Surfaces

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.

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 ${\mathbb{S}}^2$. 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 ${\mathbb{RP}}^2$, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on ${\mathbb{S}}^2$ 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.

We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow $(M^2,\partial M^2,g(t))$ 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.

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.

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

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

In this paper, we consider the Ricci flow with prescribed curvature on the finite graph $G=(V,E)$. For any $e$ in $E$, $$\frac{d\omega (t,e)}{dt}=-(\kappa (t,e)-{\kappa }^{\ast }(e))\omega (t,e),t>0,$$ where $\omega$ is the weight function, $\kappa$ is Lin-Lu-Yau Ricci curvature, and ${\kappa }^{\ast }$ 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 ${\kappa }^{\ast }$ if and only if ${\kappa }^{\ast }$ is attainable (namely, there exist weights realizing ${\kappa }^{\ast }$). In particular, we prove that the weights for constant curvature exist if and only if $$\underset{\varnothing \ne \Omega ⊊V}{\max }\frac{|E(\Omega )|}{|\Omega |}<\frac{|E|}{|V|},$$ where $E(\Omega )$ denotes the set of edges within the induced subgraph of $\Omega$, 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).

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.

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.

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.

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, $\mathrm{AVR}$ (asymptotic volume ratio) and $\mathrm{ASCD}$ (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.

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.

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.

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.

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.

In this paper we introduce the branched $\alpha$-flows on closed surfaces with Euler characteristic $\chi \le 0$. Based on the strict convexity of the branched $\alpha$-potentials, we establish the long time existence and convergence of the solutions to the branched $\alpha$-flows, which generalizes Ge and Xu's main results on the $\alpha$-flows. In addtion, we study the prescribed curvature problems under the relaxed precondition $\chi (M)\in \mathbb{Z}$ via alternative $\alpha$-flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.

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.

Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.

We show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is eventually strictly increasing along the normalized Ricci flow (NRF). More precisely, we obtain a new expression for the derivative of the Liouville entropy along an arbitrary conformal deformation in dimension 2, and we prove it is positive in the direction of the NRF for 1/6-pinched metrics. This partially answers a question of Manning from 2004. In addition, we show that the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is strictly increasing along the NRF starting from any metric of non-constant negative curvature.

In this paper, we introduce combinatorial Ricci flows (CRFs in short) in Euclidean and hyperbolic background geometries on infinite triangulations of the open disk, which are discrete analogs of Ricci flows on simply connected open surfaces. We establish well-posedness results, the existence and the uniqueness, of CRFs in both Euclidean and hyperbolic background geometries. Moreover, we prove convergence results of CRFs, which indicate a uniformization theorem for CRFs on infinite disk triangulations. As an application, we prove an existence result of circle-packing metrics with infinite prescribed cone angles in hyperbolic background geometry. To our knowledge, these are the first results of CRFs on infinite triangulations.

We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal{B}\times {\mathbb{S}}^2,g)$, where $\mathcal{B}$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal{B}$, has to be isometric to the generalized cylinder ${\mathbb{R}}^2\times {\mathbb{S}}^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products – but not products – and provide rigorous examples of the formation of generalized cylinder singularity models ${\mathbb{R}}^k\times {\mathbb{S}}^{\ell }$.

Let $(\overline{M},g_0)$ be a $2$-D compact surface with boundary $\partial M$ and its interior $M$. We show that for a large class of initial and boundary data, the initial-boundary value problem of the normalized Ricci flow $(1.10)-(1.12)$, with prescribed geodesic curvature $\psi$ on $\partial M$, has a unique solution for all $t>0$, and it converges to the complete hyperbolic metric locally uniformly in $M$. Here the natural condition that $\psi >0$ causes the main difficulty in the a priori estimates in the corresponding initial-boundary problem $(1.15)-(1.17)$ of the parabolic equations, for which an auxiliary Cauchy-Dirichlet problem is introduced. We also provide examples of the boundary data $\psi$ which fits well with the natural asymptotic behavior of the geodesic curvature, but the solution to $(1.10)-(1.12)$ fails to converge to the complete hyperbolic metric.

This paper reviews and extends the recently discovered connections between marginal and irrelevant stress-energy tensor deformations and gravity theories in arbitrary space-time dimensions. We start by discussing how $T\bar{T}$ and Root-$T\bar{T}$ deformations of two-dimensional field theories can be equivalently interpreted as the coupling of the undeformed matter sector to a gravity theory. We then extend this duality to higher-dimensional scenarios by using an approach that relies on the non-trivial eigenvalue degeneracy characterising the energy-momentum tensor of specific physical theories. We also explore incorporating dynamical degrees of freedom in the gravity sector, and show that the deformed space-time geometry induced by the $T\bar{T}$-like deformations defines a Ricci-Bourguignon flow of the metric tensor, which reduces to a Ricci flow in four dimensions. Finally, exploiting a dressing-type mechanism for the action functional characterizing a broad class of $T\bar{T}$-like deformations we study explicit examples, such as Einstein-Ricci solitons, $(d-1)$-form field theories, and spherically symmetric electrovacuum solutions.

By adapting previously known arguments concerning Ricci flow and the c-theorem, we give a direct proof that in a two-dimensional sigma-model with compact target space, scale invariance implies conformal invariance in perturbation theory. This argument, which applies to a general sigma-model constructed with a target space metric and B-field, is in accord with a more general proof in the literature that applies to arbitrary two-dimensional quantum field theories. Models with extended supersymmetry and a B-field are known to provide interesting test cases for the relation between scale invariance and conformal invariance in sigma-model perturbation theory. We give examples showing that in such models, the obstructions to conformal invariance suggested by general arguments can actually occur in models with target spaces that are not compact or complete. Thus compactness of the target space, or at least a suitable condition of completeness, is necessary as well as sufficient to ensure that scale invariance implies conformal invariance in models of this type.

Let $(M,g,\omega ,f,\lambda )$ be a Kähler gradient Ricci soliton in real dimension four. One first observes that it is an integrable Hamiltonian system in a classical sense. Indeed, all known complete examples are toric and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\text{SS}$. While another article addresses the case that these functions are functionally dependent, this one considers the independent case. The main result states that the soliton admits a toric action under a generic assumption. That is, one assumes that the system is non-degenerate and the potential function $f$ is proper. Then there is an effective, completely integrable Hamiltonian toric ${\mathbb{T}}^2$- action on $(M,\omega )$.

We study the behavior of the Cheeger isoperimetric constant under the Ricci flow on compact surfaces. For metrics on a surface diffeomorphic to $S^2$, we show that the Cheeger constant is non-decreasing along the flow. The proof uses evolution identities for parallel curves together with a viscosity formulation of the evolution of $\log h$ which accommodates for the possible switching of minimizing regions. We also give examples of nontrivial Ricci flows on topological $2$-spheres for which the Cheeger constant remains constant, demonstrating that strict monotonicity is not expected.

In this note, we provide a very simple proof of the uniformization theorem of Riemann surfaces by Ricci flow. The argument builds on a refinement of Hamilton's isoperimetric estimate for the Ricci flow on the two-sphere.

Let $P$ be a point of a compact Riemann surface $X$. We study self-adjoint extensions of the Dolbeault Laplacians in hermitian line bundles $L$ over $X$ initially defined on sections with compact supports in $X\backslash \{P\}$. We define the $\zeta$-regularized determinants for these operators and derive comparison formulas for them. We introduce the notion of the Robin mass of $L$. This quantity enters the comparison formulas for determinants and is related to the regularized $\zeta (1)$ for the Dolbeault Laplacian. For spinor bundles of even characteristic, we find an explicit expression for the Robin mass. In addition, we propose an explicit formula for the Robin mass in the scalar case. Using this formula, we describe the evolution of the regularized $\zeta (1)$ for scalar Laplacian under the Ricci flow. As a byproduct, we find an alternative proof for the Morpurgo result that the round metric minimizes the regularized $\zeta (1)$ for surfaces of genus zero.

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial $\alpha$-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures, we introduce the combinatorial $\alpha$-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial $\alpha$-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial $\alpha$-Ricci flow with surgery. As an application of the combinatorial $\alpha$-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures. We further introduce the combinatorial $\alpha$-Calabi flow with surgery and study its longtime behavior.

Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers-Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings and Luo's combinatorial Yamabe flow for vertex scalings. We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdière's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.

We study the subsequential convergence of singular solutions to the Ricci flow with prescribed constant in space geodesic curvature on compact surfaces with boundary. Furthermore, we show that in the particular case of rotational symmetry, this convergence does not depend on the sign of the geodesic curvature of the boundary.

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.

The second author and H. Yin have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a nonatomic Radon measure as a volume measure. This led to the discovery of a large array of new expanding Ricci solitons. In this paper we use the recent uniqueness theory in this context, also developed by the second author and H. Yin, to give a complete classification of all expanding Ricci solitons on surfaces. Along the way, we prove a converse to the existence theory that is not constrained to solitons: every complete Ricci flow on a surface over a time interval $(0,\epsilon )$ admits a $t\downarrow 0$ limit within the class of admissible initial data. This makes surfaces the first nontrivial setting for Ricci flow in which a bijection can be given between the entire set of complete Ricci flows over maximal time intervals $(0,T)$, and a class of initial data that induces them.