This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K¨ahler surface.The relation between the maximum of...This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K¨ahler surface.The relation between the maximum of the Kahler angle and the maximum of |H|2 on the limit flow is studied.The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.展开更多
Let M be a complete Kähler surface andbe a symplectic surface which is smoothly immersed in M.Letαbe the Kähler angle ofin M.In the previous paper Han and Li(JEMS 12:505–527,2010)2010,we study the symple...Let M be a complete Kähler surface andbe a symplectic surface which is smoothly immersed in M.Letαbe the Kähler angle ofin M.In the previous paper Han and Li(JEMS 12:505–527,2010)2010,we study the symplectic critical surfaces,which are critical points of the functional L=1 cosαdμin the class of symplectic surfaces.In this paper,we calculate the second variation of the functional L and derive some consequences.In particular,we show that,if the scalar curvature of M is positive,is a stable symplectic critical surface with cosα≥δ>0,whose normal bundle admits a holomorphic section X∈L2(),thenis holomorphic.We construct symplectic critical surfaces in C2.We also prove a Liouville theorem for symplectic critical surfaces in C2.展开更多
An efficient approach to functionalized 4,6-disubstituted-and 4,6,6-trisubstituted-1,3-oxazinan-2-ones skeleton has been developed through the reaction of semicyclic N,O-acetals 4a and 4b with 1,1-disubstituted ethyle...An efficient approach to functionalized 4,6-disubstituted-and 4,6,6-trisubstituted-1,3-oxazinan-2-ones skeleton has been developed through the reaction of semicyclic N,O-acetals 4a and 4b with 1,1-disubstituted ethylenes 5 or 8.As a result of such a[4+2]cycloaddition process,4,6,6-trisubstituted-1,3-oxazinan-2-ones 6aa,6af-6au,7ba,7bf-7bw and 6,6-spiro containing 1,3-oxazinan-2-ones 9ad,9ae,10ba-10bg were obtained in 36%-96%yields and with moderate to excellent diastereoselectivities.In addition,the synthesis of(±)-norallosedamine 12 could be conveniently achieved from the cycloadduct 7bf.展开更多
We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric.We prove that there exists a ...We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric.We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular points.展开更多
In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic ...In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos. 10901088, 11001268)
文摘This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K¨ahler surface.The relation between the maximum of the Kahler angle and the maximum of |H|2 on the limit flow is studied.The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.
基金The research was supported the National Natural Science Foundation of China,No.11131007,No.11471014The research was also supported by the Doctoral Programme Foundation of Institution of Higher Education of China,No.20110002110064.
文摘Let M be a complete Kähler surface andbe a symplectic surface which is smoothly immersed in M.Letαbe the Kähler angle ofin M.In the previous paper Han and Li(JEMS 12:505–527,2010)2010,we study the symplectic critical surfaces,which are critical points of the functional L=1 cosαdμin the class of symplectic surfaces.In this paper,we calculate the second variation of the functional L and derive some consequences.In particular,we show that,if the scalar curvature of M is positive,is a stable symplectic critical surface with cosα≥δ>0,whose normal bundle admits a holomorphic section X∈L2(),thenis holomorphic.We construct symplectic critical surfaces in C2.We also prove a Liouville theorem for symplectic critical surfaces in C2.
基金the National Natural Science Foundation of China(No.21772027 to B.-G.Wei and 21702032 to C.-M.Si)for financial support.
文摘An efficient approach to functionalized 4,6-disubstituted-and 4,6,6-trisubstituted-1,3-oxazinan-2-ones skeleton has been developed through the reaction of semicyclic N,O-acetals 4a and 4b with 1,1-disubstituted ethylenes 5 or 8.As a result of such a[4+2]cycloaddition process,4,6,6-trisubstituted-1,3-oxazinan-2-ones 6aa,6af-6au,7ba,7bf-7bw and 6,6-spiro containing 1,3-oxazinan-2-ones 9ad,9ae,10ba-10bg were obtained in 36%-96%yields and with moderate to excellent diastereoselectivities.In addition,the synthesis of(±)-norallosedamine 12 could be conveniently achieved from the cycloadduct 7bf.
基金supported by National Natural Science Foundation of China(Grant Nos.11471014 and 11471299)the Fundamental Research Funds for the Central Universities
文摘We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric.We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular points.
基金supported by theNationalNatural Science Foundation of China,Nos.11721101,12071352,12031017。
文摘In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.
