R-Minimal submanifolds in space forms

Abstract

Let x: M → R n+p (c) be an n-dimensional compact, possibly with bound- ary, submanifold in an (n + p)-dimensional space form R n+p (c). Assume that r is even and r ∈{ 0,1, ... ,n − 1}, in this paper we introduce rth mean curvature function Sr and (r + 1)-th mean curvature vector field Sr+1 .W e callM to be an r-minimal submanifold if Sr+1 ≡ 0o nM, we note that the concept of 0-minimal submani- fold is the concept of minimal submanifold. In this paper, we define a functional Jr(x) = � M Fr(S0,S2, ... ,Sr)dv of x: M → R n+p (c), by calculation of the first varia- tional 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 cal- culate the second variational formula of Jr and prove that there exists no compact without boundary stable r-minimal submanifold with Sr > 0 in the unit sphere S n+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 S n+p .