https://neckpinch.com/topics/gauss-curvature-flow/New arXiv papers on gauss curvature flow, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.18423arxiv:2610.18423Tue, 13 Oct 2026 00:00:00 GMTXinxiu Zhang — In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann (1998). We further investigate a volume-preserving shifted α-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any α>0 the solution exists for all time and converges smoothly to a geodesic sphere. The theory of modified curvature measures developed in the first part serves as a key ingredient in characterizing the Hausdorff limit shape of the flow.Other curvature flowsGauss curvature flowVolume-preserving & constrained flowsLong-time existence & convergence https://arxiv.org/abs/2608.22494arxiv:2608.22494Sun, 23 Aug 2026 00:00:00 GMTXinqun Mei, Guofang Wang, Liangjun Weng — We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.Other curvature flowsGauss curvature flowStability & uniquenessFree boundary & capillary https://arxiv.org/abs/2607.26923arxiv:2607.26923Wed, 29 Jul 2026 00:00:00 GMTTianci Luo, Yong Wei, Rong Zhou — We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the α-Gauss curvature flow ∂ₜX=−K^αν, α>0. We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If α>1/(n+2), we apply a Cayley-type conformal map that sends the extinction point to the origin of a Euclidean half-space and then normalize the enclosed half-space volume. The resulting normalized hypersurfaces converge smoothly to the unit hemisphere. The proof combines boundary identities for the spherical free boundary, a boundary-adapted Tso estimate, an almost-monotonicity formula for a half-space entropy, and uniform curvature estimates for the normalized flow.Other curvature flowsGauss curvature flowSingularitiesFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2607.00943arxiv:2607.00943Wed, 01 Jul 2026 00:00:00 GMTRafael Belli, Rafael López — In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group Nil₃ that are invariant under a one-parameter group of ambient isometries. By means of the four canonical types of Killing vector fields and the three families of invariant surfaces (vertical translations, horizontal translations, and helicoidal motions), we analyze the twelve resulting types of possible solitons. In some cases, there do not exist any invariant solitons; in others, we find explicit parametrizations, or describe their geometric properties.Other curvature flowsGauss curvature flow3-manifolds https://arxiv.org/abs/2605.17097arxiv:2605.17097Sat, 16 May 2026 00:00:00 GMTRafael Belli, Rafael López — We classify invariant surfaces in the 3-dimensional solvable Lie group sol that act as solitons for the Gauss curvature flow. We consider solitons associated with the canonical basis of Killing vector fields F₁, F₂, F₃\, where F₁ and F₂ generate horizontal translations and F₃ generates the scaling isometry. We establish rigidity results for F₃-invariant surfaces, proving that specific totally geodesic vertical planes are the only F₁- and F₂-solitons. For F₁-invariant surfaces, we establish the main geometric properties of F₂- and F₃-solitons in both the extrinsic and intrinsic Gauss curvature.Other curvature flowsGauss curvature flow3-manifoldsHomogeneous spacesStability & uniqueness https://arxiv.org/abs/2601.14659arxiv:2601.14659Wed, 21 Jan 2026 00:00:00 GMTGuanghan Li, Chenyang Liu — In this paper, we study the long-time existence and asymptotic behavior of an anisotropic capillary Gauss curvature flow. By studying this flow and proving its convergence to a stationary solution, we establish a new existence result for the capillary Orlicz-Minkowski problem without the evenness assumption, and provide a flow approach to the existence of smooth solutions.Other curvature flowsGauss curvature flowFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2510.25435arxiv:2510.25435Wed, 29 Oct 2025 00:00:00 GMTXia Zhao, Peibiao Zhao — The study of the dual curvature measures [Y. Huang, E. Lutwak, D. Yang & G. Y. Zhang, Acta. Math. 216 (2016): 325-388], which connects the cone-volume measure and Aleksandrov's integral curvature, and has created a precedent for the theoretical research of the dual Brunn-Minkowski theory. Motivated by the foregoing groundbreaking works, the present paper introduces the p-th dual k-torsional rigidity associated with a k-Hessian equation and establishes its Hadamard variational formula with 1≤ k≤ n-1, which induces the p-th dual k-torsional measure. Further, based on the p-th dual k-torsional measure, this article, for the first time, proposes the p-th dual Minkowski problem of the k-torsional rigidity which can be equivalently converted to a nonlinear partial differential equation in smooth case: \beginalign f(x)=τ(|\nabla h|^2+h^2)^{\fracp-n2}h_Ω(x)|Du(ν^-1_Ω(x))|^k+1σ_n-k(h_ij(x)+h_Ω(x)δ_ij), \endalign where τ>0 is a constant, f is a positive smooth function defined on Sⁿ⁻¹ and σₙ₋ₖ is the (n-k)-th elementary symmetric function of the principal curvature radii. We confirm the existence of smooth non-even solution to the p-th dual Minkowski problem of the k-torsional rigidity for p<n-2 by the method of a curvature flow which converges smoothly to the solution of equation (). Specially, a novel approach for the uniform lower bound estimation in the C⁰ estimation for the solution to the curvature flow is presented with the help of invariant functional Φ(Ωₜ).Other curvature flowsGauss curvature flowStability & uniquenessLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2510.05614arxiv:2510.05614Tue, 07 Oct 2025 00:00:00 GMTBeomjun Choi, Kyeongsu Choi, Dongjun Noh — In this paper, we construct a pancake-like ancient compact solution with flat sides to the Gauss curvature flow, contained in a slab. Also, we construct sausage-like ancient compact solutions to the α-Gauss curvature flow with α>1/2, asymptotic to a round cylinder.Other curvature flowsGauss curvature flowAncient solutions https://arxiv.org/abs/2508.17321arxiv:2508.17321Sun, 24 Aug 2025 00:00:00 GMTMuhittin Evren Aydin, Rafael López — A λ-translator is a surface in Euclidean space ℝ³ whose Gauss curvature K satisfies K=⟨ N, v ⟩ +λ, where N is the Gauss map, v is a fixed direction, and λ∈ ℝ. In this paper, we classify all λ-translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.Other curvature flowsGauss curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2508.05551arxiv:2508.05551Thu, 07 Aug 2025 00:00:00 GMTTristan C. Collins, Benjy Firester — We solve a general class of free boundary Monge-Ampère equations given by det D²u = λ(f(−u))/(g(u^⋆)h(∇u))χ_(u<0\) in ℝⁿ, ∇u (ℝⁿ) = P where P is a bounded convex set containing the origin, and h>0 on P. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.Gauss curvature flowRicci solitonsKähler–Ricci solitonsKähler geometryLimit spaces & RCDFree boundary & capillaryEigenvalues & spectrum https://arxiv.org/abs/2507.12863arxiv:2507.12863Thu, 17 Jul 2025 00:00:00 GMTRafael López — In this paper, we consider λ-translating solitons and λ-shrinkers of the Gauss curvature flow in Euclidean space. We prove that planes and circular cylinders are the only λ-translating solitons with constant mean curvature. We also prove that planes, spheres and circular cylinders are the only λ-shrinkers with constant mean curvature. We give a classification of the λ-translating solitons and λ-shrinkers with one constant principal curvature.Other curvature flowsGauss curvature flowRicci solitonsSelf-shrinkers & translatorsStability & uniqueness https://arxiv.org/abs/2506.09840arxiv:2506.09840Wed, 11 Jun 2025 00:00:00 GMTXinqun Mei, Guofang Wang, Liangjun Weng — In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture.Other curvature flowsGauss curvature flowSingularitiesFree boundary & capillary https://arxiv.org/abs/2505.24494arxiv:2505.24494Fri, 30 May 2025 00:00:00 GMTXia Zhao, Peibiao Zhao — P. Salani [Adv. Math., 229 (2012)] introduced the k-torsional rigidity associated with a k-Hessian equation and obtained the Brunn-Minkowski inequalities w.r.t. the torsional rigidity in ℝ³. Following this work, we first construct, in the present paper, a Hadamard variational formula for the k-torsional rigidity with 1≤ k≤ n-1, then we can deduce a k-torsional measure from the Hadamard variational formula. Based on the k-torsional measure, we propose the Minkowski problem for the k-torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the C⁰ estimation for the solution to the curvature flow is presented with the help of invariant functional Φ(Ωₜ).Other curvature flowsGauss curvature flowStability & uniquenessGeometric inequalities https://arxiv.org/abs/2505.23067arxiv:2505.23067Thu, 29 May 2025 00:00:00 GMTRongli Huang, Changzheng Qu, Zhizhang Wang, Weifeng Wo — We investigate the evolution of strictly convex hypersurfaces driven by the k-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary C² estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk and Schnürer from the second boundary value problem of Gauss curvature flow to k-Hessian curvature flow.Other curvature flowsGauss curvature flowSelf-shrinkers & translatorsBoundaries & non-compactLong-time existence & convergence https://arxiv.org/abs/2412.15612arxiv:2412.15612Fri, 20 Dec 2024 00:00:00 GMTBurcu Bektaş Demirci, Ferdağ Kahraman Aksoyak, Murat Babaarslan — In this paper, we study K^α–translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space E³. First, we investigate the condition under which two parallel surfaces can become K^α–translators moving with the same speed w. Then, we examine K^α–translators on canal surfaces and we show that if a canal surface is K^α–translator, then it must be a surface of revolution in E³. We also provide examples for moving a surface of revolution under K–flow (Gauss curvature flow) and K^(−1/2)–flow (inverse Gauss curvature flow) along a direction w=(0,0,1) and we illustrate such surfaces using Wolfram Mathematica 10.4. Finally, we prove that no K^α–translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed w, while the such rotational surfaces itself is a K^α–translator with speed w.Other curvature flowsGauss curvature flowSelf-shrinkers & translators3-manifoldsRotational symmetry & warped products https://arxiv.org/abs/2412.13557arxiv:2412.13557Wed, 18 Dec 2024 00:00:00 GMTNa Fu, Jianping Sun — This article introduces the Lₚ-Gauss dual curvature measure and proposes its related Lₚ-Gauss dual Minkowski problem as: for p,q∈ℝ, under what necessary and/or sufficient condition on a non-zero finite Borel measure μ on unit sphere does there exist a convex body K such that μ is the Lₚ Gauss dual curvature measure? If K exists, to what extent is it unique? This problem amounts to solving a class of Monge-Ampère type equations on unit sphere in smooth case: \beginalign e^{-\frac{|\nabla h_K|^2+h_K^2}2}h_K^1-p (|\nabla h_K|^2+h_K^2)^{\fracq-n2} \det(\nabla^2h_K+h_KI)=f,\qquad (0.1) \endalign where f is a given positive smooth function on unit sphere, hₖ is the support function of convex body K, ∇h_K and ∇²h_K are the gradient and Hessian of h_K on unit sphere with respect to an orthonormal basis, and I is the identity matrix. We confirm the existence of solution to the new problem with p,q>0 and the existence of smooth solution to the equation (0.1) with p ,q∈ℝ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case p,q∈ℝ with q<p is established.Other curvature flowsGauss curvature flowKähler geometryStability & uniqueness https://arxiv.org/abs/2412.10144arxiv:2412.10144Fri, 13 Dec 2024 00:00:00 GMTAlbert 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.Other curvature flowsGauss curvature flowFree boundary & capillaryHarmonic maps & DeTurck trick https://arxiv.org/abs/2410.22904arxiv:2410.22904Wed, 30 Oct 2024 00:00:00 GMTDasong 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.Mean curvature flowOther curvature flowsInverse mean curvature flowGauss curvature flowStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2411.00779arxiv:2411.00779Wed, 16 Oct 2024 00:00:00 GMTXia Zhao, Peibiao Zhao — The Minkowski problem for torsional rigidity (2-torsional rigidity) was firstly studied by Colesanti and Fimiani using variational method. Moreover, Hu also studied this problem by the method of curvature flows and obtained the existence of smooth even solutions. In addition, the smooth non-even solutions to the Orlicz Minkowski problem w. r. t q-torsional rigidity were given by Zhao et al. through a Gauss curvature flow. The dual curvature measure and the dual Minkowski problem were first posed and considered by Huang, Lutwak, Yang and Zhang in. The dual Minkowski problem is a very important problem, which has greatly contributed to the development of the dual Brunn-Minkowski theory and extended the other types dual Minkowski problem. To the best of our knowledge, the dual Minkowski problem w. r. t (q) torsional rigidity is still open because the dual (q) torsional measure is blank. Thus, it is a natural problem to consider the dual Minkowski problem for (q) torsional rigidity. In this paper, we introduce the p-th dual q-torsional measure and propose the p-th dual Minkowski problem for q-torsional rigidity with q>1. Then we confirm the existence of smooth even solutions for p<n (p≠ 0) to the p-th dual Minkowski problem for q-torsional rigidity by method of a Gauss curvature flow. Specially, we also obtain the smooth non-even solutions with p<0 to this problem.Other curvature flowsGauss curvature flowStability & uniquenessGeometric inequalities https://arxiv.org/abs/2407.05663arxiv:2407.05663Mon, 08 Jul 2024 00:00:00 GMTGenggeng Huang, Xu-Jia Wang, Yang Zhou — We study the regularity of the p-Gauss curvature flow with flat side. In our previous paper(arxiv:2403.12292), we obtained the regularity of the interface, namely the boundary of the flat part. In this paper, we study the regularity of the convex hypersurface near the interface.Other curvature flowsGauss curvature flow https://arxiv.org/abs/2406.05635arxiv:2406.05635Sun, 09 Jun 2024 00:00:00 GMTXia Zhao, Peibiao Zhao — Recently, Huang and Qin introduced the Gaussian chord measure and Lₚ-Gaussian chord measure by variational methods. Meanwhile, they posed Gaussian chord Minkowski problem for p=1 and used variational methods to obtain an origin-symmetric normalized measure solution for the Gaussian chord Minkowski problem. The smooth solution, up to now, to the Lₚ-Gaussian chord Minkowski problem is still open. Motivated by the forgoing works by Huang and Qin in, we propose in the present paper the Lₚ(p>0)-Gaussian chord Minkowski problem and log-Gaussian chord Minkowski problem, and obtain the smooth even solutions to these two types of problems by the method of a Gauss curvature flow.Other curvature flowsGauss curvature flow https://arxiv.org/abs/2406.05159arxiv:2406.05159Thu, 06 Jun 2024 00:00:00 GMTYong Wei, Bo Yang, Tailong Zhou — We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space ℍⁿ⁺¹ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time t∈ [0,∞) and converges to a geodesic sphere exponentially as t→∞ in the smooth topology. A key step is to show the L¹ oscillation decay of the Gauss curvature to its average along a subsequence of times going to the infinity, which combined with an argument using the hyperbolic curvature measure theory implies the Hausdorff convergence.Other curvature flowsGauss curvature flowVolume-preserving & constrained flowsHomogeneous spacesLong-time existence & convergence https://arxiv.org/abs/2404.19266arxiv:2404.19266Tue, 30 Apr 2024 00:00:00 GMTXia Zhao, Peibiao Zhao — The celebrated Minkowski problem for the torsional rigidity (2-torsional rigidity) was firstly studied by Colesanti and Fimiani using variational method. Moreover, Hu, Liu and Ma also studied the Minkowski problem {\it w.r.t.} 2-torsional rigidity by method of curvature flows and obtain the existence of smooth even solutions. Up to now, as far as we know, the study of the Minkowski problem for the q-torsional rigidity is still blank. In the present paper, we propose and investigate the Orlicz Minkowski problem for the q-torsional rigidity corresponding to the q-Laplace equation inspired by the foregoing works, and then confirm the existence of smooth non-even solutions to the Orlicz Minkowski problem for the q-torsional rigidity with q>1 by the method of a Gauss curvature flow.Other curvature flowsGauss curvature flowStability & uniqueness https://arxiv.org/abs/2404.10298arxiv:2404.10298Tue, 16 Apr 2024 00:00:00 GMTShujing Pan, Yong Wei — In this paper, we consider the anisotropic α-Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for all positive power α>0, if the initial hypersurface is complete noncompact and locally uniformly convex, then the solution of the flow exists for all positive time.Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsBoundaries & non-compactLong-time existence & convergence https://arxiv.org/abs/2403.12292arxiv:2403.12292Mon, 18 Mar 2024 00:00:00 GMTG. Huang, X. -J. Wang, Y. Zhou — In this paper, we prove the long time regularity of the interface in the p-Gauss curvature flow with flat side in all dimensions for p>1/n. Here the interface is the boundary of the flat part in the flow. In dimension 2, this problem was solved in for p=1 and in for p∈(1/2,1). We utilize the duality method to transform the Gauss curvature flow to a singular parabolic Monge-Ampère equation, and prove the regularity of the interface by studying the asymptotic cone of the parabolic Monge-Ampère equation in the polar coordinates.Other curvature flowsGauss curvature flowKähler geometry https://arxiv.org/abs/2401.02768arxiv:2401.02768Fri, 05 Jan 2024 00:00:00 GMTThalia D. Jeffres, Leonardo Solanilla — Earlier work of the first author examined two boundary value problems associated to the Gauss Curvature Flow on a surface of revolution generated by a positive, differentiable function on a compact interval. In this continuation, two noncompact cases are addressed.Other curvature flowsGauss curvature flowSurfacesBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2311.18606arxiv:2311.18606Thu, 30 Nov 2023 00:00:00 GMTJinrong Hu, Ping Zhang — A variant of the Gauss curvature flow for closed and convex hypersurfaces is considered. We reveal that if the initial hypersurface is pinched enough, then this property is preserved. Furthermore, based on some structure assumptions on the speed function of the shrinking flow, we show that the flow converges to a sphere. This may generalize the result of B. Chow to the possible non-homogeneous curvature flows.Other curvature flowsGauss curvature flowCurvature pinchingHomogeneous spaces https://arxiv.org/abs/2310.02668arxiv:2310.02668Wed, 04 Oct 2023 00:00:00 GMTKi-Ahm Lee, Taehun Lee — We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the C¹ regularity of free boundaries under a uniform thickness assumption.Other curvature flowsGauss curvature flowFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2308.05332arxiv:2308.05332Thu, 10 Aug 2023 00:00:00 GMTXia Zhao, Peibiao Zhao — The Lₚ chord Minkowski problem based on Chord measures and Lₚ chord measures introduced firstly by Lutwak, Xi, Yang and Zhang [38] is a very important and meaningful geometric measure problem in the Lₚ Brunn-Minkowski theory. Xi, Yang, Zhang and Zhao [45] using variational methods gave a measure solution when p > 1 and 0<p<1 in the symmetric case. Recently, Guo, Xi and Zhao [18] also obtained a measure solution for 0≤ p<1 by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the Lₚ chord Minkowski problem by replacing p with a fixed continuous function φ:(0,∞)→(0,∞), and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.Other curvature flowsGauss curvature flowLimit spaces & RCDGeometric inequalities https://arxiv.org/abs/2305.14830arxiv:2305.14830Wed, 24 May 2023 00:00:00 GMTBin Chen, Weidong Wang, Xia Zhao, Peibiao Zhao — In [Calc. Var., 57:5 (2018)], Hong-Ye-Zhang proposed the p-capacitary Orlicz-Minkowski problem and proved the existence of convex solutions to this problem by variational method for p∈(1,n). However, the smoothness and uniqueness of solutions are still open. Notice that the p-capacitary Orlicz-Minkowski problem can be converted equivalently to a Monge-Ampère type equation in smooth case: \beginalign fφ(h_K)|\nablaΨ|^p=τG \endalign for p∈(1,n) and some constant τ>0, where f is a positive function defined on the unit sphere 𝓢ⁿ⁻¹, φ is a continuous positive function defined in (0,+∞), and G is the Gauss curvature. In this paper, we confirm the existence of smooth solutions to p-capacitary Orlicz-Minkowski problem with p∈(1,n) for the first time by a class of inverse Gauss curvature flows, which converges smoothly to the solution of Equation (). Furthermore, we prove the uniqueness result for Equation () in a special case.Other curvature flowsGauss curvature flowKähler geometryStability & uniquenessLong-time existence & convergence