Spherical Geometry And Topology Of The Submanifold In Research

An important new subject in global differential geometry is the L^q—pinching(q≥n/2) problem, which mainly studies the geometric structure and topolog-ical structures of manifolds under L^q condition.We mainly study the Lⁿ-pinching and more generally L^q(q≥n/2)-pinchingproblem of submanifold in a sphere, and also the sphere theorem under theseconditions. Our main tools are Sobolev inequalities of submanifold, Morse the-ory, integration estimate. Firstly, we obtain the following theorems:Theorem A Let Mⁿ(n≥2) be an n-dimensional closed submanifold withparallel mean curvature in S^n+p(1). Denote by H and S the mean curvatureand the squared length of the sectional fundamental form of M, respectively.If‖S-nH²â€–_n＜C₁(n), where C₁(n) is an explicit positive constant dependingon n, then S=nH², i.e.M is a totally umbilical submanifold, and hence Mⁿis isometric to a sphere M=Sⁿ(1/(1+H²)^1/2).Theorem B Let Mⁿ(n≥2) be an n-dimensional complete submanifoldwith parallel mean curvature in S^n+p(1). Denote by H and S the mean cur-vature and the squared length of the sectional fundamental form of M, re-spectively. If‖S-nH²â€–_n＜C₂(n), where C₂(n) is an explicit positive constantdepending on n, then S=nH², i.e.M is a totally umbilical submanifoldand, andhence Mⁿ is isometric to a sphere M=Sⁿ(1/(1+H²)^1/2).More generally, we have:Theorem C Let Mⁿ(n≥3) be an n-dimensional closed submanifold withparallel mean curvature in S^n+p(1). Denote by H and S the mean curvatureand the squared length of the sectional fundamental form of M, respectively. If‖S-nH²â€–_q＜C(n, q, H, V), where q≥n/2, C(n, q, H, V) is an explicit positiveconstant depending on n, q, H and V. Then S=nH², i.e.M is a totally umbil-ical submanifold, and hence Mⁿ is isometric to a sphere M=Sⁿ(1/(1+H²)^1/2). In particular, the main result in [15] is just a corollary of Theorem C.We also get the following results for topological structure of submanifold ina sphere:Theorem D Let Mⁿ is a closed connected n-dimentional Riemannian mani-fold andφ: Mâ†'S^n+p(1) is an isometric immersion. There exists an explicitlygiven positive constantD(n,p)depending only on n,p,such that‖S-nH²â€–_q^n/2≥D(n,p)V^n-2q/2q sum from i=1 to n-1β_i.whereβ_i is thei-th Betti number of M, D(n, p)=pn^n/2(n-1)^(2-n)/2ω_n+p-1ω_p-1^-1, Vis the volume of M,ω_n is the volume of the unit ball in Rⁿ, q＞n/2.In particularly, M is homeomorphic to Sⁿ(1) if‖S-nH²â€–_q^n/2＜D(n,p)V^(n-2q)/2q.