caa a22 4500 463248120 CHVBK 20180405153332.0 cr unu---uuuuu 170326e20071101xx s 000 0 eng 10.1007/s10455-007-9064-x doi (NATIONALLICENCE)springer-10.1007/s10455-007-9064-x r -Minimal submanifolds in space forms [Elektronische Daten] [Linfen Cao, Haizhong Li] Let $${x: M \to R^{n+p}(c)}$$ be an n-dimensional compact, possibly with boundary, submanifold in an (n+p)-dimensional space form R n+p (c). Assume that r is even and $${r\in \{0,1,\ldots,n-1\}}$$ , in this paper we introduce rth mean curvature function S r 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 J r we show that x is a critical point of J r 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 J r and prove that there exists no compact without boundary stable r-minimal submanifold with $${S_r > 0}$$ in the unit sphere S n+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 S n+p . Springer Science + Business Media B.V., 2007 r th Mean curvature function nationallicence ( r +1)th Mean curvature vector field nationallicence L r operator nationallicence r -Minimal submanifold nationallicence Stability nationallicence Cao Linfen Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People's Republic of China aut Li Haizhong Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People's Republic of China aut Annals of Global Analysis and Geometry Springer Netherlands 32/4(2007-11-01), 311-341 0232-704X 32:4<311 2007 32 10455 https://doi.org/10.1007/s10455-007-9064-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10455-007-9064-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Cao Linfen Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People's Republic of China aut NATIONALLICENCE 700 1- Li Haizhong Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People's Republic of China aut NATIONALLICENCE 773 0- Annals of Global Analysis and Geometry Springer Netherlands 32/4(2007-11-01), 311-341 0232-704X 32:4<311 2007 32 10455 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer