Neckpinch — Harmonic maps & DeTurck trick

Combinatorial Calabi flows on infinitely triangulated surfaces

Mon, 19 Oct 2026 00:00:00 GMT

Yuerong Bian, Xu Xu — We study the combinatorial Calabi flow associated with vertex scalings on infinitely triangulated surfaces with piecewise Euclidean and piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish short-time existence of smooth solutions and prove uniqueness among solutions that are continuous in the L^∞ norm.

Almost Hermitian geometry and generalized Ricci flow

Thu, 01 Oct 2026 00:00:00 GMT

Anna Fino, Julieth Saavedra — We study almost Hermitian manifolds whose lowered Nijenhuis tensor is totally skew-symmetric and whose characteristic (Bismut) torsion is closed, the almost-SKT condition. This extends the SKT (pluriclosed) condition beyond the integrable setting. For a fixed almost complex structure, these constraints are linear in the metric. We introduce an overdetermined ellipticity condition which, on compact manifolds, makes the admissible infinitesimal deformation space finite-dimensional and yields short-time existence for a constrained Bismut–Ricci flow. In real dimension six, almost-SKT structures satisfy a dichotomy: either the almost complex structure is integrable, or the skew Nijenhuis tensor is nowhere vanishing with constant norm and the Bismut Ricci form vanishes. Hence, on compact non-integrable almost-SKT six-manifolds, the constrained Bismut–Ricci flow is stationary. In dimension eight, ellipticity always fails, while in higher dimensions Nijenhuis rigidity holds on an open dense locus. In arbitrary dimension, we characterize when total skew-symmetry of the lowered Nijenhuis tensor determines the compatible metric pointwise up to scale. On compact manifolds, with closed torsion, this implies uniqueness of the almost-SKT metric up to scaling; the constrained Bismut–Ricci flow is then an explicit homothetic solution. We also relate this flow to generalized Ricci flow and characterize the infinitesimal obstruction to lifting the evolution when the almost complex structure varies. Finally, we construct non-integrable almost-SKT structures with nowhere vanishing Nijenhuis tensor and non-flat characteristic connection on compact six-manifolds, including S¹× S²× S³ and S³× S³. In these families, the metric is fixed and the compatible almost complex structures form an S²-family. These structures have vanishing Bismut Ricci form and are therefore almost-CYT.

Smoothing polyhedral spaces via Ricci flow

Tue, 29 Sep 2026 00:00:00 GMT

Richard H. Bamler, Esther Cabezas-Rivas — We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time slices of a single orbifold Ricci flow whose metric initial condition is the given polyhedral space. The proof rests on a new short-time existence and regularization theory for Ricci flow that, under two-sided volume bounds and a generalized segment inequality, replaces pointwise lower curvature control by small scale-invariant integral control of the defect from a preserved curvature cone. Although it allows arbitrarily large pointwise violations, this theory yields an existence time and positive-time curvature estimates independent of the initial upper curvature bound. As a further application, it gives rigidity consequences for manifolds with small integral curvature defect.

A Lean Formalization of Hamilton's Three-Manifold Theorem

Fri, 21 Aug 2026 00:00:00 GMT

Bennett Chow, Yuan Liao, Ziyang Qin — We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

Ricci Solitons on the Lie Group Sol³_(m,n) and Applications

Tue, 28 Jul 2026 00:00:00 GMT

Mohammed El Amine Mekki, Ahmed Mohammed Cherif — In this paper, we study Ricci solitons on the three-dimensional solvable Lie group Sol³_(m,n) equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical Sol³ geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties of the integral curves of the Ricci soliton vector field. We also characterize harmonic linear maps from Sol³_(m,n) into Euclidean spaces.

Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

Tue, 28 Jul 2026 00:00:00 GMT

Yuerong Bian, Xu Xu — We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish the short-time existence and uniqueness of smooth solutions to the combinatorial Yamabe flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow.Furthermore, under an integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a local well-posedness theory for the hyperbolic combinatorial Yamabe flow on infinitely triangulated surfaces.

Infinite Combinatorial Yamabe Flows in Three Dimensions

Sun, 26 Jul 2026 00:00:00 GMT

Bohao Ji — In this paper, we study three-dimensional combinatorial Yamabe flows on locally finite infinite triangulations in Euclidean and hyperbolic background geometries. Under suitable non-degeneracy and bounded-degree assumptions, we establish the short-time existence and uniqueness of the original flows. We further introduce the extended flows by using the continuous extension of solid angles, and prove long-time existence for both extended flows under suitable initial assumptions.

Ricci flow for the Bures–Helstrom qubit metric

Wed, 17 Jun 2026 00:00:00 GMT

Andrew Lesniewski — The Bures–Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three–sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton–DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function Ψ=Φ² satisfies the linear forced heat equation \beginequation* D_tΨ=Ψ_ss-2, \endequation* while the fixed-lapse coordinate form contains the associated transport term. Since the Bures–Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \beginequation* g(t)=(1-4t)g_BH, \endequation* with scalar curvature 6/(1-4t) and extinction time T=1/4. Thus the metric remains inside the monotone cone for all t

Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick

Wed, 20 May 2026 00:00:00 GMT

Klaus 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].

Integrable Deformations and Stability of the Ricci Flow

Thu, 16 Apr 2026 00:00:00 GMT

Maxwell Stolarski, Alex Waldron — We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an "almost-orthogonality" property of the Ricci-DeTurck tensor, allowing us to analyze the latter directly. We obtain our main results in weighted Holder spaces and then show how to recover the L^p-stability theorems of Deruelle-Kroncke and Kroncke-Petersen.

The Ricci-DeTurck flow on complete manifolds

Tue, 24 Mar 2026 00:00:00 GMT

Jing-Bin Cai, Bing Wang — Based on the framework of Koch-Lamm and tensor heat kernel estimates, we obtain a uniform proof of the short-time existence, uniqueness, and continuous dependence for Ricci flows starting from a complete Riemannian metric with bounded curvature. A new ingredient is an effective continuous dependence estimate without the assumption of injectivity radius lower bound.

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.

On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry F⁴

Mon, 09 Mar 2026 00:00:00 GMT

Halima Boukhari, Hadjer Okbani, Ahmed Mohammed Cherif — In this paper, we consider a left-invariant Riemannian metric g on the Lie group F⁴. We classify Ricci solitons on (F⁴,g) and show that all such solitons are expanding and non-gradient. Moreover, we study the existence of harmonic maps from compact Riemannian manifolds into (F⁴,g). Finally, we characterize a class of harmonic vector fields on (F⁴,g).

Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

Sun, 08 Feb 2026 00:00:00 GMT

Linlin 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.

Ricci-harmonic flow of G₂ and Spin(7)-structures

Thu, 08 Jan 2026 00:00:00 GMT

Shubham Dwivedi — We introduce and study a new general flow of G₂-structures which we call the Ricci-harmonic flow of G₂-structures. The flow is the coupling of the Ricci flow of underlying metrics and the isometric flow of G₂-structures, but we also provide explicit lower order in the torsion terms. The lower order terms and the flow are obtained by analyzing the second order term in the Taylor series expansion of G₂-structures in normal coordinates. As such, the Ricci-harmonic flow described in the paper can be interpreted as the "heat equation" for G₂-structures. The lower order terms allow us to prove that the stationary points of the Ricci-harmonic flow are exactly torsion-free G₂-structures on compact manifolds. We study various analytic and geometric properties of the flow. We show that the flow has short-time existence and uniqueness on compact manifolds starting with an arbitrary G₂-structure and prove global Shi-type estimates. We also prove a modified local Shi-type estimates for the flow which assume bounds on the initial derivatives of the Riemann curvature tensor and the torsion but give uniform bounds on these quantities for all times. We prove a compactness theorem for the solutions of the flow and use it to prove that the Ricci-harmonic flow exists as long as the velocity of the flow remains bounded. We also study Ricci-harmonic solitons where we prove that there are no compact expanding solitons and the only steady solitons are torsion-free. We derive an analog of Hamilton's identity for gradient Ricci-harmonic solitons and prove some integral identities for the solitons. Finally, we prove a version of the Taylor series expansion for Spin(7)-structures and use it to derive the Ricci-harmonic flow of Spin(7)-structures.

A Spinorial Perelman's Functional: Critical Points and Gradient Flow

Mon, 05 Jan 2026 00:00:00 GMT

Tsz-Kiu Aaron Chow, Frederick Tsz-Ho Fong — In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L^2-gradient flow of this functional, and establish its short-time existence and uniqueness via contraction mapping methods.

Twisted Calabi functional and twisted Calabi flow

Tue, 02 Dec 2025 00:00:00 GMT

Jie He, Haozhao Li — This paper investigates the twisted Calabi functional and the associated twisted Calabi flow on compact Kähler manifolds. Our main contributions are threefold: first, we establish the convexity of the twisted Calabi functional at its critical points; second, we prove the short-time existence of the twisted Calabi flow; and third, we demonstrate the stability of this flow in the neighborhood of twisted constant scalar curvature Kähler metrics. These results provide an analytic foundation for studying the twisted Calabi flow and resolve questions about its local behavior.

Flows of conformally coclosed G₂-structures with dilaton

Wed, 26 Nov 2025 00:00:00 GMT

Spiro Karigiannis, Sébastien Picard, Caleb Suan — We study flows of G₂-structures guided by the principle of dimensional reduction: natural geometric flows in G₂-geometry reduce to natural flows in complex geometry. Our main examples are the G₂-Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The G₂-lift of the anomaly flow deforms conformally coclosed G₂-structures. We compare the G₂-anomaly flow to the G₂-Laplacian coflow, and investigate short-time existence and fixed points.

Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

Mon, 27 Oct 2025 00:00:00 GMT

José N. V. Gomes, Matheus Hudson, Carlos M. de Sousa — The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic map heat flow of a map from M to a Riemannian manifold N. In this context, we address a functional associated with this flow and calculate its variation along parameters that preserve the weighted volume measure. An extension of Hamilton's differential Harnack expression appears by considering the boundary of M evolving by mean curvature flow, which must vanish on the gradient steady soliton case. Next, we obtain a Huisken monotonicity-type formula for the mean curvature flow in the proposed background. We also show how to construct a family of mean curvature solitons and establish a characterization of such a family.

On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

Fri, 29 Aug 2025 00:00:00 GMT

Ruojing Jiang, Franco Vargas Pallete — On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric h₀, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to h₀ in a weighted Hölder norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].

BRIDGES Lectures: Flows of geometric structures, especially G₂-structures

Fri, 15 Aug 2025 00:00:00 GMT

Spiro Karigiannis — The BRIDGES meeting in gauge theory, extremal structures, and stability was held June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Spiro Karigiannis on "Flows of geometric structures, especially G₂-structures". Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information. The main theme is short time existence (STE) and uniqueness for geometric flows. We first introduce geometric structures on manifolds and geometric flows of such structures. We discuss some qualitative features of geometric flows, and consider the notions of strong and weak parabolicity. We focus on the Ricci flow, explaining carefully the DeTurck trick to establish short-time existence and uniqueness, an argument which we then extend to a general class of geometric flows of Riemannian metrics, previewing similar ideas for flows of G₂-structures. Finally, we consider geometric flows of G₂-structures. We review the basics of G₂-geometry and survey several different geometric flows of G₂-structures. In particular, we clarify in what sense STE results for the G₂ Laplacian flow differ from STE results for other geometric flows. We conclude with a summary of some recent results by the author with Dwivedi and Gianniotis, including a classification of all possible heat-type flows of G₂-structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.

An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

Wed, 13 Aug 2025 00:00:00 GMT

Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll, Lisandra Sauer, Rodrigo Soares — We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary differential equation, which allows the construction of explicit boundary barriers and it can also be applied to unbounded domains. In the setting of Hadamard manifolds, this condition relates the geometry of the domain to the admissible boundary data in a direct way. In Euclidean space, the condition leads to solvability under geometric hypotheses of a different nature from those in the classical Jenkins-Serrin theory, and in some configurations it applies where the Jenkins-Serrin method does not. A central point of the present approach is that the geometric restrictions and the boundary data enter independently. The same barrier construction can be used for graphical mean curvature flow. This yields short-time existence with prescribed boundary values even when the boundary of the domain is not mean convex. When mean convexity is present, one recovers the classical graphical setting.

The existence and uniqueness of infinite combinatorial Yamabe flows

Wed, 16 Jul 2025 00:00:00 GMT

Bohao Ji — In this paper, we study the combinatorial Yamabe flow on infinite triangulated surfaces in Euclidean background geometry, aiming for solving discrete Yamabe problem on noncompact surfaces. Under suitable conditions, we establish the short-time existence and uniqueness of the flow. We further introduce an extended version of the flow and prove its long-time existence. As an application, we prove the convergence result of the Yamabe flow in the case of hexagonal triangulations of the plane.

Short-time existence of Lagrangian mean curvature flow

Mon, 14 Jul 2025 00:00:00 GMT

Spandan Ghosh — In his paper `Conjectures on Bridgeland Stability', Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of manifolds with corners and a-corners as introduced by Joyce. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez.

Rotationally symmetric Ricci Flow on ℝⁿ⁺¹

Thu, 29 May 2025 00:00:00 GMT

Ming Hsiao — We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on ℝⁿ⁺¹ that are noncollapsed at infinity, without assuming bounded curvature. As a consequence, we construct a complete Ricci flow solution coming out of a rotationally symmetric metric, which has a cone-like singularity at the origin and no minimal hypersphere centered at the origin, using an approximation method.

Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

Wed, 23 Apr 2025 00:00:00 GMT

Pinsard Morel Lucas — Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its use in the study of the Ricci flow), showing the local existence and regularity for the solutions of this flow and hence showing that the group of symplectomorphisms of the real four-dimensional torus is locally contractible. This work follows the ideas introduced by Yann Rollin in [3], even though the moment map picture comes from different considerations.

Convergence of Ricci flow and long-time existence of Harmonic map heat flow

Thu, 03 Apr 2025 00:00:00 GMT

Kyeongsu Choi, Yi Lai — For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms. We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round Sⁿ singularity converges at least at the rate (−t)^((n+1)/(n-1)).

Uniqueness of Ricci flow with scaling invariant estimates

Wed, 26 Mar 2025 00:00:00 GMT

Man-Chun Lee — In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

Smoothness up to the free boundary for the p-Laplacian evolution equation and the α-Gauss curvature flow

Fri, 13 Dec 2024 00:00:00 GMT

Albert Chau, Ben Weinkove — The p-Laplacian evolution equation and the α-Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.

Removing scalar curvature assumption for Ricci flow smoothing

Tue, 20 Aug 2024 00:00:00 GMT

Adam Martens — In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.