Variational Problems of Surfaces in a Sphere
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Variational Problems of Surfaces in a Sphere Bang Chao YIN School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, P. R. China E-mail : [email protected] Abstract Let x : M → Sn+p (1) be an n-dimensional submanifold immersed in an (n+p)-dimensional unit sphere Sn+p (1). In this paper, we study n-dimensional submanifolds immersed in Sn+p (1) which n are critical points of the functional S(x) = M S 2 dv, where S is the squared length of the second fundamental form of the immersion x. When x : M → S2+p (1) is a surface in S2+p (1), the functional n S(x) = M S 2 dv represents double volume of image of Gaussian map. For the critical surface of S(x), we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic. Furthermore, we establish a rigidity theorem for the critical surface of S(x). Keywords
Submanifold, variation, rigidity theorem, Euler characteristic
MR(2010) Subject Classification
1
53A10, 53C24, 53C42
Introduction
Let x : M → Sn+p (1) be an n-dimensional submanifold immersed in an (n + p)-dimensional unit sphere Sn+p (1). We denote by S and H2 the squared length of the second fundamental form and the squared length of the mean curvature vector field, respectively. For the minimal submanifolds in Sn+p (1), the following rigidity theorem is well-known. Theorem 1.1 ([5, 11, 18]) Let M be an n-dimensional (n ≥ 2) compact minimal submanifold in (n + p)-dimensional unit sphere Sn+p (1). Then n S − S dv ≤ 0. (1.1) 2 − p1 M In particular, if 0≤S≤
n , 2 − p1
then either S = 0 and M is totally geodesic, or S =
n 1 2− p
. In the latter case, either p = 1 and
M is a Clifford torus, or n = 2, p = 2 and M is the Veronese surface in S4 (1). The rigidity theorem of submanifolds have been widely studied, see [5, 6, 11–14, 16–18]. In particular, Li–Li [12] improved the above Simons’ type integral inequality for minimal submanifold in Sn+p (1): 2n − S dv ≤ 0, (1.2) S 3 M Received October 9, 2019, revised February 7, 2020, accepted June 5, 2020
Yin B. C.
2
where the equality holds if and only if either S = 0 and M is totally geodesic, or S = 2n 3 and M is the Veronese surface in S4 (1). The Willmore functional is defined by W(x) = M ρn dv which is invariant under the confor mal group of Sn+p (1), where ρ2 = S − nH2 . The functional W(x) = M ρn dv is well-studied by many authors. We refer to [1, 2, 7, 13, 14] and [20]. The Euler–Lagrange equation of the Willmore functional is n−2 α (ρn−2 hα H )+ ρn−2 hβik hβkj hα 0= ij ),ij − Δ(ρ ji i,j
−
i,j,k,β
ρn−2 H β hβij hα ij
− ρn H α ,
(1.3)
i,j,β
for every α, n+1 ≤ α ≤ n+p. A submanifold satisfying (1.3) is called a Willmore submanifold. In [19], Wang got the Euler–Lagrange equation of the Willmore functional for compact ndimensional submanifold without umbilical points in an (n+p)-dimensional unit sphere Sn+p (1) by M¨ obius invariants. For an n-dimensional Willm
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