r-Minimal submanifolds in space forms |
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Authors: | Linfen Cao Haizhong Li |
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Affiliation: | (1) Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, People’s Republic of China |
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Abstract: | Let 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 , in this paper we introduce rth mean curvature function S r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , 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 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 . |
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Keywords: | rth Mean curvature function (r + 1)th Mean curvature vector field L r operator r-Minimal submanifold Stability |
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