Neckpinch — Yamabe flow

A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree

Sun, 11 Oct 2026 00:00:00 GMT

Weike Yu — In this note, we prove that on a compact Hermitian manifold with negative Gauduchon degree, the Chern-Yamabe flow introduced in our recent work [13] converges to the solution of prescribed Chern scalar curvature problem, provided that the candidate curvature function is nonpositive and not identically zero.

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.

A non-convergence phenomenon for the CR Yamabe flow

Thu, 25 Jun 2026 00:00:00 GMT

Claudio Afeltra, Andrea Pinamonti, Pak Tung Ho — We construct a contact form on a three dimensional CR manifold such that the CR Yamabe flow fails to converge. More precisely, on small Rossi deformations of the standard CR three-sphere, we exhibit an example whose corresponding CR Yamabe flow develops a one-bubble concentration regime. The construction is based on the negativity of the pseudohermitian mass on the Rossi spheres. This shows that mass positivity is not merely a technical assumption in the known convergence results for the CR Yamabe flow, but is genuinely connected to the large-time dynamics of the flow.

On the Yamabe flow in a bounded domain

Mon, 22 Jun 2026 00:00:00 GMT

Fei Fang, Zhong Tan — This paper investigates the dynamical behaviors of solutions to the Yamabe flow via the modified potential well method. We first establish the local existence and regularity of weak solutions for the flow. Several new results concerning global existence and blowup are obtained by classifying initial data into stable and unstable sets. Specifically, solutions with initial data in the stable set exist globally and extinguish in finite time, whereas those originating from unstable initial data blow up in infinite time. For certain high-energy initial data, we show that the solution decays to zero as time tends to infinity and undergoes finite-time blowup. In addition, we analyze Palais-Smale sequences to reveal the intrinsic relationship between the long-time asymptotic behavior of solutions and steady states. Finally, we derive the Pohozaev identity for the equation and prove the corresponding nonexistence theorem.

Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere

Sun, 21 Jun 2026 00:00:00 GMT

Haixia Chen, Seunghyeok Kim, Monica Musso — For every n ≥ 3, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round n-sphere Ssⁿ. These families are pairwise distinct up to conformal equivalence, and no member is conformally equivalent to a rotationally symmetric solution. At every negative time, the Ricci curvature tensor of each solution is indefinite at some point. Moreover, the associated backward limit space is a wedge sum of finitely many isometric copies of Ssⁿ. These examples show that the collection of ancient Yamabe flows on Ssⁿ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on Ss², all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on Rⁿ, whose positive entire solutions are only the standard bubbles. The construction uses a non-radial inner–outer gluing scheme. After stereographic projection, we reformulate the flow as a conformally invariant parabolic problem on Rⁿ. By exploiting Kelvin invariance and switching between the Euclidean and spherical formulations as needed, we control the non-radial modes directly without reducing the problem to one space dimension. Weighted Hölder estimates provide the pointwise control needed to establish the Type II behavior, the Ricci-sign property, conformal inequivalence, and the description of the backward limits in a straightforward manner.

Global Convergence of the Gursky-Malchiodi Q-curvature Flow

Wed, 04 Feb 2026 00:00:00 GMT

Liuwei Gong, Sanghoon Lee, Juncheng Wei — In their seminal work, Gursky and Malchiodi introduced a non-local conformal flow in dimensions n ≥ 5 to resolve the constant Q-curvature problem. They proved sequential convergence of the flow for initial metrics with positive scalar curvature and Q-curvature, provided the energy was sufficiently small. In this paper, we prove the global convergence of the flow for arbitrary initial energy under the same positivity assumptions by establishing a non-local version of the Łojasiewicz-Simon inequality for the Paneitz-Sobolev quotient along the flow. We construct test bubbles and estimate their Paneitz-Sobolev quotients, a strategy that was carried out in the celebrated work of Brendle in the context of the Yamabe flow. We develop a more geometric and systematic proof that addresses the algebraic and computational complexity inherent in the Q-curvature and the Paneitz operator. Along the way, we derive a stability inequality for the Paneitz-Sobolev quotient using a higher-order Koiso-Bochner formula established in recent work of Bahuaud, Guenther, Isenberg, and Mazzeo.

Convergence of the fractional Yamabe flow for arbitrary initial energy

Tue, 29 Jul 2025 00:00:00 GMT

Jingeon An-Lacroix, Hardy Chan, Pak Tung Ho — Since the seminal paper of Graham and Zworski (Invent. Math. 2003), conformal geometric problems are studied in the fractional setting. We consider the convergence of fractional Yamabe flow, which is previously known under small initial energy assumption. Inspired by the deep work of Brendle (J. Diff. Geom. 2005), we obtain the full convergence result for arbitrary initial energy, whenever the (fractional) positive mass conjecture is valid.

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.

Prescribing hyperbolic bordered surfaces via combinatorial flows

Tue, 15 Apr 2025 00:00:00 GMT

Shengyu Li, Zhi-Gang Wang — The aim of this paper is to investigate the fractional combinatorial Calabi flow for hyperbolic bordered surfaces. By Lyapunov theory, it is proved that the flow exists for all time and converges exponentially to a conformal factor that generates a hyperbolic surface whose lengths of boundary components are prescribed positive numbers. Furthermore, a generalized combinatorial Yamabe flow is introduced in the same geometry setting, with the long time existence and convergence established. This result yields an algorithm for searching bordered surfaces, which may accelerate convergence speed.

Gap Theorem on locally conformally flat manifold

Fri, 11 Apr 2025 00:00:00 GMT

Ming Hsiao, Man-Chun Lee — In this work, we study a gap phenomenon in locally conformally flat Riemannian manifolds with non-negative Ricci curvature. We construct complete solutions to the Yamabe flow that exhibit instantaneous bounded curvature as they evolve. Using this, we demonstrate that if the curvature decays quickly enough in an integral sense, then the manifold must be flat. This partially generalizes the results of Chen-Zhu and Ma.

Generalized Yamabe Flows

Tue, 03 Dec 2024 00:00:00 GMT

Jørgen Olsen Lye, Boris Vertman, Mannaim Gennaro Vitti — In this work we introduce a family of conformal flows generalizing the classical Yamabe flow. We prove that for a large class of such flows long-time existence holds, and the arguments are in fact simpler than in the classical case. Moreover, we establish convergence for the case of negative scalar curvature and expect a similar statement for the positive and the flat cases as well.

On the existence and classification of k-Yamabe gradient solitons

Wed, 09 Oct 2024 00:00:00 GMT

Maria Fernanda Espinal, Mariel Sáez — In this paper we classify rotationally symmetric conformally flat admissible solitons to the k-Yamabe flow, a fully non-linear version of the Yamabe flow. For n≥ 2k we prove existence of complete expanding, steady and shrinking solitons and describe their asymptotic behavior at infinity. For n<2k we prove that steady and expanding solitons are not admissible. The proof is based on the careful analysis of an associated dynamical system.

Convergence rate of the Q-curvature flow

Thu, 04 Jul 2024 00:00:00 GMT

Pak Tung Ho, Sanghoon Lee — Carlotto, Chodosh and Rubinstein have studied the convergence rate of the Yamabe flow. Inspired by their result, we study the convergence rate of the Q-curvature flow in this paper. In particular, we provide an example of a slowly converging Q₆-curvature flow in dimension 6, in constrast to the dimension 2 case, where the Q-curvature flow always converges exponentially.

A dual Yamabe flow and related integral flows

Tue, 26 Dec 2023 00:00:00 GMT

Jingang Xiong — We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a dual Q curvature we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.

Deformation of discrete conformal structures on surfaces

Tue, 05 Dec 2023 00:00:00 GMT

Xu Xu — 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.

Flows of piecewise analytic vector fields in convex polytope decompositions

Mon, 04 Dec 2023 00:00:00 GMT

Tianqi Wu — We prove that for a convex polytope decomposition of a domain in ℝⁿ, an integral curve of a piecewise analytic vector field is chopped by the decomposition into finitely many pieces. As a consequence, we prove the finiteness of the number of the edge flips in a discrete Yamabe flow.

Some properties of hyperbolic Yamabe solitons

Tue, 24 Oct 2023 00:00:00 GMT

Adara M. Blaga, Cihan Özgür — We define the hyperbolic Yamabe flow and obtain some properties of its stationary solutions, namely, of hyperbolic Yamabe solitons. We consider immersed submanifolds as hyperbolic Yamabe solitons and prove that, under certain assumptions, a hyperbolic Yamabe soliton hypersurface is a pseudosymmetric or a metallic shaped hypersurface. We characterize the hyperbolic Yamabe soliton factor manifolds of a multiply twisted, multiply warped, doubly warped, and warped product manifold and provide a classification for a complete gradient hyperbolic Yamabe soliton factor manifold. We also determine the conditions for the factor manifolds to be hyperbolic Yamabe solitons if the manifold is a hyperbolic Yamabe soliton and illustrate this result for a physical model of the universe, namely, for the Robertson–Walker spacetime.

On locally conformally flat manifolds with positive pinched Ricci curvature

Sun, 12 Mar 2023 00:00:00 GMT

Liang Cheng — By using the Yamabe flow, we prove that if (Mⁿ,g), n≥3, is an n-dimensional locally conformally flat complete Riemannian manifold Rc≥ εRg>0, where ε>0 is a uniformly constant, then Mⁿ must be compact. Our result shows that Hamilton's pinching conjecture also holds for higher dimensional case if we assume additionally the metric is locally conformally flat.

Existence of the (α,β)-Ricci-Yamabe flow on closed manifolds

Tue, 07 Feb 2023 00:00:00 GMT

Liangdi Zhang — On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the (α,β)-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed (α,β)-Ricci-Yamabe flow by estimating its curvatures.

Convergence rate of the weighted Yamabe flow

Wed, 28 Sep 2022 00:00:00 GMT

Pak Tung Ho, Jinwoo Shin, Zetian Yan — The weighted Yamabe flow was the geometric flow introduced to study the weighted Yamabe problem on smooth metric measure spaces. Carlotto, Chodosh and Rubinstein have studied the convergence rate of the Yamabe flow. Inspired by their result, we study in this paper the convergence rate of the weighted Yamabe flow.

The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

Thu, 14 Jul 2022 00:00:00 GMT

Gilles Carron, Eric Chen, Yi Wang — We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant Y≤ 0. Previous work by the second and third named authors showed that while the Yamabe flow always converges in a global weighted sense when Y>0, the flow must diverge when Y≤ 0. We show here in the Y≤ 0 case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.

Combinatorial Yamabe flow on hyperbolic bordered surfaces

Mon, 18 Apr 2022 00:00:00 GMT

Shengyu Li, Xu Xu, Ze Zhou — This paper studies the combinatorial Yamabe flow on hyperbolic bordered surfaces. We show that the flow exists for all time and converges exponentially fast to conformal factor which produces a hyperbolic surface whose lengths of boundary components are equal to prescribed positive numbers. This provides an algorithm to such problems.

Convergence of the Weighted Yamabe Flow

Wed, 10 Nov 2021 00:00:00 GMT

Zetian Yan — We introduce the weighted Yamabe flow ∂g/∂t=(r^m_φ−R^m_φ)g, ∂φ/∂t=m/2(R^m_φ−r^m_φ) on a smooth metric measure space (Mⁿ, g, e^(−φ)rm dvol_g, m), where R^m_φ denotes the associated weighted scalar curvature, and r^m_φ denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension n satisfies ngeqslant 3.

Infinite-time incompleteness of noncompact Yamabe flow

Fri, 05 Nov 2021 00:00:00 GMT

Jin Takahashi, Hikaru Yamamoto — We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.

A new class of discrete conformal structures on surfaces with boundary

Sat, 16 Oct 2021 00:00:00 GMT

Xu Xu — We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo-Luo's rigidity of the discrete conformal structures and Guo's rigidity of vertex scaling on surface with boundary. As a result, new convexities of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow-Luo's combinatorial Ricci flow and Luo's combinatorial Yamabe flow on closed surfaces, we further introduce combinatorial Ricci flow and combinatorial Calabi flows to deform the new discrete conformal structures on surfaces with boundary. The basic properties of these combinatorial curvature flows are established. These combinatorial curvature flows provide effective algorithms for constructing hyperbolic metrics on surfaces with totally geodesic boundary components of prescribed lengths.

Combinatorial Calabi flows on surfaces with boundary

Mon, 04 Oct 2021 00:00:00 GMT

Yanwen Luo, Xu Xu — Motivated by Luo's combinatorial Yamabe flow on closed surfaces and Guo's combinatorial Yamabe flow on surfaces with boundary, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.

Fractional combinatorial Calabi flow on surfaces

Thu, 29 Jul 2021 00:00:00 GMT

Tianqi Wu, Xu Xu — Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for Thurston's circle packings, Luo's combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurston's Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings on polyhedral surfaces, we do surgery on the fractional combinatorial Calabi flow by edge flipping under the Delaunay condition to handle the potential singularities along the flow. Using the discrete conformal theory established by Gu et al., we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow with surgery.