https://neckpinch.com/topics/anisotropic-flows/New arXiv papers on anisotropic & nonlocal flows, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.35064arxiv:2610.35064Mon, 26 Oct 2026 00:00:00 GMTZe-Yu Ye, Xiao-Liu Wang — In this paper, we study the singularity behavior of immersed locally convex closed plane curves evolving under generalized anisotropic Gage area-preserving flows with slow curvature speeds of the form κ_σ^α, where 0<α≤ 1. Here κ_σ denotes the anisotropic curvature associated with a prescribed anisotropy function σ. We prove that every finite-time blow-up is of Type-II provided that the Euclidean length of the evolving curve remains uniformly bounded away from zero. This result holds without any symmetry assumptions on either the initial curve or the anisotropy. When the initial signed area is nonzero, this length lower bound is automatically satisfied, and thus any finite-time singularity is necessarily of Type-II. By means of Hamilton's rescaling method combined with a differential Harnack estimate, we show that along a suitable blow-up subsequence, the rescaled curves at the selected times converge locally to the profile of a translating solution of the corresponding anisotropic curvature flow V=κ_σ^α. We also establish sufficient conditions for finite-time singularity formation in terms of the initial signed area and anisotropic length. Moreover, we construct locally convex initial curves arbitrarily close to the Wulff curve for which a finite-time Type-II singularity occurs.Other curvature flowsVolume-preserving & constrained flowsAnisotropic & nonlocal flowsSelf-shrinkers & translatorsSingularitiesHarnack inequalities https://arxiv.org/abs/2609.24431arxiv:2609.24431Mon, 21 Sep 2026 00:00:00 GMTZhishuai Liu, Guoxin Wei — In this paper, we study the anisotropic-length-preserving weighted anisotropic curvature flow for smooth convex closed plane curves. For any smooth, embedded, closed, convex initial curve, the flow exists smoothly for all time, and the anisotropic curvature converges smoothly to that of the boundary of the corresponding homothetic Wulff shape as time tends to infinity.Other curvature flowsAnisotropic & nonlocal flowsLong-time existence & convergence https://arxiv.org/abs/2609.14866arxiv:2609.14866Mon, 14 Sep 2026 00:00:00 GMTAbdou Bousso, Ameth Ndiaye — We study the iterated action of the Lie derivative on the curvature tensor of a Riemannian manifold with rank-one anisotropic curvature, whose Riemann tensor is expressed via the Kulkarni-Nomizu product as R = λ(ξ^flat⊗ξ^flat)owedge g. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost θ-Yamabe soliton structure. Furthermore, we show that if the associated potential vector field X is a symmetry of the Ricci tensor of a fixed order k (i.e., 𝓛_X^k Ric = 0), the geometric problem reduces to solving a partial differential equation of order k+1 along the flow. Finally, under the assumption that X is a conformal vector field (𝓛_X g = 2φg) whose infinitesimal flow preserves the line distribution 𝓓=Span\ξ\ (with [X,ξ]=aξ for a∈ℝ), we prove that several key geometric problems (such as establishing the relation 𝓛_X^k R = R, determining the minimal order k for X to be a Lie curvature symmetry, or satisfying 𝓛_X^(k+1)R = f 𝓛_X^k R for a continuous function f) are equivalent to a scalar differential problem governed by the operator D_X = X + 6φ+ 2a.Anisotropic & nonlocal flowsRicci solitonsAlmost & η-Ricci solitons https://arxiv.org/abs/2608.00293arxiv: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://arxiv.org/abs/2607.02067arxiv: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://arxiv.org/abs/2606.29058arxiv:2606.29058Sat, 27 Jun 2026 00:00:00 GMTEzequiel Barbosa, Rosivaldo Gonçalves, Luan de Figueiredo — We study a parabolic obstacle problem for surfaces evolving by anisotropic mean curvature flow subject to an obstacle constraint. Given a convex obstacle and initial data, we seek an evolving surface minimizing an anisotropic energy functional while remaining above the obstacle; as a special case, this framework includes the anisotropic Stefan problem, where the free boundary represents a phase transition interface with direction-dependent surface tension. The central tool is the Cahn–Hoffman transform S(x) = A^(−1/2)x, which maps the Wulff ellipsoid x : x^T A⁻¹x ≤ 1\ to the Euclidean unit ball and converts the anisotropic problem into an equivalent isotropic one with a generalized Robin-type condition on the free boundary. We prove optimal regularity of the solution (C^(1,α) in space and C^(0,α/2) in time up to the free boundary) and C^(1,α)-regularity of the evolving free boundary at non-degenerate points. The parabolic Hausdorff dimension of the space-time singular set is shown to be at most n − 1.Mean curvature flowAnisotropic & nonlocal flowsFree boundary & capillary https://arxiv.org/abs/2606.06911arxiv:2606.06911Fri, 05 Jun 2026 00:00:00 GMTEric Kim — We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on φ that the flat flow solution from regular initial data coincides with a classical ODE evolution, extending the results of to the area-preserving setting. We also show that for arbitrary initial data, the flat flow converges exponentially in time to a disjoint union of Wulff shapes, and under a non-bubbling assumption, the flow eventually becomes regular. Both of these results are novel for area-preserving crystalline flow of general sets, i.e. without assuming geometric properties such as convexity or star-shapedness. A key ingredient of independent interest is that planar almost-minimizers are Lipschitz φ-regular, which we prove by exploiting a sharp minimality estimate for distinguished line segments, as opposed to the excess decay argument given in. The novelty of our approach lies in the application of φ-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.Other curvature flowsVolume-preserving & constrained flowsAnisotropic & nonlocal flowsStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2602.07989arxiv:2602.07989Sun, 08 Feb 2026 00:00:00 GMTLinlin Fan, Peibiao Zhao — Wang, Weng and Xia[Math. Ann. 388 (2024), no. 2] studied a mean curvature type flow for the smooth, embedded capillary hypersurfaces with a constant contact angle θ∈(0,π) and confirmed the existence of solutions by the standard PDE theory. In the present paper, we study a fractional mean curvature flow for C^(1,1)-regular hypersurfaces with a capillary-type boundary condition and obtain the short time existence by the fixed point argument.Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsBoundaries & non-compactFree boundary & capillaryHarmonic maps & DeTurck trick https://arxiv.org/abs/2510.22146arxiv:2510.22146Sat, 25 Oct 2025 00:00:00 GMTCan Cui, Nung Kwan Yip — Over a bounded strictly convex domain in ℝⁿ with smooth boundary, we establish a priori gradient estimate for an anisotropic mean curvature flow with prescribed contact angle and Neumann boundary conditions. The estimates require careful analysis of the degeneracy property of the anisotropic mean curvature operator. As a result, for both problems, we can infer that the solutions converge to one that is translation invariant in time.Mean curvature flowAnisotropic & nonlocal flowsBoundaries & non-compactFree boundary & capillary https://arxiv.org/abs/2510.22136arxiv:2510.22136Sat, 25 Oct 2025 00:00:00 GMTCan Cui, Nung Kwan Yip — In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in ℝ². We establish a prior gradient estimate for smooth solutions to this boundary value problem. The same approach can also handle Dirichlet boundary condition in ℝⁿ, n≥ 2. For both problems, we prove that the solutions converge to one that is translation invariant in time.Mean curvature flowAnisotropic & nonlocal flowsSurfacesBoundaries & non-compactFree boundary & capillary https://arxiv.org/abs/2510.08168arxiv:2510.08168Thu, 09 Oct 2025 00:00:00 GMTChaoqun Gao, Yong Wei, Rong Zhou — We first establish a local gradient estimate for anisotropic p-harmonic functions. A key feature of our estimate is that the constant remains bounded as p→ 1; consequently, in the limit p→ 1, this estimate yields the local gradient estimate for weak solutions of the inverse anisotropic mean curvature flow (IAMCF). As an application, we show that the weak IAMCF is asymptotic to the expanding Wulff shape solution at the infinity, thereby extending the result of Huisken and Ilmanen in [8] to the anisotropic case.Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows https://arxiv.org/abs/2509.06125arxiv:2509.06125Sun, 07 Sep 2025 00:00:00 GMTYuchuan Yang, Selim Esedoglu — We consider a PDE system that describes curvature motion of networks with a dynamic boundary condition known as triple junction drag. This model arises in the study of grain boundary evolution in polycrystalline materials. In this system, the surface tension coefficients depend on the crystallographic orientations of the grains, which are allowed to rotate. We prove existence and uniqueness of solutions to this system in the parabolic Hölder class C^(2+α,1+α/2). Moreover, we extend our existence result to accommodate a wider class of initial networks by relaxing the compatibility conditions on the angles and curvatures at the triple junction. As an important by-product of our result, we demonstrate the possibility of a new type of topological change during the evolution of the network. We also revisit the question of stability of stationary networks and how it is affected by the choice of surface tensions.Other curvature flowsAnisotropic & nonlocal flowsStability & uniquenessBoundaries & non-compactPhase fields & multiphase flows https://arxiv.org/abs/2508.03134arxiv:2508.03134Tue, 05 Aug 2025 00:00:00 GMTAndrea Kubin, Domenico Angelo La Manna, Enrico Pasqualetto — In this article, we introduce a variational algorithm, in the spirit of the minimizing movements scheme, to model the volume-preserving anisotropic mean curvature flow in 2D. We show that this algorithm can be used to prove the existence of classical solutions. Moreover, we prove that this algorithm converges to the global solution of the equation.Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsWeak & singular flowsLong-time existence & convergence https://arxiv.org/abs/2506.05951arxiv:2506.05951Fri, 06 Jun 2025 00:00:00 GMTDaniele De Gennaro — We prove that the minimizing movements scheme á la Almgren-Taylor-Wang converges towards level-set solutions to a nonlinear version of nonlocal curvature flows with time-depending forcing term, in the rather general framework of variational curvatures introduced in. The nonlinearity involved is assumed to satisfy minimal assumptions, namely continuity, monotonicity, and vanishing at zero. Under additional assumptions only on the curvatures involved, we establish uniqueness for level-set solutions.Other curvature flowsAnisotropic & nonlocal flowsWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2504.06162arxiv:2504.06162Tue, 08 Apr 2025 00:00:00 GMTFilippo Cagnetti, Massimiliano Morini, Dario Reggiani — In a novel distributional approach has been introduced to provide a well-posed formulation of a class of crystalline mean curvature flows. In this paper, such an approach is extended to the nonlocal setting. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the (N-1)-dimensional Minkowski pre-content.Mean curvature flowOther curvature flowsAnisotropic & nonlocal flows https://arxiv.org/abs/2503.20524arxiv:2503.20524Wed, 26 Mar 2025 00:00:00 GMTAndrea Chiesa, Karel Svadlenka — We extend the analysis by Esedoglu and Otto (2015) of thresholding energies for the celebrated multiphase Bence-Merriman-Osher algorithm for computing mean curvature flow of interfacial networks, to the case of differing space-dependent anisotropies. In particular, we address the special setting of an obstacle problem, where anisotropic particles move on an inhomogeneous substrate. By suitable modification of the surface energies we construct an approximate energy that uses a single convolution kernel and is monotone with respect to the convolution width. This allows us to prove that the approximate energies Γ-converge to their sharp interface counterpart.Mean curvature flowAnisotropic & nonlocal flowsHomogeneous spacesPhase fields & multiphase flows https://arxiv.org/abs/2501.10341arxiv:2501.10341Fri, 17 Jan 2025 00:00:00 GMTBohdan Bulanyi, Berardo Ruffini — We establish the convergence of threshold dynamics-type approximation schemes to propagating fronts evolving according to an anisotropic mean curvature motion in the presence of a forcing term depending on both time and position, thus generalizing the consistency result obtained in [Ishii-Pires-Souganidis, 1999] by extending the results obtained in [Caffarelli-Souganidis, 2010] for α∈ [1,2) to anisotropic kernels and in the presence of a driving force. The limit geometric evolution is of a variational type and can be approximated, at a large scale, by eikonal-type equations modeling dislocations dynamics. We prove that it preserves convexity under suitable convexity assumptions on the forcing term and that convex evolutions of compact sets are unique. If the initial set is bounded and sufficiently large, and the driving force is constant, then the corresponding generalized front propagation is asymptotically similar to the Wulff shape.Mean curvature flowAnisotropic & nonlocal flows https://arxiv.org/abs/2410.03700arxiv:2410.03700Tue, 24 Sep 2024 00:00:00 GMTL. Bubuianu, E. Nurlan, J. O. Seti, S. Vacaru, E. V. Veliev — We elaborate on nonmetric geometric flow theory and metric-affine gravity with applications in modern cosmology. Two main motivations for our research follow from the facts that 1) cosmological models for f(Q) modified gravity theories, MGTs, are efficient for describing recent observational data provided by the James Webb Space Telescope; and 2) the statistical thermodynamic properties of such nonmetric locally anisotropic cosmological models can be studied using generalizations of the concept of G. Perelman entropy. We derive nonmetric distorted R. Hamilton and Ricci soliton equations in such canonical nonholonomic variables when corresponding systems of nonlinear PDEs can be decoupled and integrated in general off-diagonal forms. This is possible if we develop and apply the anholonomic frame and connection deformation method involving corresponding types of generating functions and generating sources encoding nonmetric distortions. Using such generic off-diagonal solutions (when the coefficients of metrics and connections may depend generically on all spacetime coordinates), we model accelerating cosmological scenarios with quasi-periodic gravitational and (effective) matter fields; and study topological and nonlinear geometric properties of respective dark energy and dark matter, DE and DM, models. As explicit examples, we analyze some classes of nonlinear symmetries defining topological quasicrystal, QC, phases which can modified to generate other types of quasi-periodic and locally anisotropic structures. The conditions when such nonlinear systems possess a behaviour which is similar to that of the Lambda cold dark matter (ΛCDM) scenario are stated. We conclude that nonmetric geometric and cosmological flows can be considered as an alternative to the ΛCDM concordance models and speculate on how such theories can be elaborated.Anisotropic & nonlocal flowsRicci solitonsPerelman entropyPhysics https://arxiv.org/abs/2405.08296arxiv:2405.08296Tue, 14 May 2024 00:00:00 GMTEric Kim, Dohyun Kwon — We study the motion of sets by anisotropic curvature under a volume constraint in the plane. We establish the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of Wulff shapes of equal area, the critical point of the anisotropic perimeter functional. This is an anisotropic analogue of the results in the isotropic case studied in. The novelty of our approach is in using the Cahn-Hoffman map to parametrize boundary components as small perturbations of the Wulff shape. In addition, we show that certain reflection comparison symmetries are preserved by the flat flow, which lets us obtain uniform bounds on the distance between the convergent profile and the initial data.Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsLong-time existence & convergence https://arxiv.org/abs/2404.14402arxiv:2404.14402Mon, 22 Apr 2024 00:00:00 GMTLeon Bungert, Tim Laux, Kerrek Stinson — We connect adversarial training for binary classification to a geometric evolution equation for the decision boundary. Relying on a perspective that recasts adversarial training as a regularization problem, we introduce a modified training scheme that constitutes a minimizing movements scheme for a nonlocal perimeter functional. We prove that the scheme is monotone and consistent as the adversarial budget vanishes and the perimeter localizes, and as a consequence we rigorously show that the scheme approximates a weighted mean curvature flow. This highlights that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. In our analysis, we introduce a variety of tools for working with the subdifferential of a supremal-type nonlocal total variation and its regularity properties.Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows https://arxiv.org/abs/2404.10298arxiv:2404.10298Tue, 16 Apr 2024 00:00:00 GMTShujing Pan, Yong Wei — In this paper, we consider the anisotropic α-Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for all positive power α>0, if the initial hypersurface is complete noncompact and locally uniformly convex, then the solution of the flow exists for all positive time.Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsBoundaries & non-compactLong-time existence & convergence https://arxiv.org/abs/2404.05267arxiv:2404.05267Mon, 08 Apr 2024 00:00:00 GMTLaiyuan Gao, Horst Martini, Deyan Zhang — A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant k-order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is k-symmetric.Other curvature flowsAnisotropic & nonlocal flowsLong-time existence & convergence https://arxiv.org/abs/2404.01668arxiv:2404.01668Tue, 02 Apr 2024 00:00:00 GMTEsther Cabezas-Rivas, Salvador Moll, Marcos Solera — We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in ℝ^N with no ellip\-ticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.Mean curvature flowOther curvature flowsInverse mean curvature flowAnisotropic & nonlocal flowsHarnack inequalitiesWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2403.09902arxiv:2403.09902Thu, 14 Mar 2024 00:00:00 GMTShokhrukh Yu. Kholmatov — We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in [Bellettini, Kholmatov: J. Math. Pures Appl. (2018)] we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young's law, and also the existence of a 1/2-Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness and the consistency with the smooth flow.Other curvature flowsAnisotropic & nonlocal flowsWeak & singular flowsHomogeneous spaces https://arxiv.org/abs/2403.04725arxiv:2403.04725Thu, 07 Mar 2024 00:00:00 GMTAntonin Chambolle, Daniele De Gennaro, Massimiliano Morini — We consider here a fully discrete variant of the implicit variational scheme for mean curvature flow [AlmTayWan,LucStu], in a setting where the flow is governed by a crystalline surface tension defined by the limit of pairwise interactions energy on the discrete grid. The algorithm is based on a new discrete distance from the evolving sets, which prevents the occurrence of the spatial drift and pinning phenomena identified in [MisiatsYip16,BraGelNov] in a similar discrete framework. We provide the first rigorous convergence result holding in any dimension, for any initial set and for a large class of purely crystalline anisotropies, in which the spatial discretization mesh can be of the same order or coarser than the time step.Mean curvature flowOther curvature flowsAnisotropic & nonlocal flows https://arxiv.org/abs/2401.15358arxiv:2401.15358Sat, 27 Jan 2024 00:00:00 GMTGiovanni Bellettini, Shokhrukh Kholmatov, Firdavsjon Almuratov — We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically shrinking solutions having one bounded component. We also provide an example of network shrinking to a segment with multiplicity two.Other curvature flowsAnisotropic & nonlocal flowsPhase fields & multiphase flows https://arxiv.org/abs/2310.05596arxiv:2310.05596Mon, 09 Oct 2023 00:00:00 GMTMichael Gößwein, Matteo Novaga, Paola Pozzi — In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon gradient inequality and use this knowledge to derive stability results for the flow. Precisely, in our main theorem we show that for any initial data, which are C^(2,α)-close to a (local) energy minimizer, the flow exists globally and converges to a possibly different energy minimum.Curve shorteningAnisotropic & nonlocal flowsStability & uniquenessPhase fields & multiphase flows https://arxiv.org/abs/2307.03912arxiv:2307.03912Sat, 08 Jul 2023 00:00:00 GMTVesa Julin, Domenico Angelo La Manna — We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci imply the C^(1+α)-regularity of the flow and then provide a regularity argument which improves this into C^(2+α)-regularity of the flow. The regularity step from C^(1+α) into C^(2+α) does not rely on convexity and can probably be adopted to more general setting.Mean curvature flowAnisotropic & nonlocal flowsSingularitiesLong-time existence & convergence https://arxiv.org/abs/2306.09009arxiv:2306.09009Thu, 15 Jun 2023 00:00:00 GMTHaizhong Li, Ruixuan Li, Changwei Xiong — We derive various sharp upper bounds for the p-capacity of a smooth compact set K in the hyperbolic space ℍⁿ and the Euclidean space ℝⁿ. Firstly, using the inverse mean curvature flow, for the mean convex and star-shaped set K in ℍⁿ, we obtain sharp upper bounds for the p-capacity Capₚ(K) in three cases: (1) n≥ 2 and p=2, (2) n=2 and p≥ 3, (3) n=3 and 1<p≤ 3; Using the unit-speed normal flow, we prove a sharp upper bound for Capₚ(K) of a convex set K in ℍⁿ for n≥ 2 and p>1. Secondly, for the compact set K in ℝ³, using the weak inverse mean curvature flow, we get a sharp upper bound for the p-capacity (1<p<3) of the set K with connected boundary; Using the inverse anisotropic mean curvature flow, we deduce a sharp upper bound for the anisotropic p-capacity (1<p<3) of an F-mean convex and star-shaped set K in ℝ³.Mean curvature flowOther curvature flowsInverse mean curvature flowAnisotropic & nonlocal flows https://arxiv.org/abs/2304.13454arxiv:2304.13454Wed, 26 Apr 2023 00:00:00 GMTGiovanni Bellettini, Shokhrukh Yu. Kholmatov — We consider the geometric evolution of a network in the plane, flowing by anisotropic curvature. We discuss local existence of a classical solution in the presence of several smooth anisotropies. Next, we discuss some aspects of the polycrystalline case.Other curvature flowsAnisotropic & nonlocal flows