus, we analyze the evolution of network structures influenced by random perturbations. We derive a stochastic differential equation (SDE) for the network edges and utilize numerical simulations to study the stability, long-term behavior, and pattern formation in these systems. Our results offer new insights into the dynamics of complex networks under stochastic influences and open pathways for future research in stochastic geometry.Mean curvature flowStability & uniquenessNumerics & visualization https://neckpinch.com/papers/2403.09876/arxiv:2403.09876Thu, 14 Mar 2024 00:00:00 GMTSigurd Angenent, Evan Patrick Davis, Ellie DeCleene, Paige Ellingson, Ziheng Feng, Edgar Gevorgyan, Aris Lemmenes, Alex Moon, Tyler Joseph Tommasi, Yamin Zhou — We study possible tangles that can occur in singularities of solutions to plane Curve Shortening Flow. We exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. It seems that there are many examples of this kind and that a complete classification presents a problem similar to the problem of classifying all knots in ℝ³. As a particular example, we introduce the so-called n-loop curves, which generalize Matt Grayson's Figure-Eight curve, and we conjecture a generalization of the Coiculescu-Schwarz asymptotic bow-tie result, namely, a vanishing n-loop, when rescaled anisotropically to fit a square bounding box, converges to a "squeezed bow-tie," i.e. the curve \(x, y) : |x|≤ 1, y=± xⁿ⁻¹\∪\(± 1, y) : |y|≤ 1\. As evidence in support of the conjecture, we provide a formal asymptotic analysis on one hand, and a numerical simulation for the cases n=3 and n=4 on the other.Curve shorteningSingularitiesNumerics & visualization https://neckpinch.com/papers/2403.06427/arxiv:2403.06427Mon, 11 Mar 2024 00:00:00 GMTDavid Garfinkle, James Isenberg, Dan Knopf, Haotian Wu — Kröncke has shown that the Fubini-Study metric is an unstable generalized stationary solution of Ricci flow [Krö20]. In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini-Study metric develop local singularities modeled by the blowdown soliton discovered in [FIK03].Ricci flowSingularitiesStability & uniquenessNumerics & visualization https://neckpinch.com/papers/2401.13426/arxiv:2401.13426Wed, 24 Jan 2024 00:00:00 GMTVeit Krause, Axel Voigt — Wrinkling instabilities of thin elastic sheets can be used to generate periodic structures over a wide range of length scales. Viscosity of the thin elastic sheet or its surrounding medium has been shown to be responsible for dynamic processes. While this has been explored for solid as well as liquid thin elastic sheets we here consider wrinkling of fluid deformable surfaces, which show a solid-fluid duality and have been established as model systems for biomembranes and cellular sheets. We use this hydrodynamic theory and numerically explore the formation of wrinkles and their coarsening, either by a continuous reduction of the enclosed volume or the continuous increase of the surface area. Both lead to almost identical results for wrinkle formation and the coarsening process, for which a universal scaling law for the wavenumber is obtained for a broad range of surface viscosity and rate of change of volume or area. However, for large Reynolds numbers and small changes in volume or area wrinkling can be suppressed and surface hydrodynamics allows for global shape changes following the minimal energy configurations of the Helfrich energy for corresponding reduced volumes.Reduced volumeNumerics & visualization https://neckpinch.com/papers/2311.08537/arxiv:2311.08537Tue, 14 Nov 2023 00:00:00 GMTPatrick Guidotti — We revisit the well-known Curve Shortening Flow for immersed curves in the d-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is valid for all times and is insensitive to singularities. The construction is characterized by discretization in time and the approximant, while still exhibiting the possibile formation of finitely many singularities at a finite set of singular times, exists globally and is well behaved and simpler to analyze than a solution of the CSF. A solution of the latter is obtained in the limit. Estimates for a natural (geometric) norm involving length and total absolute curvature allow passage to the limit. Many classical qualitative results about the flow can be recovered by exploiting the simplicity of the approximant and new ones can be proved. The construction also suggests a numerical procedure for the computation of the flow which proves very effective as demonstrated by a series of numerical experiments scattered throughout the paper.Curve shorteningSingularitiesNumerics & visualization https://neckpinch.com/papers/2310.19870/arxiv:2310.19870Mon, 30 Oct 2023 00:00:00 GMTJames Halverson, Fabian Ruehle — We develop a general theory of flows in the space of Riemannian metrics induced by neural network gradient descent. This is motivated in part by recent advances in approximating Calabi-Yau metrics with neural networks and is enabled by recent advances in understanding flows in the space of neural networks. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel, a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman's formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi-Yau metrics, as is expected since the associated neural networks do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of neural network metric flows therefore explains why neural networks are better at learning Calabi-Yau metrics than fixed kernel methods, such as the Ricci flow.Ricci flowRicci-flat & Calabi–YauPoincaré conjecture3-manifoldsKähler geometryDiscrete & graph RicciNumerics & visualization https://neckpinch.com/papers/2308.12527/arxiv:2308.12527Thu, 24 Aug 2023 00:00:00 GMTHosea Wondo, Zhou Zhang — In this paper, we show that the singularity type of solutions to the Käher-Ricci flow on a numerically effective manifold does not depend on the initial metric. More precisely if there exists a type III solution to the Kähler-Ricci flow, then any other solution starting from a different initial metric will also be Type III. This generalises previous results by Y. Zhang for the semi-ample case.Ricci flowKähler–Ricci flowSingularitiesKähler geometryLong-time existence & convergenceAnalytic minimal model programNumerics & visualization https://neckpinch.com/papers/2306.12745/arxiv:2306.12745Thu, 22 Jun 2023 00:00:00 GMTRory Conboye — The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise flat approximations of smooth manifolds is adapted to avoid an inherent numerical instability. These adaptations have also been used in a related paper to show the convergence of the piecewise flat Ricci flow to known smooth Ricci flow solutions for a variety of manifolds.Ricci flowStability & uniquenessNumerics & visualization https://neckpinch.com/papers/2212.05892/arxiv:2212.05892Wed, 30 Nov 2022 00:00:00 GMTAarjav Jain, Challenger Mishra, Pietro Liò — Neural networks with PDEs embedded in their loss functions (physics-informed neural networks) are employed as a function approximators to find solutions to the Ricci flow (a curvature based evolution) of Riemannian metrics. A general method is developed and applied to the real torus. The validity of the solution is verified by comparing the time evolution of scalar curvature with that found using a standard PDE solver, which decreases to a constant value of 0 on the whole manifold. We also consider certain solitonic solutions to the Ricci flow equation in two real dimensions. We create visualisations of the flow by utilising an embedding into ℝ³. Snapshots of highly accurate numerical evolution of the toroidal metric over time are reported. We provide guidelines on applications of this methodology to the problem of determining Ricci flat Calabi–Yau metrics in the context of String theory, a long standing problem in complex geometry.Ricci flowRicci-flat & Calabi–YauKähler geometryScalar curvaturePhysicsDiscrete & graph RicciNumerics & visualization https://neckpinch.com/papers/2210.14705/arxiv:2210.14705Wed, 26 Oct 2022 00:00:00 GMTDavide De Biasio, Julian Freigang, Dieter Lust, Toby Wiseman — Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field. We argue that this flow is well-posed for static spacetimes with pure electric or magnetic potentialsand show it preserves both non-extremal and extremal black hole horizons. In the latter case we find the flow of the near horizon geometry decouples from that of the exterior. The Schwarzschild black hole is an unstable fixed point of Ricci flow for static spacetimes. Here we consider flows of the Reissner-Nordström (RN) fixed point. The magnetic RN solution becomes a stable fixed point of the flow for sufficient charge. However we find that the electric RN black hole is always unstable. Numerically solving the flow starting with a spherically symmetric perturbation of a non-extremal RN solution, we find similar behaviour in the electric case to the Ricci flows of perturbed Schwarzschild, namely the horizon shrinks to a singularity in finite time or expands forever. In the magnetic case, a perturbed unstable RN solution has a similar expanding behaviour, but a perturbation that decreases the horizon size flows to a stable black hole solution rather than a singularity. For extremal RN we solve the near horizon flow for spherical symmetry exactly, and see in the electric case two unstable directions which flow to singularities in finite flow time. However, even turning these off, and fixing the near horizon geometry to be that of RN, we numerically show that the flows appear to become singular in the vicinity of its horizon.Ricci flowSingularitiesStability & uniquenessRotational symmetry & warped productsPhysicsNumerics & visualization https://neckpinch.com/papers/2206.01890/arxiv:2206.01890Sat, 04 Jun 2022 00:00:00 GMTDavid Garfinkle, James Isenberg, Dan Knopf, Haotian Wu — In previous work [GIKW21], we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally-symmetric, complete, noncompact embedded hypersurfaces constructed in [IW19, IWZ21] are stable. More precisely, it is shown in that paper that for small rotationally-symmetric perturbations of initial embeddings near the "tip", numerical simulations of MCF of such initial embeddings develop the same Type-II singularities with the same "bowl soliton" blowup behaviors in a neighborhood of the singularity. It is also shown in that work that for small rotationally-symmetric perturbations of the initial embeddings that are sufficiently far away from the tip, MCF develops Type-I "neckpinch" singularities. In this work, we again use numerical simulations to show that MCF subject to initial perturbations that are not rotationally symmetric behaves asymptotically like it does for rotationally-symmetric perturbations. In particular, if we impose sinusoidal angular dependence on the initial embeddings, we find that for perturbations near the tip, evolutions by MCF asymptotically lose their angular dependence – becoming round – and develop Type-II bowl soliton singularities. As well, if we impose sinusoidal angular dependence on the initial embeddings for perturbations sufficiently far from the tip, the angular dependence again disappears as Type-I neckpinch singularities develop. The numerical analysis carried out in this work is an adaptation of the "overlap" method introduced in [GIKW21] and permits angular dependence.Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessBoundaries & non-compactRotational symmetry & warped productsNumerics & visualization https://neckpinch.com/papers/2204.12878/arxiv:2204.12878Wed, 27 Apr 2022 00:00:00 GMTKlaus Deckelnick, Robert Nürnberg — Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave phenomena in solid-liquid interfaces. We introduce a semidiscrete finite difference method for the approximation of hyperbolic curvature flow and prove error bounds for natural discrete norms. We also present numerical simulations, including the onset of singularities starting from smooth strictly convex initial data.Curve shorteningOther curvature flowsSingularitiesNumerics & visualization https://neckpinch.com/papers/2110.06027/arxiv:2110.06027Tue, 12 Oct 2021 00:00:00 GMTReto Buzano, Huy The Nguyen, Mario B. Schulz — In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfacesNumerics & visualization https://neckpinch.com/papers/2110.04605/arxiv:2110.04605Sat, 09 Oct 2021 00:00:00 GMTKlaus Deckelnick, Robert Nürnberg — We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to e.g.\ geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal H¹–error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.Curve shorteningOther curvature flowsAnisotropic & nonlocal flowsStability & uniquenessHarmonic maps & DeTurck trickNumerics & visualization https://neckpinch.com/papers/2109.09688/arxiv:2109.09688Mon, 20 Sep 2021 00:00:00 GMTEder Correa — In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the homogeneous Kähler-Ricci flow on rational homogeneous varieties. This description enables us to compute explicitly the maximal existence time for any homogeneous solution and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between numerical invariants associated to ample divisors and numerical invariants arising from solutions of the homogeneous Kähler-Ricci flow. In the particular setting of full flag varieties, we prove that the numerical invariants obtained from solutions of the homogeneous Kähler-Ricci flow can be related to certain well-known invariants which appear in some different contexts, including the global Seshadri constant of ample line bundles, the maximum possible radius of embeddings of symplectic and Kähler balls, and the log canonical threshold of ample ℚ-divisors. From this, we obtain constraints in terms of the scalar and Ricci curvatures, respectively, for symplectic embeddings of open Euclidean balls, and for the triviality of analytic multiplier ideal sheaves defined by ample ℚ-divisors.Ricci flowKähler–Ricci flowKähler geometryHomogeneous spacesEigenvalues & spectrumNumerics & visualization https://neckpinch.com/papers/2109.03677/arxiv:2109.03677Wed, 08 Sep 2021 00:00:00 GMTFabio Nunes da Silva, Keti Tenenblat — We show that the solutions to the curvature flow (CF) for curves on the 2-dimensional light cone are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperboles are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant c from being the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. The curve is a soliton solution when c=0. We prove that, for each vector v there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Moreover, at each end of such a curve, the curvature is either unbounded or it tends to 0 or to the constant c. Explicitly given soliton solutions are included and some self-similar solutions on the light cone, are visualized.Other curvature flowsInverse mean curvature flowAncient solutions3-manifoldsSurfacesPhysicsNumerics & visualization