Neckpinch — Inverse mean curvature flow

The quermassintegral inequalities for k-convex spiral-star-shaped domains

Mon, 26 Oct 2026 00:00:00 GMT

Junfang Li — We extend the quermassintegral inequalities of our earlier work from star-shaped domains to a substantially broader class of domains in Euclidean space, which we call spiral-star-shaped. A smooth domain is spiral-star-shaped if its boundary satisfies ⟨ X+A X,ν⟩>0 for some fixed skew-symmetric linear map A, where X is the position vector and ν is the outward unit normal. The vector field X+A X is conformal Killing, with dilation component X and rotational component A X; its trajectories combine expansion and rotation, giving rise to the spiral terminology. When A=0, it reduces to ordinary star-shapedness, while nonzero A allows domains that are not star-shaped in the usual radial sense. For smooth k-convex spiral-star-shaped domains, we prove long-time existence and exponential convergence to a round sphere for a locally constrained inverse curvature flow. The positive spiral support function provides the geometric control needed to treat initial hypersurfaces that need not be star-shaped in the usual sense. Monotonicity of the quermassintegrals along the flow yields sharp quermassintegral inequalities in this enlarged class, with equality if and only if the domain is a ball.

A Positive Mass Type Theorem for a Singular Toroidal Static Model

Wed, 23 Sep 2026 00:00:00 GMT

Zhixin Wang — We study positive mass and related Brown–York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi–Tam type construction to obtain a Brown–York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.

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.

Sharp p-Capacity Estimates via Quermassintegrals in Hyperbolic Space

Tue, 18 Aug 2026 00:00:00 GMT

Xiaoshang Jin, Yao Wan, Jie Xiao — This paper establishes sharp upper bounds for p-capacities Cap_(12m+1, an interpolating radius combines the 2m-th curvature-excess radius with the L^∞ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.

Weinstock inequalities for outward-minimizing domains

Wed, 12 Aug 2026 00:00:00 GMT

Chaoqun Gao, Yong Wei, Rong Zhou — We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.

The Penrose inequality with charge for 2-convex initial data sets

Fri, 31 Jul 2026 00:00:00 GMT

Tuan Dolmen — We prove the Penrose inequality with charge under the 2-convexity condition recently introduced by Dong. More precisely, given a complete, connected and asymptotically flat Einstein-Maxwell initial data set (M,g,k; E,B) satisfying the charged dominant energy and the 2-convexity conditions, with divergence-free electromagnetic vector fields (E,B) and a connected outermost past apparent horizon Σ that satisfies |Σ| ≥ 4πq² - where q is the total charge - we show that the following inequality for the ADM mass m holds: m≥ √((|Σ|)/16π) + q² √(fracπ|Σ|), with equality if and only if k ≡ 0 and (M,g;E,B) is isometric to a canonical slice of sub-extremal Reissner-Nordström spacetime. Building on Dong's proof of the uncharged case, we use his P-inverse mean curvature flow and its weak formulation, which only depends on (g,P) and hence applies to the charged setting unchanged. The novelty of our work is the modification of the monotonicity formula to account for the additional charge term. For time-symmetric data (k ≡ 0), the flow reduces to the classical inverse mean curvature flow and our monotonicity formula to Jang's monotonicity of the charged Hawking mass, recovering the charged Riemannian Penrose inequality.

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.

The Penrose conjecture for initial data sets satisfying a 2-convexity condition

Wed, 22 Jul 2026 00:00:00 GMT

Conghan Dong — Let (M³, g, k) be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon Σ. We prove the Penrose conjecture, namely that m_ADM(g) ≥ √(|Σ|)/(16 π), under the assumptions of the dominant energy condition and the 2-convexity condition that the sum of the two smallest eigenvalues of k is nonnegative. The main tool is the σ-inverse mean curvature flow, together with a monotonicity formula developed in.

Infinity-harmonic functions and inverse mean curvature flow clusters

Tue, 07 Jul 2026 00:00:00 GMT

Kai Xu — An ∞-harmonic function is a viscosity solution of ∇² u(∇u,∇u)=0, or equivalently, an absolute minimizer of \|∇u\|_(L^∞). We prove a variety of new structural and regularity results in two dimensions, including: 1. ∞-harmonic functions in domains of ℝ² are C^(1,1/3). 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating ∞-harmonic functions to inverse mean curvature flow (IMCF) clusters – which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the p→∞ limit of the classical duality between p-harmonic and q-harmonic functions in ℝ², where 1/p+1/q=1.

Higher regularity of the inverse anisotropic mean curvature flow

Thu, 02 Jul 2026 00:00:00 GMT

Chaoqun Gao, Yong Wei, Rong Zhou — We prove an anisotropic analogue of the higher regularity theorem of Huisken and Ilmanen for inverse mean curvature flow. For an arbitrary smooth Minkowski norm, we first prove a Huisken–Ilmanen type Harnack estimate for smooth closed strictly star-shaped solutions. We then construct global smooth solutions starting from C¹ strictly star-shaped hypersurfaces with bounded nonnegative weak anisotropic mean curvature. Combining this construction with the asymptotic theory for weak inverse anisotropic mean curvature flow, we show that weak solutions starting from bounded smooth initial sets become smooth outside a compact set.

The σ-inverse mean curvature flow and the generalized Penrose conjecture

Tue, 26 May 2026 00:00:00 GMT

Conghan Dong — Let (M³, g, k) be a complete asymptotically flat initial data set satisfying the dominant energy condition, and let m denote its ADM mass. The generalized Penrose conjecture asserts that the area of an outermost generalized apparent horizon N⊂ M satisfies |N| ≤ 16 πm². In this paper, we establish this inequality for each connected component of N in the case where k is proportional to the metric g. Our approach is based on a new geometric evolution, which we call the σ-inverse mean curvature flow, together with a novel monotonicity formula that may be of independent interest.

Spectral Obstructions to Contracting Transport Maps on Curved Spaces

Sat, 23 May 2026 00:00:00 GMT

Shrey Aryan — Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is 1-Lipschitz. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition CD(ρ,∞). Recently, Beck and Jerison [BJ21] raised related questions on the round hemisphere. The existence of a contracting transport map implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. In this work, we construct counterexamples to the corresponding spectral comparisons on spheres and on weighted manifolds satisfying CD(1,∞) in dimensions d≥4, yielding obstructions to contracting transport maps. In dimensions d≥5, the weighted counterexamples can be chosen to satisfy Ric_g≥ 0 and ∇_g² V ≥ g separately. In dimension two, we use inverse mean curvature flow to construct a contracting transport map from the suitably rescaled round sphere to every closed connected Riemannian surface satisfying the same positive Ricci curvature lower bound. This implies the spectral comparison in dimension two. Together with the recent counterexample in dimension three by Lin, Wang, and Xu [LWX26], this settles the spherical spectral comparison in every dimension d≥2. Using the same method, we also construct a contracting transport from the uniform probability measure on a hemisphere onto the normalized uniform measure on any geodesically convex subset of positive volume, thereby answering affirmatively the remaining case of a conjecture by Beck and Jerison [BJ21] following the work of Fathi, Fradelizi, Gozlan, and Zugmeyer [FFGZ26].

Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

Tue, 19 May 2026 00:00:00 GMT

Mattia Fogagnolo, Giorgio Gatti, Alessandra Pluda — We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a C⁰ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem for continuous Riemannian metrics with nonnegative scalar curvature in such IMCF sense.

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.

Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

Thu, 12 Feb 2026 00:00:00 GMT

Theodora Bourni, José M. Espinar, Aakash Mishra — We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow, and as a consequence we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become star-shaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend these results to the non-compact setting in two cases. First, assuming the asymptotic boundary of the solution consists of a single point, we show that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere. Second, assuming the asymptotic boundary consists of two points, we prove that the flow eventually becomes a global graph over a hyperbolic cylinder with uniform gradient bounds; this is achieved through an explicit cylindrical barrier construction analogous to the horospherical one.

Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

Mon, 22 Dec 2025 00:00:00 GMT

Kuicheng Ma — In this paper, we study the behavior of some locally constrained inverse curvature flow in de Sitter space, with initial value any closed spacelike k-convex hypersurface satisfying some pinching condition. Assume further the Heintze-Karcher inequality for any closed spacelike mean convex hypersurface in de Sitter space, we derive a class of Alexandrov-Fenchel inequalities.

Inverse curvature flow of closed Legendre curves

Mon, 06 Oct 2025 00:00:00 GMT

Takashi Kagaya, Masatomo Takahashi — In this paper, we deal with an inverse curvature flow of ℓ-convex Legendre curves. Since the Legendre curve is a natural generalization of regular curve, the flow is a generalization of the classical inverse curvature flow of regular curves. For the initial value problem, we study on the unique existence of the flow in global time, the monotonicity of the number of the singular cusps with respect to t > 0 and the asymptotic behavior of the flow as t → ∞. Regarding the asymptotics, the flow asymptotically converges to one of the self similar solutions by scaling appropriately, and the convergence is completely categorized depending on the initial curve.

Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

Sun, 28 Sep 2025 00:00:00 GMT

Jingshi Cui, Peibiao Zhao — In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold M for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature W of M satisfies W ∈ (-∞, W₀ )∪ \ 0\ ∪ (W₀, +∞), where W₀ <0 and W₀ >0 are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when W ≥ 0, wherea it is a geodesic of nonvanishing curvature when W is a negative constant. Specially, in the first Heisenberg group M(0), we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in M(0) and provide a new proof of the fact that the total curvature of γ⊂ M(0) with strictly positive curvature equals 2π.

On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes

Mon, 22 Sep 2025 00:00:00 GMT

Brian Harvie, Ye-Kai Wang — We establish several geometric characterizations and rigidity results for 3-dimensional asymptotically locally hyperbolic (ALH) static spaces with horizon boundary. Notably, a 3-dimensional ALH static space with toroidal infinity and strictly non-spherical horizons is isometric to a toroidal Kottler metric. Furthermore, we show that the surface gravity of a static horizon with spherical infinity is bounded below by √3, with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincaré-Einstein fillings of S² × S¹(λ) arising from these spaces have length parameter λ≤ (√3)⁻¹, which supports a recent conjecture of Chang-Yang-Zhang[18]. Finally, static horizons with hyperbolic infinity and non-negative Chruściel-Herzlich mass obey the reverse Riemannian Penrose inequality. In conjunction with work of Ge-Wang-Wu-Xia [29], we use this fact to obtain uniqueness of static ALH graphs with hyperbolic infinities. These results follow from a generalization of the Minkowski inequality in AdS-Schwarzschild space due to Brendle-Hung-Wang [15]. Using optimal coefficients for the sub-static Heintze-Karcher inequality from [24], we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with negative cosmological constant. Another fundamental tool developed in this paper is a regularity theorem for IMCF in ALH manifolds. Specifically, we prove that a weak solution of IMCF in an ALH 3-manifold with horizon boundary is eventually smooth. This extends the regularity theorem for a spherical infinity due to Shi-Zhu [52].

On an area-preserving inverse curvature flow for plane curves

Wed, 30 Jul 2025 00:00:00 GMT

Zezhen Sun, Yuting Wu — In this paper, we study a 1/κⁿ-type area-preserving non-local flow of convex closed plane curves for any n>0. We show that the flow exists globally, the length of evolving curve is non-increasing, and the limiting curve will be a circle in the C^(∞) metric as time t→∞.

Inverse curvature flows for capillary hypersurfaces in the unit ball

Wed, 16 Jul 2025 00:00:00 GMT

Shujing Pan, Bo Yang — In this paper, we study inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball. We establish the existence and convergence results for a class of such flows. As an application, we derive a family of Alexandrov Fenchel inequalities for weakly convex hypersurfaces with free boundary.

A locally constrained inverse Hessian quotient flow in de Sitter space

Tue, 15 Jul 2025 00:00:00 GMT

Kuicheng Ma — In this paper, an Alexandrov-Fenchel inequality is established for closed 2-convex spacelike hypersurface in de Sitter space by investigating the behavior of the locally constrained inverse curvature flow \beginalign \frac{\partial }{\partial t}x=\bigg(u-\fracλ'E_1E_2\bigg)ν,\nonumber \endalign which provides a partial answer to the conjecture raised by Hu and Li in.

New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

Sat, 28 Jun 2025 00:00:00 GMT

Jie Wu — In this paper, we establish a broad class of new sharp Alexandrov-Fenchel inequalities involving general convex weight functions for static convex hypersurfaces in hyperbolic space. Additionally, we derive new weighted Minkowski-type inequalities for static convex hypersurfaces in hyperbolic space ℍⁿ and for convex hypersurfaces in the sphere Sⁿ. The tools we shall use are the locally constrained inverse curvature flows in hyperbolic space and in the sphere.

Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

Thu, 20 Mar 2025 00:00:00 GMT

Kwok-Kun Kwong, Yong Wei — We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under weaker convexity assumptions compared to approaches based on inverse curvature flows. The use of weighted formulas provides flexibility in deriving inequalities with different weight functions. Furthermore, our estimates are more quantitative as they include a distance term measuring the domain's deviation from a reference ball. We also analyze the stability of a weighted geometric inequality from a recent paper via analysis of the support function on the sphere and show that, with an optimal choice of the origin, this inequality is stronger than the classical isoperimetric inequality.

Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

Tue, 25 Feb 2025 00:00:00 GMT

Liam Mazurowski, Jintian Zhu — We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let (M³,g) be an asymptotically flat manifold with scalar curvature R≥ 0. Then, for each constant c>0, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. Likewise, let (M³,g) be an asymptotically hyperbolic manifold with scalar curvature R≥ −6. Then, for each constant c>2, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any T the inverse mean curvature flow emerging out of a point p far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times t∈ (-∞,T].

3-Manifolds with positive scalar curvature and bounded geometry

Thu, 13 Feb 2025 00:00:00 GMT

Otis Chodosh, Yi Lai, Kai Xu — We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be ℝ³. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.

Evolution of the Torsional Rigidity under Geometric Flows

Tue, 26 Nov 2024 00:00:00 GMT

Vicent Gimeno i Garcia, Fernán González-Ibáñez — This paper explores the behavior of the torsional rigidity of a precompact domain as the ambient manifold evolves under a geometric flow. Specifically, we derive bounds on torsional rigidity under the Ricci Flow for Heisenberg spaces and homogeneous spheres. Additionally, we establish bounds under the Inverse Mean Curvature Flow for strictly convex, free-boundary, disk-type hypersurfaces within a ball. In this latter case, by extending the analysis to the maximal existence time of the flow, we obtain inequalities of comparison with the flat disk for both volume and torsional rigidity.

Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

Tue, 12 Nov 2024 00:00:00 GMT

Frederick Tsz-Ho Fong — In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree −1 (including the inverse mean curvature flow) in warped product spaces I ×_φ Mⁿ, where Mⁿ is a compact homogeneous manifold and φ" ≥ 0, must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than −1 and with an extra assumption φ' ≥ 0. Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric.

Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

Sun, 10 Nov 2024 00:00:00 GMT

Luca Benatti, Alessandra Pluda, Marco Pozzetta — We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of p-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In 3-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of p-capacitary potentials. We prove that they strongly converge in W^(1,q)ₗoc as p→ 1⁺ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of p-capacitary potentials.

Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds

Wed, 30 Oct 2024 00:00:00 GMT

Dasong Li, John Man Shun Ma — In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow, including the mean curvature flow, inverse mean curvature flow, Gauss curvature flow and so on.