Annals of Global Analysis and Geometry

, Volume 32, Issue 4, pp 311–341

r-Minimal submanifolds in space forms

Research Paper

DOI: 10.1007/s10455-007-9064-x

Cite this article as:
Cao, L. & Li, H. Ann Glob Anal Geom (2007) 32: 311. doi:10.1007/s10455-007-9064-x

Abstract

Let \({x: M \to R^{n+p}(c)}\) 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 \({r\in \{0,1,\ldots,n-1\}}\) , in this paper we introduce rth mean curvature function Sr and (r + 1)-th mean curvature vector field \({\vec{S}_{r+1}}\) . We call M to be an r-minimal submanifold if \({\vec{S}_{r+1}\equiv 0}\) on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional \({J_r(x)=\int_M F_r(S_0,S_2,\ldots,S_r)dv}\) of \({x: M \to R^{n+p}(c)}\) , 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 S0 = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere Sn+p.

Keywords

rth Mean curvature function (r + 1)th Mean curvature vector field Lr operator r-Minimal submanifold Stability 

Mathematics Subject Classification (2000)

Primary 53C40 Secondary 53C42 

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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