Neckpinch — Surfaces

Variations of the harmonic entropy and linear drift on negatively curved surfaces

Mon, 26 Oct 2026 00:00:00 GMT

Karen Butt, Alena Erchenko, Tristan Humbert — For closed negatively curved surfaces, we obtain formulas for the derivatives of the harmonic entropy of its geodesic flow and the linear drift of its Brownian motion along conformal deformations. We deduce that the harmonic entropy of the geodesic flow of a closed surface of non-constant negative curvature (resp. linear drift of the Brownian motion on its universal cover) is strictly decreasing (resp. increasing) along the normalized Ricci flow. We also prove that any hyperbolic metric on a surface admits a neighborhood in the space of conformal negatively curved metrics in which the linear drift is greater or equal to the Liouville entropy, with equality if and only if the metric is hyperbolic.

Algebraic and Symmetrized Affine star-Ricci Solitons for the Bott Connection on Three-Dimensional Lorentzian Lie Groups

Wed, 14 Oct 2026 00:00:00 GMT

Yuxin Ding — We fix a spacelike two-dimensional left-invariant distribution and its associated almost contact structure on a three-dimensional Lorentzian Lie group, and investigate the star-Ricci contraction of the curvature of the Bott connection. We compute the complete tensor for seven standard bracket families and determine all real parameter solutions of the general algebraic equation and its self-adjoint subclass. Using the symmetric part of the tensor, we also define a Bott affine star-Ricci soliton equation and classify all left-invariant vector field solutions. Every affine solution is steady, while two non-unimodular families admit branches with a nonzero symmetrized tensor. The proofs provide complete component equations, treat all degenerate parameter branches, and establish sufficiency by explicit substitution. We also distinguish almost contact structures from contact metric structures.

Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds

Sun, 20 Sep 2026 00:00:00 GMT

Naotoshi Fujihara, Toru Kajigaya, Albert Wood — We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies ρ= Cω+ ndd^cf, and which satisfy C>0. With analogous assumptions on the flow, we obtain subconvergence to an f-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that (M,g,f) is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.

Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow

Thu, 17 Sep 2026 00:00:00 GMT

Ping-Hung Lee — We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.

Ricci flow with metric torsion on surfaces of positive Euler characteristic

Mon, 14 Sep 2026 00:00:00 GMT

Shubham Dwivedi — We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the 2-sphere thus confirming a conjecture of Branding–Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on S². Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on ℝP}², which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on S² whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz–Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.

Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications

Thu, 27 Aug 2026 00:00:00 GMT

Fábio Nunes da Silva, Edwin Salinas Reyes — We show that the spacelike solutions to the curvature flow for curves on the De Sitter space are in correspondence with the solutions to the inverse curvature flow on the 2-dimensional hyperbolic space, that on the De Sitter space, the timelike solutions to the curvature flow are in correspondence with the timelike solutions to the inverse curvature flow, and that the solutions curve shortening flow on the 2-dimensional hyperbolic space are in correspondence with the solutions to the spacelike solutions to the inverse curvature flow on the De Sitter space. We prove that, for spacelike curves on the De Sitter space, the curvature flow is a gradient-type flow for the arc-length functional. We observe that a spacelike or timelike curve on the De Sitter space is a soliton solution to the curvature flow (resp. inverse curvature flow) if and only if its curvature (resp. inverse of its curvature) can be written as the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. We prove that for each vector v, there exists a 2-parameter family of timelike (spacelike) soliton solutions to the curvature flow and to the inverse curvature flow on the De Sitter space. We show that there exists no non-trivial complete timelike soliton. There exist non-trivial complete spacelike solutions. As a consequence of curvature flow, we obtain the behavior of the soliton solutions to the inverse curvature flow on the De Sitter space and hyperbolic space.

Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

Thu, 27 Aug 2026 00:00:00 GMT

Kyeongho Bang, Eric Chen, Wenkui Du — We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow (M², ∂M², g(t)) on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.

Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. II: The two-dimensional space case

Tue, 11 Aug 2026 00:00:00 GMT

Roman Cherniha, John R. King — The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.

Contracting Transport Maps on Riemannian Manifolds

Thu, 30 Jul 2026 00:00:00 GMT

Shrey Aryan, Bang-Xian Han, Zhuo-Nan Zhu — We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions n≥3 show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every n≥2, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere.

Calibration energy and mean curvature flow

Wed, 03 Jun 2026 00:00:00 GMT

Tatsuya 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.

Notes on harmonic-Ricci flow on surface

Thu, 07 May 2026 00:00:00 GMT

Xiang-Zhi Cao — In this note, we want to establish several formulas about functionals along harmonic Ricci flow on surface with boundary

On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane

Mon, 13 Apr 2026 00:00:00 GMT

Zhishuai Liu, Guoxin Wei — In this paper, we study the area-preserving and length-preserving κ^α-type curvature flows of smooth, closed, convex curves in the two-dimensional hyperbolic plane ℍ² for α<0 and prove that convexity is preserved along the flows. Assuming that the flows exist for all time, we show that the evolving curves converge smoothly to geodesic circles. Furthermore, we also derive a sufficient condition for global existence of the flows.

Liouville theorem on p-biharmonic map from gradient Ricci soliton

Tue, 17 Mar 2026 00:00:00 GMT

Xiangzhi Cao — In this paper, we are devoted to obtain some results on p-biharmonic map from gradient Ricci soliton, especially on two dimensional cigar soliton.

The Ricci flow with prescribed curvature on graphs

Wed, 11 Mar 2026 00:00:00 GMT

Yong Lin, Shuang Liu — In this paper, we consider the Ricci flow with prescribed curvature on the finite graph G=(V,E). For any e in E, (dω(t,e))/dt = −(κ(t,e)−κ^*(e))ω(t,e), t > 0, where ω is the weight function, κ is Lin-Lu-Yau Ricci curvature, and κ^* is the prescribed curvature. By imposing invariance of the graph distance with respect to time t, the Ricci flow introduced above characterizes the weight evolution governed by the Lin-Lu-Yau curvature. We first establish the existence and uniqueness of the solution to this equation on general graphs. Furthermore, for graphs with girth of at least 6, we prove that the Ricci flow converges exponentially to weights of κ^* if and only if κ^* is attainable (namely, there exist weights realizing κ^*). In particular, we prove that the weights for constant curvature exist if and only if max_(∅ ≠ Ωsubsetneq V) (|E(Ω)|)/(|Ω|) < (|E|)/(|V|), where E(Ω) denotes the set of edges within the induced subgraph of Ω, and |A| is the cardinality of the set A. Viewing edge weights as metrics on surface tilings with girth of at least 5 or the duals of triangulations with vertex degrees exceeding 5, we demonstrate that our constant Lin-Lu-Yau curvature flow serves as an analog to the 2D combinatorial Ricci flow for piecewise constant curvature metrics, thereby providing an affirmative answer to Question 2 posed by Chow and Luo (J Differ Geom, 63(1) 2002).

Dynamical stability of various convex graphical translators

Thu, 29 Jan 2026 00:00:00 GMT

Junyoung Park — In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.

Quantitative Spectral Stability for an Embedded Annulus under Coupled Curve Shortening and 2D Ricci Flows

Wed, 21 Jan 2026 00:00:00 GMT

Mohammadjavad Habibivostakolaei — We study the spectral stability of Dirichlet eigenvalues on an embedded annulus whose boundary evolves by curve shortening flow while the ambient surface evolves under the two dimensional Ricci flow using variational formulas, Rellich–type identities, and harmonic capacity methods, we relate eigenvalue variations to geometric deficit and modulus. We establish quantitative bounds comparing the spectrum of the evolving annulus with that of a flat cylinder of equal modulus. As a consequence, we obtain geometric stability and a spectral gap estimate controlled by the deficit functional.

Path Connectivity of Anosov Metrics on Surfaces

Tue, 13 Jan 2026 00:00:00 GMT

Guilherme Brandão Guglielmo, R. Ruggiero — We construct a class of Riemannian metrics in closed surfaces of genus greater than one, having Anosov geodesic flows, and some regions of positive curvature, such that for each such surface, there exists a smooth curve of conformal deformations that preserves the Anosov property and connects the surface with a Riemannian metric of negative curvature. The conformal deformation does not arise from geometric flows like the Ricci flow, since it is known that such flows might generate conjugate points in the presence of points of positive curvature in the surface.

Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

Mon, 05 Jan 2026 00:00:00 GMT

Hamid Reza Salimi Moghaddam — In this article, we investigate when a left-invariant Riemannian metric on a Lie group is a Ricci soliton, under the assumption that the derived algebra has dimension at most two. We establish computable necessary and sufficient conditions for a given left-invariant Riemannian metric to be a Ricci soliton. As applications, we obtain several examples of Ricci nilsolitons and apply our results to indecomposable Lie groups of dimension at least five with two-dimensional derived algebra.

K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

Wed, 03 Dec 2025 00:00:00 GMT

Charles Cifarelli, Carlos Esparza — Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration (π: X → Y, ξ). In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric g implies K-polystability of (π: X → Y, ξ), in the case that the Ricci curvature of g decays at infinity. As an application, we give a non-existence result: if M is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space X of the cube root of K_M is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.

Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in ℍ³ as Translation Surfaces

Wed, 26 Nov 2025 00:00:00 GMT

Tarcios Andrey Ferreira, João Paulo dos Santos — We consider the hyperbolic three-space in the half-space model endowed with a metric Lie group structure. In this setting, translation surfaces are defined as products of two curves α and β with respect to the Lie group operation. We investigate minimal surfaces and solitons to the mean curvature flow arising from specific types of products of these curves. In particular, we provide classification results for minimal surfaces, hyperbolic translators, and conformal solitons to the mean curvature flow.

Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds

Mon, 03 Nov 2025 00:00:00 GMT

Yuang Shi — We consider noncompact complete Kähler manifolds with nonnegative bisectional curvature. Our main results are: 1. Precise relations among refined minimal degree of polynomial growth holomorphic functions and holomorphic volume forms, AVR (asymptotic volume ratio) and ASCD (average of scalar curvature decay) are established. 2. The Lyapunov asymptotic behavior of the Kähler-Ricci flow can be described in terms of polynomial growth holomorphic functions. This provides a unifying perspective that bridges the two distinct proofs of Yau's uniformization conjecture by Liu and Chau-Lee-Tam. These resolve two conjectures made by Yang.

The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe

Wed, 29 Oct 2025 00:00:00 GMT

Severin Bunk, Miguel Pino Carmona, C. S. Shahbazi — We prove well-posedness of the analytic Cauchy problem for gradient generalized Ricci solitons on an abelian bundle gerbe and solve the initial data equations on every compact Riemann surface. Along the way, we provide a novel characterization of the self-similar solutions of the generalized Ricci flow by means of families of automorphisms of the underlying abelian bundle gerbe covering families of diffeomorphisms isotopic to the identity.

Two dimensional anisotropic mean curvature flow with contact angle condition

Sat, 25 Oct 2025 00:00:00 GMT

Can 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.

Sigma model renormalisation group flows, singularities and some remarks on cosmology

Tue, 16 Sep 2025 00:00:00 GMT

Georgios Papadopoulos — We investigate the properties of the renormalisation group (RG) flow of two-dimensional sigma models with a generic metric coupling by utilising known results for the Ricci flow. We point out that on many occasions the RG flow develops singularities, due to strong coupling behaviour, before it reaches a UV or an IR fixed point. We illustrate our analysis with several examples. We give particular emphasis to type I singularities, where the length of the curvature of the sigma model target space grows at most as |t-T|⁻¹ as the flow parameter t approaches the singularity at T. For these, the geometry near the singularity is described in terms of a shrinking Ricci soliton that exhibits a cosmological constant even though the original RG flow does not. Assuming that the spacetime satisfies an RG flow equation, we use the Ricci solitons to introduce a cosmological constant in a string theory setting. This can allow for different cosmological constants at different regions of spacetime. In particular, we point out how the de-Sitter space is a solution of the theory. We also raise the question on whether the techniques used to prove the geometrisation conjecture can be applied to prove the homogeneity and isotropy of the universe at large scales.

Contractive transport maps from S² to nearly spherical surfaces with positive Ricci curvature

Tue, 19 Aug 2025 00:00:00 GMT

Jordan Serres — We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-Émery criterion.

Discrete conformal structures on surfaces with boundary (III) – Deformation

Thu, 24 Jul 2025 00:00:00 GMT

Xu Xu, Chao Zheng — The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works, we established, respectively, a classification of these discrete conformal structures and results on their rigidity and existence. Building on this foundation, the present work focuses on the deformation theory of discrete conformal structures on surfaces with boundary. Specifically, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow, and establish the longtime existence and global convergence of solutions to these combinatorial curvature flows. These results yield effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

Branched α-combinatorial Ricci flows on closed surfaces with Euler characteristic χ≤ 0

Fri, 30 May 2025 00:00:00 GMT

Wenjun Li, Rongyuan Liu, Guohao Chen, Aijin Lin — In this paper we introduce the branched α-flows on closed surfaces with Euler characteristic χ≤ 0. Based on the strict convexity of the branched α-potentials, we establish the long time existence and convergence of the solutions to the branched α-flows, which generalizes Ge and Xu's main results on the α-flows. In addtion, we study the prescribed curvature problems under the relaxed precondition χ(M)∈ ℤ via alternative α-flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.

A prescribed curvature flow on hyperbolic surfaces with infinite topological type

Mon, 26 May 2025 00:00:00 GMT

Xinrong Zhao, Puchun Zhou — In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce a prescribed curvature flow-a discrete analogue of the Ricci flow on noncompact surfaces-specifically adapted to the setting of infinite cellular decompositions. We establish the well-posedness of the flow and prove two convergence results under certain conditions. Our approach resolves the prescribed total geodesic curvature problem for a broad class of surfaces with infinite cellular decompositions, yielding, in certain cases, smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. Moreover, the proposed flow provides a method for constructing hyperbolic metrics from appropriate initial data.

The mean curvature flow of subgroups on Lie groups of dimension three

Fri, 23 May 2025 00:00:00 GMT

Romina M. Arroyo, Gabriela P. Ovando, Mariel Sáez — In this work we study the existence of solutions to the Mean Curvature Flow for which the initial condition has the structure of a two-dimensional Lie subgroup within a Lie group of dimension three. We consider Lie groups with a fixed left-invariant metric and first observe that if the Lie group is unimodular, then every Lie subgroup is a minimal surface (hence a trivial solution). For this reason we focus on non-unimodular Lie groups, finding the evolution of every Lie subgroup of dimension 2 (within a 3 dimensional Lie group). These evolutions are self-similar for abelian subgroups (i.e. evolve by isometries), but not self-similar in the other cases.

Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

Mon, 19 May 2025 00:00:00 GMT

Miroslav 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.