In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F : S1 × [0, T ) → R^2 which satisfies the followin...In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F : S1 × [0, T ) → R^2 which satisfies the following evolution equation δ^2F /δt^2 (u, t) = k(u, t)N(u, t)-▽ρ(u, t), ∨(u, t) ∈ S^1 × [0, T ) with the initial data F (u, 0) = F0(u) and δF/δt (u, 0) = f(u)N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F (u, t), f(u) and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by ▽ρ Δ=(δ^2F /δsδt ,δF/δt) T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1.展开更多
This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloc...This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.展开更多
Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestra...Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.展开更多
Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curva...Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case展开更多
Under investigation in this paper is a hyperbolic mean curvature flow for convex evolving curves.Firstly,in view of Lie group analysis,infinitesimal generators,symmetry groups and an optimal system of symmetries of th...Under investigation in this paper is a hyperbolic mean curvature flow for convex evolving curves.Firstly,in view of Lie group analysis,infinitesimal generators,symmetry groups and an optimal system of symmetries of the considered hyperbolic mean curvature flow are presented.At the same time,some group invariant solutions are computed through reduced equations.In particular,we construct explicit solutions by applying the power series method.Furthermore,the convergence of the solutions of power series is certificated.Finally,conservation laws of the hyperbolic mean curvature flow are established via Ibragimov's approach.展开更多
First, we review the authors' recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric soluti...First, we review the authors' recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.展开更多
Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of sm...Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CPm.The purpose of this paper is to develop the work due to Pipoli and Sinestrari,and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space.Namely,the authors prove that if the initial submanifold in CPm satisfies a suitable pinching condition,then the mean curvature flow converges to a round point in finite time,or converges to a totally geodesic submanifold as t→∞.Consequently,they obtain a differentiable sphere theorem for submanifolds in the complex projective space.展开更多
This paper is devoted to the study of the vortex dynamics of the Cauchy problem for a parabolic Ginzburg Landau system which simulates inhomogeneous type II superconducting materials and three-dimensional superconduct...This paper is devoted to the study of the vortex dynamics of the Cauchy problem for a parabolic Ginzburg Landau system which simulates inhomogeneous type II superconducting materials and three-dimensional superconducting thin films having variable thickness. We will prove that the vortex of the problem is moved by a codimension k mean curvature flow with external force field. Besides, we will show that the mean curvature flow depends strongly on the external force, having completely different phenomena from the usual mean curvature flow.展开更多
A surface E is a graph in R^4 if there is a unit constant 2-form ω on R^4 such that <e_1∧e_2.ω>≥v_0>0 where{e_1.e_2}is an orthonormal frame on Σ.We prove that.if v_0≥on the initial snrface,then the mean...A surface E is a graph in R^4 if there is a unit constant 2-form ω on R^4 such that <e_1∧e_2.ω>≥v_0>0 where{e_1.e_2}is an orthonormal frame on Σ.We prove that.if v_0≥on the initial snrface,then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution.A surface Σ is a graph in M_1×M_2 where M_1 and M_2 are Riemann surfaces. if<e_1∧e_2.ω>≥v_0>0 where w_1 is a Khler form on M_1.We prove that.if M is a Khler-Einstein surface with scalar curvature R.v_0≥ on the initial surface,then the mean curvature flow has a global solution and it sub-converges to a minimal surface,if.in addition.R≥0 it converges to a totally geodesic surface which is holomorphic.展开更多
In this paper, for the Lorentz manifold M^(2)× R with M^(2) a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains...In this paper, for the Lorentz manifold M^(2)× R with M^(2) a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M^(2), which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.展开更多
A smooth, compact and strictly convex hypersurface evolving in R^n+1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is...A smooth, compact and strictly convex hypersurface evolving in R^n+1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.展开更多
In this paper we prove an so-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫∑0 |A|^2 ≤ ε0 and the initial area u0(∑0...In this paper we prove an so-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫∑0 |A|^2 ≤ ε0 and the initial area u0(∑0) is not large, then along the mean curvature flow, we have ∫∑t |A|^2 ≤ ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.展开更多
In this paper,the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold M^(n)×R,where M^(n) is an n-...In this paper,the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold M^(n)×R,where M^(n) is an n-dimensional(n≥2)complete Riemannian manifold with nonnegative Ricci curvature,and R is the Euclidean 1-space.展开更多
This paper proves the existence of rotationally symmetric solutions to a curvature flow in image processing. The flow includes the level sets flow and the mean curvature flow projected onto the normal. Sharp estimates...This paper proves the existence of rotationally symmetric solutions to a curvature flow in image processing. The flow includes the level sets flow and the mean curvature flow projected onto the normal. Sharp estimates are obtained for these solutions..展开更多
By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow ...By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C^(2)-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.展开更多
In this paper,we study the mean curvature flow with oblique derivative boundary conditions.We prove the longtime existence by choosing a suitable auxiliary function.Also,we prove the asymptotic behavior of the mean cu...In this paper,we study the mean curvature flow with oblique derivative boundary conditions.We prove the longtime existence by choosing a suitable auxiliary function.Also,we prove the asymptotic behavior of the mean curvature flow with zero oblique derivative boundary data which is a generalization of Huisken’s original result about prescribed perpendicular contact angle.展开更多
Mean curvature flow and its singularities have been paid attention extensively in recent years. The present article reviews briefly their certain aspects in the author's interests.
We study the evolution of convex hypersurfaces H(., t) with initial H(., 0) = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H(., t), H0 is a smooth, cl...We study the evolution of convex hypersurfaces H(., t) with initial H(., 0) = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H(., t), H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^* 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^*, then H(.,t) remains uniformly convex and expands to infinity as t →∞ and its scaling, H(-, t)e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.展开更多
We consider the inverse curvature flows of smooth,closed and strictly convex rotation hypersurfaces in space forms M_(κ)^(n+1)with speed function given by F^(-α),whereα∈(0,1]forκ=0,-1,α=1 forκ=1 and F is a smoo...We consider the inverse curvature flows of smooth,closed and strictly convex rotation hypersurfaces in space forms M_(κ)^(n+1)with speed function given by F^(-α),whereα∈(0,1]forκ=0,-1,α=1 forκ=1 and F is a smooth,symmetric,strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces.We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value,and prove the long time existence and convergence of the flows.No second derivatives conditions are required on F.展开更多
In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow’s Harnack inequali...In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow’s Harnack inequality and entropy estimate.展开更多
基金Kong and Wang was supported in part by the NSF of China (10671124)the NCET of China (NCET-05-0390)the work of Liu was supported in part by the NSF of China
文摘In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F : S1 × [0, T ) → R^2 which satisfies the following evolution equation δ^2F /δt^2 (u, t) = k(u, t)N(u, t)-▽ρ(u, t), ∨(u, t) ∈ S^1 × [0, T ) with the initial data F (u, 0) = F0(u) and δF/δt (u, 0) = f(u)N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F (u, t), f(u) and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by ▽ρ Δ=(δ^2F /δsδt ,δF/δt) T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1.
基金supported in part by a grant from China Scholarship Councilthe National Natural Science Foundation of China(11301006)the Anhui Provincial Natural Science Foundation(1408085MA01)
文摘This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.
基金supported partially by the National Natural Science Foundation of China (10871171)the Chinese-Hungarian Sci. and Tech. cooperation (for 2007-2009)
文摘Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.
基金supported by the NSFC(11101267,11271132)the Innovation Program of Shanghai Municipal Education Commission(13YZ087)the Science and Technology Program of Shanghai Maritime University(20120061)
文摘Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case
基金Supported by the Natural Science Foundation of Shanxi(202103021224068).
文摘Under investigation in this paper is a hyperbolic mean curvature flow for convex evolving curves.Firstly,in view of Lie group analysis,infinitesimal generators,symmetry groups and an optimal system of symmetries of the considered hyperbolic mean curvature flow are presented.At the same time,some group invariant solutions are computed through reduced equations.In particular,we construct explicit solutions by applying the power series method.Furthermore,the convergence of the solutions of power series is certificated.Finally,conservation laws of the hyperbolic mean curvature flow are established via Ibragimov's approach.
基金Supported by Natural Science Foundation of China (10631020, 10871061)the Grant for Ph.D Program of Ministry of Education of Chinasupported by Innovation Propject for the Development of Science and Technology (IHLB) (201098)
文摘First, we review the authors' recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.
基金supported by the National Natural Science Foundation of China(Nos.12071424,11531012,12201087).
文摘Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CPm.The purpose of this paper is to develop the work due to Pipoli and Sinestrari,and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space.Namely,the authors prove that if the initial submanifold in CPm satisfies a suitable pinching condition,then the mean curvature flow converges to a round point in finite time,or converges to a totally geodesic submanifold as t→∞.Consequently,they obtain a differentiable sphere theorem for submanifolds in the complex projective space.
基金Supported by National 973-Project and Basic Research Grant of Tsinghua University
文摘This paper is devoted to the study of the vortex dynamics of the Cauchy problem for a parabolic Ginzburg Landau system which simulates inhomogeneous type II superconducting materials and three-dimensional superconducting thin films having variable thickness. We will prove that the vortex of the problem is moved by a codimension k mean curvature flow with external force field. Besides, we will show that the mean curvature flow depends strongly on the external force, having completely different phenomena from the usual mean curvature flow.
基金supported in part by a Sloan fellowship and an NSERC grant for Chenby a grant from NSF of China for Li.by a grant from NSF of USA for Tian
文摘A surface E is a graph in R^4 if there is a unit constant 2-form ω on R^4 such that <e_1∧e_2.ω>≥v_0>0 where{e_1.e_2}is an orthonormal frame on Σ.We prove that.if v_0≥on the initial snrface,then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution.A surface Σ is a graph in M_1×M_2 where M_1 and M_2 are Riemann surfaces. if<e_1∧e_2.ω>≥v_0>0 where w_1 is a Khler form on M_1.We prove that.if M is a Khler-Einstein surface with scalar curvature R.v_0≥ on the initial surface,then the mean curvature flow has a global solution and it sub-converges to a minimal surface,if.in addition.R≥0 it converges to a totally geodesic surface which is holomorphic.
基金This work was supported by the National Natural Science Foundation of China(Nos.11401131,11101132,11926352)the Fok Ying-Tung Education Foundation(China)Hubei Key Laboratory of Applied Mathematics(Hubei University)。
文摘In this paper, for the Lorentz manifold M^(2)× R with M^(2) a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M^(2), which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.
基金supported by National Natural Science Foundation of China(Grant No.10971055)Funds for Disciplines Leaders of Wuhan(Grant No.Z201051730002)+1 种基金Project of Hubei Provincial Department of Education(Grant No.T200901)supported by Fundao Ciênciae Tecnologia(FCT)through a doctoral fellowship SFRH/BD/60313/2009
文摘A smooth, compact and strictly convex hypersurface evolving in R^n+1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.
基金supported by National Natural Science Foundation of China (Grant No. 10901088)
文摘In this paper we prove an so-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫∑0 |A|^2 ≤ ε0 and the initial area u0(∑0) is not large, then along the mean curvature flow, we have ∫∑t |A|^2 ≤ ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.
基金supported by the National Natural Science Foundation of China(Nos.11801496,11926352)the Fok Ying-Tung Education Foundation(China)Hubei Key Laboratory of Applied Mathematics(Hubei University)。
文摘In this paper,the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold M^(n)×R,where M^(n) is an n-dimensional(n≥2)complete Riemannian manifold with nonnegative Ricci curvature,and R is the Euclidean 1-space.
基金Supported by the National Key Basic Research and Devedepment (973) Program of China (No. G19990751) and the Trans-Century Training Program Foundation for the Talents of China
文摘This paper proves the existence of rotationally symmetric solutions to a curvature flow in image processing. The flow includes the level sets flow and the mean curvature flow projected onto the normal. Sharp estimates are obtained for these solutions..
基金supported by National Natural Science Foundation of China(Grant Nos.11671015 and 11731001)。
文摘By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C^(2)-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.
基金supported by National Natural Science Foundation of China(Grant No.11471188)。
文摘In this paper,we study the mean curvature flow with oblique derivative boundary conditions.We prove the longtime existence by choosing a suitable auxiliary function.Also,we prove the asymptotic behavior of the mean curvature flow with zero oblique derivative boundary data which is a generalization of Huisken’s original result about prescribed perpendicular contact angle.
文摘Mean curvature flow and its singularities have been paid attention extensively in recent years. The present article reviews briefly their certain aspects in the author's interests.
基金Acknowledgements The author would like to thank professor Huaiyu Jian for his comments and suggestions about this work. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11131005, 11271118, 11301034) and the Doctoral Programme Foundation of Institution of Higher Education of China.
文摘We study the evolution of convex hypersurfaces H(., t) with initial H(., 0) = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H(., t), H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^* 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^*, then H(.,t) remains uniformly convex and expands to infinity as t →∞ and its scaling, H(-, t)e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.
基金Supported by the National Key R and D Program of China(Grant No.2020YFA0713100)National Natural Science Foundation of China(Grant Nos.11971244 and 11871283)+1 种基金Natural Science Foundation of Tianjin,China(Grant No.19JCQNJC14300)Research(Grant No.KY0010000052)from University of Science and Technology of China。
文摘We consider the inverse curvature flows of smooth,closed and strictly convex rotation hypersurfaces in space forms M_(κ)^(n+1)with speed function given by F^(-α),whereα∈(0,1]forκ=0,-1,α=1 forκ=1 and F is a smooth,symmetric,strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces.We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value,and prove the long time existence and convergence of the flows.No second derivatives conditions are required on F.
基金Supported by the National Natural Science Foundation of China(No.11971355)Natural Science Foundation of Zhejiang Province(No.LY22A010007)。
文摘In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow’s Harnack inequality and entropy estimate.