In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is pr...In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature estimates.Then the author proves a long time existence result.The asymptotic behavior of these solutions when t→∞is also studied.展开更多
Let M be a compact Riemann surface, h（x） a positive smooth function on M, and Ф（x） a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ（u）=1/2∫M｜△u｜2eФdVg＋...Let M be a compact Riemann surface, h（x） a positive smooth function on M, and Ф（x） a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ（u）=1/2∫M｜△u｜2eФdVg＋8πc∫MueФ-8πclog∫Mheu＋ФdVg.We give a sufficient condition under which J achieves its minimum for c ≤ infx∈MeФ（X）.展开更多
In this paper,we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(m),which is the case where the generalized Gauss mapΦis ramified over a famil...In this paper,we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(m),which is the case where the generalized Gauss mapΦis ramified over a family of hypersurfaces{Q_(j)}_(j=1)^(q)in P^(m-1)(C)located in the N-subgeneral position.In addition,we investigate the Gauss curvature estimate for the K-quasiconformal harmonic surfaces immersed in R^(3)whose Gauss maps are ramified over a family of hypersurfaces located in the N-subgeneral position.展开更多
We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric...We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric.We also construct explicitly some conical metrics whose curvature is not integrable.展开更多
In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is ...In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is an n-dimensional complete hypersurface in Sn＋1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations展开更多
基金the National Natural Science Foundation of China(No.10531090).
文摘In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature estimates.Then the author proves a long time existence result.The asymptotic behavior of these solutions when t→∞is also studied.
文摘Let M be a compact Riemann surface, h（x） a positive smooth function on M, and Ф（x） a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ（u）=1/2∫M｜△u｜2eФdVg＋8πc∫MueФ-8πclog∫Mheu＋ФdVg.We give a sufficient condition under which J achieves its minimum for c ≤ infx∈MeФ（X）.
基金supported by the NFSC(11971182,12271189)the NFS of Fujian Province of China(2019J01066,2021J01304)。
文摘In this paper,we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(m),which is the case where the generalized Gauss mapΦis ramified over a family of hypersurfaces{Q_(j)}_(j=1)^(q)in P^(m-1)(C)located in the N-subgeneral position.In addition,we investigate the Gauss curvature estimate for the K-quasiconformal harmonic surfaces immersed in R^(3)whose Gauss maps are ramified over a family of hypersurfaces located in the N-subgeneral position.
基金Support by the Project of Stable Support for Youth Team in Basic Research Field,CAS(Grant No.YSBR-001)NSFC(Grant Nos.12271495,11971450 and 12071449).
文摘We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric.We also construct explicitly some conical metrics whose curvature is not integrable.
文摘In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is an n-dimensional complete hypersurface in Sn＋1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations
