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	r-Minimal submanifolds in space forms
	Cao, Linfen ; Li, Haizhong
	2010-05-06 ; 2010-05-06
关键词	rth mean curvature function (r+1)th mean curvature vector field L-r operator r-minimal submanifold stability CONSTANT SCALAR CURVATURE COMPACT SUBMANIFOLDS MEAN CURVATURE 1ST EIGENVALUE HYPERSURFACES STABILITY LAPLACIAN SPHERE Mathematics
中文摘要	Let x : M -> Rn+p(c) be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form Rn+p(c). Assume that r is even and r is an element of {0,1,..., n - 1}, in this paper we introduce rth mean curvature function Sr and ( r + 1)-th mean curvature vector field Sr+1. We call M to be an r-minimal submanifold if Sr+1 = 0 on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional J(r)(x) = integral(M) F-r(S-0, S-2,..., S-r) dv of x : M -> Rn+p(c), by calculation of the first variational formula of Jr we show that x is a critical point of Jr if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of Jr and prove that there exists no compact without boundary stable r-minimal submanifold with S-r > 0 in the unit sphere Sn+p. When r = 0, noting S-0 = 1, our result reduces to Simons' result: there exists no compact without boundary stable minimal submanifold in the unit sphere Sn+p.
语种	英语 ; 英语
出版者	SPRINGER ; DORDRECHT ; VAN GODEWIJCKSTRAAT 30, 3311 GZ DORDRECHT, NETHERLANDS
内容类型	期刊论文
源URL	[http://hdl.handle.net/123456789/13990]
专题	清华大学
推荐引用方式 GB/T 7714	Cao, Linfen,Li, Haizhong. r-Minimal submanifolds in space forms[J],2010, 2010.
APA	Cao, Linfen,&Li, Haizhong.(2010).r-Minimal submanifolds in space forms..
MLA	Cao, Linfen,et al."r-Minimal submanifolds in space forms".(2010).

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