设F1和F2分别是光滑流形M1和M2上的芬斯勒度量,共形双扭曲积芬斯勒度量是在乘积流形M=M1×M2上赋予的芬斯勒度量F2=e2σ(f22F12+f12F22), 其中f1、f2和 σ分别是M1 、M2和M上的正值光滑函数。本文证明了半C-可约共形双扭曲积芬斯勒...设F1和F2分别是光滑流形M1和M2上的芬斯勒度量,共形双扭曲积芬斯勒度量是在乘积流形M=M1×M2上赋予的芬斯勒度量F2=e2σ(f22F12+f12F22), 其中f1、f2和 σ分别是M1 、M2和M上的正值光滑函数。本文证明了半C-可约共形双扭曲积芬斯勒度量是类C2芬斯勒度量。Let F1 and F2 be two Finsler metrics on smooth manifold M1 and M2,respectively.The conformally doubly warped product Finsler metric F2=e2σ(f22F12+f12F22) is a Finsler metric endowed on the M=M1×M2 ,where f1、f2 and σ are positive smooth functions on M1 、M2 and M, respectively.It is proved that semi-C-reducible conformally doubly warped product Finsler metric is a C2-like Finsler metric.展开更多
An upper estimate of the new curvature entropy is provided,via the integral inequality of a concave function.For two origin-symmetric convex bodies in R^(n),this bound is sharper than the log-Minkowski inequality of c...An upper estimate of the new curvature entropy is provided,via the integral inequality of a concave function.For two origin-symmetric convex bodies in R^(n),this bound is sharper than the log-Minkowski inequality of curvature entropy.As its application,a novel proof of the log-Minkowski inequality of curvature entropy in the plane is given.展开更多
Two sharp Chernoff type inequalities are derived for star bodies in R2,one is an extension of the dual Chernoff-Ou-Pan inequality,and the other is the reverse Chernoff type inequality.Furthermore,we establish a genera...Two sharp Chernoff type inequalities are derived for star bodies in R2,one is an extension of the dual Chernoff-Ou-Pan inequality,and the other is the reverse Chernoff type inequality.Furthermore,we establish a generalized dual symmetric mixed Chernoff inequality for two planar star bodies.As a direct consequence,a new proof of the dual symmetric mixed isoperimetric inequality is presented.展开更多
本文建立了一个新的几何量——沿外法线向量的曲率轨迹,探讨了其几何意义及其与曲率中心轨迹的关系。并利用新的几何量建立了一组参数不等式:α∫γκ2ds+β∫02πρ2(θ)dθ+λ∫02πρβ2(θ)dθ+δL2+σA+ ω| A˜|+μ| A^|+ν∫02π...本文建立了一个新的几何量——沿外法线向量的曲率轨迹,探讨了其几何意义及其与曲率中心轨迹的关系。并利用新的几何量建立了一组参数不等式:α∫γκ2ds+β∫02πρ2(θ)dθ+λ∫02πρβ2(θ)dθ+δL2+σA+ ω| A˜|+μ| A^|+ν∫02πρβ^2(θ)dθ+ζ(ρM−ρm)2+ξL^2≥0同时,本文还通过所建立的等周不等式推导出了一些新的几何Bonnesen型不等式,并研究了这些不等式的稳定性。In this paper, we establish a new geometric quantity-locus of curvature along outer normal vector. Its geometric meaning and its relationship with the curvature center locus are discussed. And, we use the new geometric quantity to establish a family of parametric inequalities:α∫γκ2ds+β∫02πρ2(θ)dθ+λ∫02πρβ2(θ)dθ+δL2+σA+ ω| A˜|+μ| A^|+ν∫02πρβ^2(θ)dθ+ζ(ρM−ρm)2+ξL^2≥0And we also use our isoperimetric inequalities to derive some new geometric Bonnesen-type in equalities. Furthermore, we investigate the stability property of such inequalities.展开更多
Let(M,g)be a compact Riemann surface with unit area,h a smooth function on M.The Kazdan-Warner problem is that under what kind of conditions on h the equationΔu=8π-8πhe^(u) has a solution.In this survey article,we ...Let(M,g)be a compact Riemann surface with unit area,h a smooth function on M.The Kazdan-Warner problem is that under what kind of conditions on h the equationΔu=8π-8πhe^(u) has a solution.In this survey article,we shall review the development of this problem along the variational method.展开更多
On a compact Riemann surface with finite punctures P_(1),…P_(k),we define toric curves as multivalued,totallyunramified holomorphic maps to P^(n)with monodromy in a maximal torus of PSU(n+1).Toric solutions to SU(n+1...On a compact Riemann surface with finite punctures P_(1),…P_(k),we define toric curves as multivalued,totallyunramified holomorphic maps to P^(n)with monodromy in a maximal torus of PSU(n+1).Toric solutions to SU(n+1)Todasystems on X\{P_(1);…;P_(k)}are recognized by the associated toric curves in.We introduce character n-ensembles as-tuples of meromorphic one-forms with simple poles and purely imaginary periods,generating toric curves on minus finitelymany points.On X,we establish a correspondence between character-ensembles and toric solutions to the SU(n+1)system with finitely many cone singularities.Our approach not only broadens seminal solutions with two conesingularities on the Riemann sphere,as classified by Jost-Wang(Int.Math.Res.Not.,2002,(6):277-290)andLin-Wei-Ye(Invent.Math.,2012,190(1):169-207),but also advances beyond the limits of Lin-Yang-Zhong’s existencetheorems(J.Differential Geom.,2020,114(2):337-391)by introducing a new solution class.展开更多
文摘设F1和F2分别是光滑流形M1和M2上的芬斯勒度量,共形双扭曲积芬斯勒度量是在乘积流形M=M1×M2上赋予的芬斯勒度量F2=e2σ(f22F12+f12F22), 其中f1、f2和 σ分别是M1 、M2和M上的正值光滑函数。本文证明了半C-可约共形双扭曲积芬斯勒度量是类C2芬斯勒度量。Let F1 and F2 be two Finsler metrics on smooth manifold M1 and M2,respectively.The conformally doubly warped product Finsler metric F2=e2σ(f22F12+f12F22) is a Finsler metric endowed on the M=M1×M2 ,where f1、f2 and σ are positive smooth functions on M1 、M2 and M, respectively.It is proved that semi-C-reducible conformally doubly warped product Finsler metric is a C2-like Finsler metric.
基金supported by the NSFC(12171378)supported by the Characteristic innovation projects of universities in Guangdong province(2023K-TSCX381)+3 种基金supported by the Young Top-Talent program of Chongqing(CQYC2021059145)the Major Special Project of NSFC(12141101)the Science and Technology Research Program of Chongqing Municipal Education Commission(KJZD-K202200509)the Natural Science Foundation Project of Chongqing(CSTB2024NSCQ-MSX0937).
文摘An upper estimate of the new curvature entropy is provided,via the integral inequality of a concave function.For two origin-symmetric convex bodies in R^(n),this bound is sharper than the log-Minkowski inequality of curvature entropy.As its application,a novel proof of the log-Minkowski inequality of curvature entropy in the plane is given.
基金supported by the Postgraduate Scientic Research Innovation Project of Chongqing Normal University(YKC24010)Chunna Zeng's research was supported by the Major Special Project of the National Natural Science Foundation of China(12141101)+2 种基金the Young Top-Talent program of Chongqing(CQYC2021059145)the Technology Research Foundation of Chongqing Educational committee(KJZD-K202200509)the Natural Science Foundation Project of Chongqing(CSTB2024NSCQ-MSX0937)。
文摘Two sharp Chernoff type inequalities are derived for star bodies in R2,one is an extension of the dual Chernoff-Ou-Pan inequality,and the other is the reverse Chernoff type inequality.Furthermore,we establish a generalized dual symmetric mixed Chernoff inequality for two planar star bodies.As a direct consequence,a new proof of the dual symmetric mixed isoperimetric inequality is presented.
文摘本文建立了一个新的几何量——沿外法线向量的曲率轨迹,探讨了其几何意义及其与曲率中心轨迹的关系。并利用新的几何量建立了一组参数不等式:α∫γκ2ds+β∫02πρ2(θ)dθ+λ∫02πρβ2(θ)dθ+δL2+σA+ ω| A˜|+μ| A^|+ν∫02πρβ^2(θ)dθ+ζ(ρM−ρm)2+ξL^2≥0同时,本文还通过所建立的等周不等式推导出了一些新的几何Bonnesen型不等式,并研究了这些不等式的稳定性。In this paper, we establish a new geometric quantity-locus of curvature along outer normal vector. Its geometric meaning and its relationship with the curvature center locus are discussed. And, we use the new geometric quantity to establish a family of parametric inequalities:α∫γκ2ds+β∫02πρ2(θ)dθ+λ∫02πρβ2(θ)dθ+δL2+σA+ ω| A˜|+μ| A^|+ν∫02πρβ^2(θ)dθ+ζ(ρM−ρm)2+ξL^2≥0And we also use our isoperimetric inequalities to derive some new geometric Bonnesen-type in equalities. Furthermore, we investigate the stability property of such inequalities.
文摘Let(M,g)be a compact Riemann surface with unit area,h a smooth function on M.The Kazdan-Warner problem is that under what kind of conditions on h the equationΔu=8π-8πhe^(u) has a solution.In this survey article,we shall review the development of this problem along the variational method.
基金supported by the National Natural Science Foundation of China(11931009,12271495,11971450,and 12071449)Anhui Initiative in Quantum Information Technologies(AHY150200)the Project of Stable Support for Youth Team in Basic Research Field,Chinese Academy of Sciences(YSBR-001).
文摘On a compact Riemann surface with finite punctures P_(1),…P_(k),we define toric curves as multivalued,totallyunramified holomorphic maps to P^(n)with monodromy in a maximal torus of PSU(n+1).Toric solutions to SU(n+1)Todasystems on X\{P_(1);…;P_(k)}are recognized by the associated toric curves in.We introduce character n-ensembles as-tuples of meromorphic one-forms with simple poles and purely imaginary periods,generating toric curves on minus finitelymany points.On X,we establish a correspondence between character-ensembles and toric solutions to the SU(n+1)system with finitely many cone singularities.Our approach not only broadens seminal solutions with two conesingularities on the Riemann sphere,as classified by Jost-Wang(Int.Math.Res.Not.,2002,(6):277-290)andLin-Wei-Ye(Invent.Math.,2012,190(1):169-207),but also advances beyond the limits of Lin-Yang-Zhong’s existencetheorems(J.Differential Geom.,2020,114(2):337-391)by introducing a new solution class.
