S~2×S~2中的拉格朗日曲面
摘要
在4维流形里,K（a|¨）hler-Einstein的乘积空间S~2×S~2是除复空间形式C~2,CP~2和畃畈2外较为重要的几何对象,其中的曲面论研究也非常丰富. S~2×S~2中的很多类拉格朗日(Lagrangian)曲面:极小,平行平均曲率, Hamiltonian稳定,Hamiltonian极小等,都存在重要的研究结果.
本文通过子流形的各种理论和方法,主要研究S~2×S~2中满足一个几何等式|σ|~2 = 3|H|~2的Lagrangian曲面.从S~2×S~2的Lagrangian曲面上的Jacobian函数出发,我们研究了满足|σ|~2 = 3|H|~2的曲面的局部性质,并证明在一些特殊的几何条件下,这样的曲面退化为全测地的球面M0或全测地环面T.另外我们还给出了这样的曲面的可积性方程.
此外我们还研究了满足|σ|~2 = 3|H|~2的曲面的整体性质,给出了曲面的拓扑限制,并构造了一个Lagrangian球面的例子.该Lagrangian球面是非全测地的,这意味着S~2×S~2中满足该几何不等式的Lagrangian球面并非只有全测地的M0.因此这个例子表明了这一类曲面的分类问题中存在的障碍.将本文的结论与I. Castro和F. Urbano关于S~2×S~2中极小Lagrangian曲面的研究结果相比较,我们看到这两类曲面具有许多相似的性质.
Among the class of 4-dimensional Riemannian manifolds, the product spaceS~2×S~2, which is K（a|¨）hler-Einstein with constant scalar curvature, may be the mostimportant space aside from the 2-dimensional complex space forms. The methods andresults on the surfaces in S~2×S~2are very rich, especially on the Lagrangian surfaces ofdi?erent families: minimal, with parallel mean curvature tensor, Hamiltonian-stable,Hamiltonian-minimal, etc.
In this paper, we mainly study the Lagrangian surfaces in S~2×S~2achieving theminimum of a geometric inequality. We derive some local properties of such La-grangian surfaces via studying the associated Jacobian function in depth, and show thatunder some typical geometric conditions, there exist no surfaces satisfying |σ|~2 = 3|H|~2except the totally geodesic sphere M0 or the totally geodesic torus T.
In addition, we also study the global problems of the Lagrangian surfaces with|σ|~2 = 3|H|~2. We derive a restriction on the genus of such surfaces, and constructan example of a Lagrangian sphere. The sphere is not totally geodesic, which showsM0 is not the only case of Lagrangian surfaces satisfying the geometric equality, incontrast with the torus. The sample implies the obstruction of the classification of suchLagrangian surfaces in S~2×S~2. Comparing the results of this paper with the ones ofminimal Lagrangian surfaces by I. Castro and F. Urbano, we see these two families ofLagrangian surfaces in S~2×S~2have many similarities.
本文通过子流形的各种理论和方法,主要研究S~2×S~2中满足一个几何等式|σ|~2 = 3|H|~2的Lagrangian曲面.从S~2×S~2的Lagrangian曲面上的Jacobian函数出发,我们研究了满足|σ|~2 = 3|H|~2的曲面的局部性质,并证明在一些特殊的几何条件下,这样的曲面退化为全测地的球面M0或全测地环面T.另外我们还给出了这样的曲面的可积性方程.
此外我们还研究了满足|σ|~2 = 3|H|~2的曲面的整体性质,给出了曲面的拓扑限制,并构造了一个Lagrangian球面的例子.该Lagrangian球面是非全测地的,这意味着S~2×S~2中满足该几何不等式的Lagrangian球面并非只有全测地的M0.因此这个例子表明了这一类曲面的分类问题中存在的障碍.将本文的结论与I. Castro和F. Urbano关于S~2×S~2中极小Lagrangian曲面的研究结果相比较,我们看到这两类曲面具有许多相似的性质.
Among the class of 4-dimensional Riemannian manifolds, the product spaceS~2×S~2, which is K（a|¨）hler-Einstein with constant scalar curvature, may be the mostimportant space aside from the 2-dimensional complex space forms. The methods andresults on the surfaces in S~2×S~2are very rich, especially on the Lagrangian surfaces ofdi?erent families: minimal, with parallel mean curvature tensor, Hamiltonian-stable,Hamiltonian-minimal, etc.
In this paper, we mainly study the Lagrangian surfaces in S~2×S~2achieving theminimum of a geometric inequality. We derive some local properties of such La-grangian surfaces via studying the associated Jacobian function in depth, and show thatunder some typical geometric conditions, there exist no surfaces satisfying |σ|~2 = 3|H|~2except the totally geodesic sphere M0 or the totally geodesic torus T.
In addition, we also study the global problems of the Lagrangian surfaces with|σ|~2 = 3|H|~2. We derive a restriction on the genus of such surfaces, and constructan example of a Lagrangian sphere. The sphere is not totally geodesic, which showsM0 is not the only case of Lagrangian surfaces satisfying the geometric equality, incontrast with the torus. The sample implies the obstruction of the classification of suchLagrangian surfaces in S~2×S~2. Comparing the results of this paper with the ones ofminimal Lagrangian surfaces by I. Castro and F. Urbano, we see these two families ofLagrangian surfaces in S~2×S~2have many similarities.
引文
[1] Borrelli V, Chen BY, Morvan J. Une caracte′rization ge′ome′trique de la sphere de Whitney.Comptes Rendus de l’Acade′mie des Sciences, Se′rie I, Mathe′matique, 1995, 321: 1485-1490
[2] Castro I, Torralbo F, Urbano F. On Hamiltonian stationary Lagrangian spheres in non-Einstein Ka¨hler surfaces. arXiv:math.DG/07064390 v1, 2007
[3] Castro I, Urbano F. Twistor holomorphic Lagrangian surfaces in the complex projective andhyperbolic planes. Annals of Global Analysis and Geometry, 1995, 13: 59-67
[4] Castro I, Urbano F. Lagrangian surfaces in the complex Euclidean plane with conformalMaslov form. To?hoku Math. J, 1993, 45: 565-582
[5] Castro I, Urbano F. Minimal Lagrangian surfaces in畓2×畓2. Commun. Anal. Geom., 200715: 217-248
[6] Castro I, Urbano F. Surfaces with parallel mean curvature vector in畓2×畓2 and畈2×畈2.arXiv:math.DG/08071808 v2, 2008
[7] Chen BY, Nagano T. Totally geodesic submanifolds of symmetric spaces, I. Duke Mathe-matical Journal, 1977, 44: 745-755
[8] Chern SS, Wolfson J. Minimal surfaces by moving frames. American Journal of Mathemat-ics, 1983, 105: 59-83
[9] Iriyeh H, Ono H, Sakai T. Integral geometry and Hamiltonian volume minimizing propertyof a totally geodesic Lagrangian torus in畓2×畓2. Proc. Japan Acad. Ser. A Math. Sci., 2003,79: 167-170
[10] Li H, Ma H, Su L. Lagrangian spheres in the 2-dimensional complex space forms. IsraelJournal of Mathematics, 2008, 166: 113-124
[11] Li H, Vrancken L. A basic inequality and new characterization of Whitney spheres in acomplex space form. Israel Journal of Mathematics, 2005, 146: 223-242
[12] Ma H, Ohnita Y. On Lagrangian submanifolds in complex hyperquadrics and isoparametrichypersurfaces in spheres. Mathematische Zeitschrift, 2009, 261:749-785
[13] Oh YG. Second variation and stabilities of minimal Lagrangian submanifolds in Kaehlermanifolds. Inventiones Mathematicae, 1990, 101: 501-519
[14] Oh YG. Volume minimization of Lagrangian submanifolds under Hamiltonian deformation.Mathematische Zeitschrift, 1993, 212: 175-192
[2] Castro I, Torralbo F, Urbano F. On Hamiltonian stationary Lagrangian spheres in non-Einstein Ka¨hler surfaces. arXiv:math.DG/07064390 v1, 2007
[3] Castro I, Urbano F. Twistor holomorphic Lagrangian surfaces in the complex projective andhyperbolic planes. Annals of Global Analysis and Geometry, 1995, 13: 59-67
[4] Castro I, Urbano F. Lagrangian surfaces in the complex Euclidean plane with conformalMaslov form. To?hoku Math. J, 1993, 45: 565-582
[5] Castro I, Urbano F. Minimal Lagrangian surfaces in畓2×畓2. Commun. Anal. Geom., 200715: 217-248
[6] Castro I, Urbano F. Surfaces with parallel mean curvature vector in畓2×畓2 and畈2×畈2.arXiv:math.DG/08071808 v2, 2008
[7] Chen BY, Nagano T. Totally geodesic submanifolds of symmetric spaces, I. Duke Mathe-matical Journal, 1977, 44: 745-755
[8] Chern SS, Wolfson J. Minimal surfaces by moving frames. American Journal of Mathemat-ics, 1983, 105: 59-83
[9] Iriyeh H, Ono H, Sakai T. Integral geometry and Hamiltonian volume minimizing propertyof a totally geodesic Lagrangian torus in畓2×畓2. Proc. Japan Acad. Ser. A Math. Sci., 2003,79: 167-170
[10] Li H, Ma H, Su L. Lagrangian spheres in the 2-dimensional complex space forms. IsraelJournal of Mathematics, 2008, 166: 113-124
[11] Li H, Vrancken L. A basic inequality and new characterization of Whitney spheres in acomplex space form. Israel Journal of Mathematics, 2005, 146: 223-242
[12] Ma H, Ohnita Y. On Lagrangian submanifolds in complex hyperquadrics and isoparametrichypersurfaces in spheres. Mathematische Zeitschrift, 2009, 261:749-785
[13] Oh YG. Second variation and stabilities of minimal Lagrangian submanifolds in Kaehlermanifolds. Inventiones Mathematicae, 1990, 101: 501-519
[14] Oh YG. Volume minimization of Lagrangian submanifolds under Hamiltonian deformation.Mathematische Zeitschrift, 1993, 212: 175-192