偶数维黎曼流形直径估计及曲面法向演化问题
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摘要
本文研究偶数维Riemannian流形的直径及曲面法向演化问题,共分四节.
第一二节为本文的引言与预备知识.
第三节首先介绍了Hausdorff距离及Gromov-Hausdorff收敛的概念.设M~(2n)是具有度量g的2n维紧致无边单连通的Riemannian流形,S~(2n)是欧氏空间R~(2n+1)中的单位球面.若流形M~(2n)满足截面曲率K_M∈(0,1],体积0(?).最后利用Hausdor(?)收敛得到了这类流形直径更精确的一个上界估计及更宽的一个gap现象.
第四节介绍了超曲面的概念.设M~n是R~(n+1)中紧致无边连通的超曲面.若M~n满足初始状态是严格凸的,得到它沿外法向演化的渐近性态在某种意义下是R~(n+1)中n维超球面.
This dissertation consists of four sections. We shall investigate the diameter on even dimensionalRiemannian manifolds and asymptotic analysis to the flow of supersurfaces along their normal direction.
The first and second sections are introduction and preliminaries respectively.In the third section, we first introduce the concept of Hausdorff distance and Gromov-Hausdorffconvergence. Let M~(2n) be a 2n-dimensional compact, simply connected Riemannian manifold without boundary and S~(2n) be the unit sphere in Euclidean space R~(2n+1). We derive an upper bound of the diameter in this note whenever the manifold concerned satisfies that the sectional curvature K_M varies in (0,1] and the volume V(M) is not larger than 2(l +η)V(B_(?)π) for some positive numberηdepending only on n, where B_(?)πis the geodesic ball on S~(2n) with radius (?)π. A gap phenomenon of the manifold concerned will be given out. Then we give a lower bound of the first eigenvalue of Laplacian operator on manifold M, which is,λ_1 is biggerthan (?). Finally, by using the method of Hausdorff convergence, we get a more concise estimation of the diameter and prove the existence of a broader gap on the manifold mentioned.
In the forth section, we introduce the concept of hypersurface. Let M~n be a n(≥3)-dimensionalcompact, connected hypersurface without boundary in R~(n+1). We derive asymptotic analysis of this hypersurface along its outer normal direction which is a super-sphere in some sense whenever the initial hypersurface concerned satisfies that it is strictly convex hypersurface.
第一二节为本文的引言与预备知识.
第三节首先介绍了Hausdorff距离及Gromov-Hausdorff收敛的概念.设M~(2n)是具有度量g的2n维紧致无边单连通的Riemannian流形,S~(2n)是欧氏空间R~(2n+1)中的单位球面.若流形M~(2n)满足截面曲率K_M∈(0,1],体积0
第四节介绍了超曲面的概念.设M~n是R~(n+1)中紧致无边连通的超曲面.若M~n满足初始状态是严格凸的,得到它沿外法向演化的渐近性态在某种意义下是R~(n+1)中n维超球面.
This dissertation consists of four sections. We shall investigate the diameter on even dimensionalRiemannian manifolds and asymptotic analysis to the flow of supersurfaces along their normal direction.
The first and second sections are introduction and preliminaries respectively.In the third section, we first introduce the concept of Hausdorff distance and Gromov-Hausdorffconvergence. Let M~(2n) be a 2n-dimensional compact, simply connected Riemannian manifold without boundary and S~(2n) be the unit sphere in Euclidean space R~(2n+1). We derive an upper bound of the diameter in this note whenever the manifold concerned satisfies that the sectional curvature K_M varies in (0,1] and the volume V(M) is not larger than 2(l +η)V(B_(?)π) for some positive numberηdepending only on n, where B_(?)πis the geodesic ball on S~(2n) with radius (?)π. A gap phenomenon of the manifold concerned will be given out. Then we give a lower bound of the first eigenvalue of Laplacian operator on manifold M, which is,λ_1 is biggerthan (?). Finally, by using the method of Hausdorff convergence, we get a more concise estimation of the diameter and prove the existence of a broader gap on the manifold mentioned.
In the forth section, we introduce the concept of hypersurface. Let M~n be a n(≥3)-dimensionalcompact, connected hypersurface without boundary in R~(n+1). We derive asymptotic analysis of this hypersurface along its outer normal direction which is a super-sphere in some sense whenever the initial hypersurface concerned satisfies that it is strictly convex hypersurface.
引文
[1] 白正国等.黎曼几何初步[M].北京:高等教育出版社,2004.
[2] Petersen P. Riemannian Geometry [M]. New York: Springer-Verlag, 1998.
[3] K. Grove, V. P. Petersen. Manifolds near the boundary of existence [J]. Diff. Geom., 1991, 33:379-394.
[4] S. Peters. Convergence of Riemannian manifolds [J]. Compositio Math., 1987, 62: 3-16.
[5] 王培合.完备Riemannian流形上的Laplace算子及相关问题[D].上海:华东师范大学数学系,2003,12-14.
[6] J. Q. Zhong, H. C. Yang. On the estimates of the first eigenvalue of a compact Riemannian manifold [J]. Sci. Sinica, Ser. A, 1984, 27(12): 1265-1273.
[7] 陈维桓,李兴校.黎曼几何引论[M].北京:北京大学出版社,2004.
[8] 王培合,李傅山,姜娜.曲线法向流及其相关问题[J].曲阜师范大学学报,2008.
[9] J. Cheeger, D. G. Ebin. Comparison theorems in Riemannian geometry [J|. North-Hoiland, 1975.
[10] L. Coghlan, Y. Itokawa. A sphere theorem for reverse volume pinching on even-dimension manifolds [J]. Proc. Amer. Math. Soc, 1991, Ⅲ (3): 815-819.
[11] J. Y. Wu. Hausdorff convergence and sphere theorem [J]. Proc. Sym. Pure Math., 1993, 54.685-692.
[12] Y. L. Wen. A note on pinching sphere theorem [J]. C. R. Acad. Sci. Paris. Ser.I, 2004, 338:229-234.
[13] K. Grove, K. Shiohama. A generalized sphere theorem [J]. Ann. of Math., 1977, 106. 201-211. [14] K. Pawel. On the spectral gap for compact manifolds [J]. Diff. Geom., 1992, 36 (2): 315-330.
[15] L. Pawel, T. Andreis. Applications of eigenvalue techniques to geometry [J]. Contemporary geometry, 1991: 21-52, Univ. Ser. Math., Plenum, 1991.
[16] T. Sakai. Riemannian geometry [M]. Tokyo: Shokabo Publishing Co. Ltd., 1992.
[17] K. Shiohama. A sphere theorem for manifolds of positive Ricci curvature [J]. Tansaction Amer. Math. Soc, 1983,27: 811-819.
[18] J. Y. Wu. A diameter pinching sphere theorem [J]. Proc. Amer. Math. Soc.、 1990,197 (3): 796-802.
[19] D. G. Yang. Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature [J]. Pacific J. Math., 1999, 190 (2): 383-398.
[20] H. C. Yang. Estimates of the first eigenvalue for a compact Riemann manifold [J]. Sci. China Ser A, 1990, 33(1): 39-51.
[21] C. Y. Xia. Rigidity and sphere theorem for manifolds with positive Ricci curvature [J]. Manuscript Math., 1994,85: 79-87.
[22] M. Gromov. Groups of polynomial growth and expanding maps [J]. Inst. Hautes Etudes Sci. Pub.Math., 1981, 53: 183-215.
[23] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces [J]. Birkh(?)user, 1999.
[24] Y. L. Wen. A gap phenomenon on even dimension Riemannian manifolds and eigenvalue estimate [J]. Preprint.
[25] Millman, R. S. Parker, G. D. Elements of differential geometry [J]. Prentice-Hall, Englewood Cliffs,N.J., 1977.
[26] Hsiung, C. C. A first course in differentialgeometry [M]. John Wiley Sons, Inc., 1981.
[27] M. Gage. An isoperimetric inequality with applicationsto curve shortening [J]. Duke Math, j.,1983,50: 1225-1229.
[28] Bonnesen, T. Fenchel. W. Theorie derkonvexen Korper [M]. Berlin: Springer, 1934.
[29] 潘生亮.关于凸曲面的一些注记及其对曲线流的一些应用[J].数学年刊,2000,28:53-56.
[30] 彭家贵,陈卿.微分几何[M].北京:高等教育出版社,2002.
[2] Petersen P. Riemannian Geometry [M]. New York: Springer-Verlag, 1998.
[3] K. Grove, V. P. Petersen. Manifolds near the boundary of existence [J]. Diff. Geom., 1991, 33:379-394.
[4] S. Peters. Convergence of Riemannian manifolds [J]. Compositio Math., 1987, 62: 3-16.
[5] 王培合.完备Riemannian流形上的Laplace算子及相关问题[D].上海:华东师范大学数学系,2003,12-14.
[6] J. Q. Zhong, H. C. Yang. On the estimates of the first eigenvalue of a compact Riemannian manifold [J]. Sci. Sinica, Ser. A, 1984, 27(12): 1265-1273.
[7] 陈维桓,李兴校.黎曼几何引论[M].北京:北京大学出版社,2004.
[8] 王培合,李傅山,姜娜.曲线法向流及其相关问题[J].曲阜师范大学学报,2008.
[9] J. Cheeger, D. G. Ebin. Comparison theorems in Riemannian geometry [J|. North-Hoiland, 1975.
[10] L. Coghlan, Y. Itokawa. A sphere theorem for reverse volume pinching on even-dimension manifolds [J]. Proc. Amer. Math. Soc, 1991, Ⅲ (3): 815-819.
[11] J. Y. Wu. Hausdorff convergence and sphere theorem [J]. Proc. Sym. Pure Math., 1993, 54.685-692.
[12] Y. L. Wen. A note on pinching sphere theorem [J]. C. R. Acad. Sci. Paris. Ser.I, 2004, 338:229-234.
[13] K. Grove, K. Shiohama. A generalized sphere theorem [J]. Ann. of Math., 1977, 106. 201-211. [14] K. Pawel. On the spectral gap for compact manifolds [J]. Diff. Geom., 1992, 36 (2): 315-330.
[15] L. Pawel, T. Andreis. Applications of eigenvalue techniques to geometry [J]. Contemporary geometry, 1991: 21-52, Univ. Ser. Math., Plenum, 1991.
[16] T. Sakai. Riemannian geometry [M]. Tokyo: Shokabo Publishing Co. Ltd., 1992.
[17] K. Shiohama. A sphere theorem for manifolds of positive Ricci curvature [J]. Tansaction Amer. Math. Soc, 1983,27: 811-819.
[18] J. Y. Wu. A diameter pinching sphere theorem [J]. Proc. Amer. Math. Soc.、 1990,197 (3): 796-802.
[19] D. G. Yang. Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature [J]. Pacific J. Math., 1999, 190 (2): 383-398.
[20] H. C. Yang. Estimates of the first eigenvalue for a compact Riemann manifold [J]. Sci. China Ser A, 1990, 33(1): 39-51.
[21] C. Y. Xia. Rigidity and sphere theorem for manifolds with positive Ricci curvature [J]. Manuscript Math., 1994,85: 79-87.
[22] M. Gromov. Groups of polynomial growth and expanding maps [J]. Inst. Hautes Etudes Sci. Pub.Math., 1981, 53: 183-215.
[23] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces [J]. Birkh(?)user, 1999.
[24] Y. L. Wen. A gap phenomenon on even dimension Riemannian manifolds and eigenvalue estimate [J]. Preprint.
[25] Millman, R. S. Parker, G. D. Elements of differential geometry [J]. Prentice-Hall, Englewood Cliffs,N.J., 1977.
[26] Hsiung, C. C. A first course in differentialgeometry [M]. John Wiley Sons, Inc., 1981.
[27] M. Gage. An isoperimetric inequality with applicationsto curve shortening [J]. Duke Math, j.,1983,50: 1225-1229.
[28] Bonnesen, T. Fenchel. W. Theorie derkonvexen Korper [M]. Berlin: Springer, 1934.
[29] 潘生亮.关于凸曲面的一些注记及其对曲线流的一些应用[J].数学年刊,2000,28:53-56.
[30] 彭家贵,陈卿.微分几何[M].北京:高等教育出版社,2002.