关于Udo Simon猜测的注记
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摘要
本文主要研究了关于Udo Simon猜测中k=4的情形,获得以下主要结果:设Ψ:S~2→S~n为线性满的极小浸入,若Gauss曲率K满足1/10≤K≤1/6且不是常数,则n=6,且Ψ的准线φ_0至少有2个不同的分歧点.同时给出它的一个推论,如果1/7＜K≤1/6,则K是常数1/6.
全文共分三个部分,第一节引言中介绍本文所研究问题的历史背景,及本文所用主要方法和所得的主要结果:第二节利用调和序列方法研究了极小浸入Ψ:S~2→S~n,给出了一些基本公式和引理;第三节首先用种新的方法证明了k=2和3的情形,然后给出Udo Simon猜测中k=4的情形时一些主要结果.
In this paper we research Udo Simon conjecture in the case k=4.We obtain the main results as follows:Letψ:S~2→S~n be a linearly full minimal immersion with induced Gaussian curvature.If K is not constant and 1/10≤K≤1/6,then n=6,and the directrixφ_0 ofψhas at least 2 distinct ramified points. Consequently if 1/7＜K≤1/6,then K is constant 1/6.
The paper is divided into three sections.In section 1,as an introduction,the historical background of the relevant problems,and the main method used in this article and the principal results.In section 2,we study minimal immersionψ:S~2→S~n by the method of harmonic sequences,get some basic formulas and basic lemmas.Firstly,In section 3 we use another method to prove the case k=2,3. Secondly,we give the main results about Udo Simon conjecture in the case k=4.
全文共分三个部分,第一节引言中介绍本文所研究问题的历史背景,及本文所用主要方法和所得的主要结果:第二节利用调和序列方法研究了极小浸入Ψ:S~2→S~n,给出了一些基本公式和引理;第三节首先用种新的方法证明了k=2和3的情形,然后给出Udo Simon猜测中k=4的情形时一些主要结果.
In this paper we research Udo Simon conjecture in the case k=4.We obtain the main results as follows:Letψ:S~2→S~n be a linearly full minimal immersion with induced Gaussian curvature.If K is not constant and 1/10≤K≤1/6,then n=6,and the directrixφ_0 ofψhas at least 2 distinct ramified points. Consequently if 1/7＜K≤1/6,then K is constant 1/6.
The paper is divided into three sections.In section 1,as an introduction,the historical background of the relevant problems,and the main method used in this article and the principal results.In section 2,we study minimal immersionψ:S~2→S~n by the method of harmonic sequences,get some basic formulas and basic lemmas.Firstly,In section 3 we use another method to prove the case k=2,3. Secondly,we give the main results about Udo Simon conjecture in the case k=4.
引文
[1]Takahashi.T.,Minimal immersions of Riemannian manifolds[J].Math.Soc.Japan.,1966,18:380-38
[2]Calabi.E.,Minimal immersions of surfaces in Euclidean spheres[J].Diff.Geom.,1967,1:111-125.
[3]Kenmotsu.K,On Minimal immersions of R~2 into S~3[J].Math.Soc.Japan.,1976,28:182-191.
[4]Bryant.R.L,Minimal surfaces of constant curvature in S~n[J],Trans.Amer.Math.Soc.,1985,290:259-271.
[5]Lawson.H.B.Local rigidity theorems for minimal hypersurfaces[J],Ann.Math,1967,89(2):187-197.
[6]Simon.U.,Eigenvalues of the Laplacian and minimal immersions into spheres[J].In:L.A.Co rdero(ed.),Diff.Geometry,Research Notes Math.1985,131(Pitman Adv.Publ.Program):115-120.
[7]Simon.U.,Kozlowski.M,Minimal immersions of 2-manifolds into spheres[J],Math.Z.,1984,186:377-382.
[8]Gray.A.,Minimal varieties and almost Hermitian submanifolds[J].Michigan Math.,1965,12:273-287.
[9]Dill.F.,Verstraelen.L.and Vrancken.L.,On almost complex surfaces of the nearly K(a|¨)hler 6-sphere Ⅱ[J],Kodai.Math.,1987,10:261-271.
[10]Dill.F.,Verstraelen.L.and Vrancken.L.,On problems of Simon.U concerning minimal submanifolds of the nearly K(a|¨)hler 6-sphere[J],Bull(New Ser.)of the Amer.Math.Soc.,1988,19:433-438.
[11]Dill.F.,Opozda.B.,Verstraelen.L.and Vrancken.L.,On totally real 3-dimensional submani-nfolds of the nearly K(a|¨)hler 6-sphere[J].Proc Amer.Math.Soc.,1987,99:741-749.
[12]Shen.Y.B.,The Riemannian geomet ry of minimal surfaces in complex space forms[J],Acta.Math.,Sinica,New Series,1996,12:298-313.
[13]Ogata.T.,Curvature pinching theorem for minimal Surfaces with constant K(a|¨)hler angle in complex projective spaces Ⅰ,Ⅱ[J],TEhoku.Math.,1993,45:271-283.
[14]黎镇琦,欧阳崇珍,作用在向量丛值Abel形式上的两个微分算子及其应用[J].数学年刊,1998.1:83-92
[15]Bolton,J.,Jensen,G.R.,Rigoli,M.,and Woodward,L.M.,On conformal minimal immersions of S~2 into CP~n[J],Math.Ann,1988,279:599-620.
[16]Chern,S.S.,Wolfson,J.G.Harmonic maps of the two-sphere into a complex Grassmann manifold Ⅱ[J],Ann.of Math.,1987,125:301-335.
[17]Li,Z.Q.,counterexamples to the conjecture on minimal S~2 in CP~n with constant K(a|¨)hler an gle[J],Manuscript math.,1995,88:417-431.
[18]Li,Z.Q.,constant curved minimal 2-sphere in G(2,4)[J],Manuscript math.,1999,100:305-316.
[2]Calabi.E.,Minimal immersions of surfaces in Euclidean spheres[J].Diff.Geom.,1967,1:111-125.
[3]Kenmotsu.K,On Minimal immersions of R~2 into S~3[J].Math.Soc.Japan.,1976,28:182-191.
[4]Bryant.R.L,Minimal surfaces of constant curvature in S~n[J],Trans.Amer.Math.Soc.,1985,290:259-271.
[5]Lawson.H.B.Local rigidity theorems for minimal hypersurfaces[J],Ann.Math,1967,89(2):187-197.
[6]Simon.U.,Eigenvalues of the Laplacian and minimal immersions into spheres[J].In:L.A.Co rdero(ed.),Diff.Geometry,Research Notes Math.1985,131(Pitman Adv.Publ.Program):115-120.
[7]Simon.U.,Kozlowski.M,Minimal immersions of 2-manifolds into spheres[J],Math.Z.,1984,186:377-382.
[8]Gray.A.,Minimal varieties and almost Hermitian submanifolds[J].Michigan Math.,1965,12:273-287.
[9]Dill.F.,Verstraelen.L.and Vrancken.L.,On almost complex surfaces of the nearly K(a|¨)hler 6-sphere Ⅱ[J],Kodai.Math.,1987,10:261-271.
[10]Dill.F.,Verstraelen.L.and Vrancken.L.,On problems of Simon.U concerning minimal submanifolds of the nearly K(a|¨)hler 6-sphere[J],Bull(New Ser.)of the Amer.Math.Soc.,1988,19:433-438.
[11]Dill.F.,Opozda.B.,Verstraelen.L.and Vrancken.L.,On totally real 3-dimensional submani-nfolds of the nearly K(a|¨)hler 6-sphere[J].Proc Amer.Math.Soc.,1987,99:741-749.
[12]Shen.Y.B.,The Riemannian geomet ry of minimal surfaces in complex space forms[J],Acta.Math.,Sinica,New Series,1996,12:298-313.
[13]Ogata.T.,Curvature pinching theorem for minimal Surfaces with constant K(a|¨)hler angle in complex projective spaces Ⅰ,Ⅱ[J],TEhoku.Math.,1993,45:271-283.
[14]黎镇琦,欧阳崇珍,作用在向量丛值Abel形式上的两个微分算子及其应用[J].数学年刊,1998.1:83-92
[15]Bolton,J.,Jensen,G.R.,Rigoli,M.,and Woodward,L.M.,On conformal minimal immersions of S~2 into CP~n[J],Math.Ann,1988,279:599-620.
[16]Chern,S.S.,Wolfson,J.G.Harmonic maps of the two-sphere into a complex Grassmann manifold Ⅱ[J],Ann.of Math.,1987,125:301-335.
[17]Li,Z.Q.,counterexamples to the conjecture on minimal S~2 in CP~n with constant K(a|¨)hler an gle[J],Manuscript math.,1995,88:417-431.
[18]Li,Z.Q.,constant curved minimal 2-sphere in G(2,4)[J],Manuscript math.,1999,100:305-316.