Submanifold On The Overall Geometry And Geometric Analysis

In this thesis, we mainly study problems on global geometry and geometricanalysis of Riemannian submanifolds, including vanishing theorem for homologygroups, topological sphere theorem, L² harmonic 1-forms, finiteness of end andthe spectrum of the Laplacian.In 1973, by using the Federer-Fleming existence theorem and the techniquesfrom the calculus of variations in the geometric measure theory, H. B. Lawson andJ. Simons [30] proved the non-existence theorem for stable integral current in acompact Riemannian submanifold isometrically immersed into a unit sphere andvanishing theorem for homology groups. In 1984, Y. L. Xin [47] generalized theLawson-Simon's nonexistence theorem for stable integral current and vanishingtheorem for homology groups to the case of compact submanifolds in Euclideanspace, and gave several important applications. In 1997, making use of Lawson-Simons-Xin nonexistence theorem for stable integral currents, K. Shiohama andH. W. Xu proved a topological sphere theorem for complete submanifolds in thesimply connected space forms with nonnegative constant curvature [43]. In thisthesis, we first extend the vanishing theorem due to Lawson, Simons and Xinto the case of compact submanifolds of a hyperbolic space. Thus, by using thenew vanishing theorem for homology groups, we prove the topological spheretheorem for complete submanifolds in a hyperbolic space. Hence we generalizethe Shiohama-Xu topological sphere theorem.The well-known Berstein theorem asserts that an entire minimal graph Mⁿ(?)Rⁿ⁺¹ must be an n-dimensional hyperplane if n≤7. There have been many worksto extend the above Berstein theorem by assuming that the complete minimalhypersurface be stable. Utilizing a result of Schoen-Yau [40], Cao-Shen-Zhuproved that a complete stable minimal hypersurface Mⁿ of Rⁿ⁺¹ with n≥3must have only one end [5]. Li-Wang further showed that each end of a completeminimal submanifold Mⁿ(n≥3) must be non-parabolic. And they proved thatif M has finite index, then the dimension of the space of L² harmonic 1-forms on M is finite, and M must have finitely many ends in [35]. Let Mⁿ(n≥3)be an n-dimensional complete noncompact oriented submanifold in an (n+p)-dimensional Euclidean space R^n+p. In this thesis, we prove that if the secondfundamental form B of M satisfies integral from n=M|B|ⁿ＜C(n), where positive constantC(n) depends only on n, then there are no notrivial L² harmonic 1-forms on M,and M must have only one end. We also extend this result to the case that theambient spaces are complete simply connected space forms with nonzero constantcurvature. On the other hand, Let Mⁿ be an z-dimensional complete noncompactoriented submanifold with finite total curvature in an (n+p)-dimensional simplyconnected space form F^n+p(c) of constant curvature c. In this thesis, we provethat if M satisfies one of the following: (â…°) n≥3, c=0 and integral from n=M Hⁿ＜∞; (â…±)n≥5, c=-1 and H＜1-2/n^1/2; (â…²) n≥3, c=1 and H is bounded, where Hdenotes the mean curvature of M. then the dimension of the space of L² harmonic1-forms on M is finite. Moreover, in the case of (â…°) or (â…±), M must have finitelymany ends. We still study a complete noncompact space-like submanifold ina pseudo-Euclidean space R_p^n+p, and obtain some sufficient conditions for thespace-like submanifold to have finite many ends in this thesis.Recently, Cheng-Cheung-Zhou studied the global behavior of complete non-compact weakly stable constant mean curvature hypersurfaces in general Rie-mannian manifolds. In particular, they proved some one-end theorems in spaceforms[14]. Let Mⁿ be an n-dimensional complete noncompact oriented weaklystable constant mean curvature hypersurface in an (n+1)-dimensional Rieman-nian manifold Nⁿ⁺¹ with the (n-1)th-Ricci curvature satisfying Ric_(n-1) (N)≥(n-1)c≥0. In this thesis, we prove the non-existence theorem for boundedharmonic functions on M, and M must have only one end. In addition, we alsoextend the above result to the case that the ambient space is the hyperbolicspace.The spectrum of the Laplacian on a Riemannian manifold is an importantanalytic invariant and is deeply connected with geometry and topology of a man-ifold. Estimating the lower and upper bounds of the eigenvalue in terms of geo-metric and topological quantities of manifold is a very important research topic.Up to now, people only know the spectrum of the Laplacian on a few simple manifolds, for instance, the Euclidean sphere Sⁿ and the production of severalspheres. In this thesis, we totally depict the spectrum of the Laplacian on theclosed hypersurface with constant mean curvature and constant scalar curvaturein S⁴, i.e., compute all eigenvalues of such closed hypersurface. In particular,the first eigenvalue of the Laplacian on the closed minimal hypersurface withconstant scalar curvature in S⁴ is 3.