Abstract.
Let \( x{:}\ M\rightarrow S^{n+p} \) be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S n + p, M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional \(\int_M(S-nH^2)^{\frac{n}{2}}d\nu\), where \( S=\sum_{\alpha,i,\,j}(h^\alpha_{ij})^2 \) is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S n + p. In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional \(\int_M(S-nH^2)d\nu\). Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.
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Guo, Z., Li, H. A variational problem for submanifolds in a sphere. Mh Math 152, 295–302 (2007). https://doi.org/10.1007/s00605-007-0476-2
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DOI: https://doi.org/10.1007/s00605-007-0476-2