Abstract
Let M 2 be a compact Willmore surface in the (2 + p)-dimensional unit sphere S 2 + p. Denote by H and S the mean curvature and the squared length of the second fundamental form of M 2, respectively. Set ρ 2 = S−2H 2. In this note, we proved that there exists a universal positive constant C, such that if ∥ρ 2∥2<C, then ρ 2=0 and M 2 is a totally umbilical sphere.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let M be a compact surface in the (2 + p)-dimensional unit sphere S 2 + p . Choose a local orthonormal frame field {e 1,e 2,…,e 2 + p } in S 2 + p such that, restricted to M, the {e 1,e 2} are tangent to M. The following convention of indices are used throughout.
Denote by H and S the mean curvature and the squared length of the second fundamental form of M, respectively. Then, we have
where \(h_{ij}^{\alpha }\) is the component of the second fundamental tensor of M.
Let ρ 2 = S−2H 2. In fact, if we set \(\tilde {h}^{\alpha }_{ij}=h^{\alpha }_{ij}-\delta _{ij}H^{\alpha }\), by a direct computation, one has
So, ρ 2≥0, and ρ vanishes at the umbilical points of M.
The Willmore functional is defined by
Here the integration is with respect to the area measure of M. In [3], Chen proved that this functional is invariant under conformal transformations of S 2 + p.
DEFINITION
x:M→S 2 + p is called a Willmore surface if it is a critical surface of the Willmore functional W(x).
It was proved by Bryant [1] and Weiner [7] that M is a Willmore surface if and only if
i.e.,
where \({\Delta }^{\perp } H^{\alpha }=\sum \limits _{k}H^{\alpha }_{kk}\).
From (1.1), we know that all minimal surfaces in S 2 + p are Willmore surfaces. So, the Veronese surface must be the Willmore surface. Moreover, Pinkall [4] constructed many compact non-minimal flat Willmore surfaces in S 3, and Castro and Urbano [2] constructed many compact non-minimal Willmore surfaces in S 4.
In [6], Li obtained the following rigidity theorem for Willmore surfaces in a unit sphere.
Theorem A.
Let M be a compact Willmore surface in S 2+p . Then
where
In particular, if
then either ρ 2 = 0 and M is totally umbilical, or ρ 2 =B. In the latter case, p=2 and M is the Veronese surface or p=1 and \(M=S^{1}\left (\frac {1}{\sqrt {2}}\right )\times S^{1}\left (\frac {1}{\sqrt {2}}\right )\).
Applying Theorem A and the Sobolev inequality, we proved the following result (see [9]).
Theorem B.
Let M be a compact Willmore surface in S 2+p . There exists a positive constant \(\tilde {C}(H_{0})\) , defined by
such that if
then M is a totally umbilical surface, where H 0 = maxx∈MH and B is defined in Theorem A.
We shall improve the constant of Theorem B and obtain the following global pinching theorem for compact Willmore surfaces in S 2 + p.Main theorem. Let M be a compact Willmore surface in the unit sphere S 2 + p. There exists an explicit positive constant
such that if
then ρ 2=0and M is a totally umbilical sphere, where B is defined in Theorem A.
Remark 1.
By a simple calculation, we know that the pinching constant in Theorem B \(\tilde {C}(H_{0})\rightarrow 0\) as H→∞. But the pinching constant C in our main theorem is independent of mean curvature H. So C is superior to \(\tilde {C}(H_{0})\).
2 Basic lemmas
In this section, we introduce several useful lemmas
Lemma 2.1.
Let x:M 2 →S 2+p be a surface in a unit sphere. We have the following inequality:
Proof.
We can see from \(\rho ^{2}={\sum }_{\alpha ,i,j}(\tilde {h}_{ij}^{\alpha })^{2}\) and the Cauchy–Schwarz inequality that
at all points where ρ ≠ 0 and hence by analyticity at all the points. □
Lemma 2.2.
[10]. Let x:M→S 2+p be a surface. Then
where equality holds if and only if p=1 or p≥2, \({\sum }_{\alpha }(\tilde {h}_{11}^{\alpha })^{2}={\sum }_{\alpha }(\tilde {h}_{12}^{\alpha })^{2}\) and \({\sum }_{\alpha }\tilde {h}_{11}^{\alpha }\tilde {h}_{12}^{\alpha }=0\) . HereBis defined in Theorem A.
Lemma 2.3.
Let x:M→S 2+p be a surface.
where \(|\nabla ^{\perp }\mathbf {H}|^{2}={\sum }_{\alpha ,i}(H^{\alpha }_{i})^{2}\).
Proof.
By a simple calculation, we have
In [6], Li proved
Substituting (2.6 ) into ( 2.5 ), we obtain ( 2.4). □
Lemma 2.4.
[10]. Let x:M 2 →S 2+p be a compact Willmore surface in a unit sphere. Then
Lemma 2.5.
Let x:M→S 2+p be a compact surface.
Proof.
By Stoke formula, we have
We obtain (2.8 ) by putting ( 2.10 ) into ( 2.9). □
Lemma 2.6.
Let M be a compact 2-dimensional surface in S 2+p . Then for any g∈C 1 (M), g≥0, t>0, g satisfies
where \(A=\frac {12\sqrt {3}}{\sqrt {\pi }}\).
Proof.
Replacing g by g 2, we get
where t∈R +. So, we have
i.e.,
This proves Lemma 2.6. □
3 Proof of the main theorem
In this section, we give the proof of our main theorem. From Lemma 2.1 in [10] and (2.3), we have
Integrating the above inequality and using Lemma 2.5, we get
where 0<η<1. From (2.1 ), ( 2.7 ) and ( 3.1), we have
Substituting (2.11 ) into ( 3.2), we get
As ε→0, this implies
Choose \(t=\displaystyle \frac {2(1-\eta )}{\eta }\), then \(1-\eta -\displaystyle \frac {\eta t}{2}=0\). So we have
i.e.,
We take \(\eta =2-\sqrt {2}\). This together with (3.3) yields
which implies ∥ρ 2∥2=0 for
i.e., S=2H 2 and M is a totally umbilical Willmore surface. This completes the proof of the main theorem.
As we all know, minimal surfaces must be Willmore surfaces, so we obtain the following corollary.
COROLLARY
Let M be a compact minimal surface in the unit sphere S 2 + p. There exists a positive constant
such that if
then S = 0 and M is a totally geodesic, where B is defined in Theorem A.
References
Bryant R, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984) 23–53
Castro I and Urbano F, Willmore surfaces of R 4 and the Whitney sphere, Ann. Global Anal. Geom. 19 (2001) 153–175
Chen B Y, Some conformal invariants of submanifolds and their application, Boll. Un. Mat. Ital. 10 (1974) 380–385
Chen B Y, Geometry of submanifolds (1973) (New York: Marcel Dekker)
Hoffman D and Spruck J, Sobolev and isoperimetric inequalities for Riemannian submanifold, Comm. Pure Appl. Math. 27 (1974) 715–727
Li H Z, Willmore surfaces in S n, Ann. Global Anal. Geom. 21 (2002) 203–213
Weiner J, On a problem of Chen, Willmore et al., Indiana Univ. Math. J. 27 (1978) 19–35
Xu H W, \(L^{\frac {n}{2}}\)-pinching theorems for submanifolds with parallel mean curvature in a sphere, J. Math. Soc. Japan 46 (1994) 503–515
Xu H W and Yang D Y, A global pinching theorem for Willmore submanifolds in a sphere, Preprint (2008)
Yu C C and Hsu Y J, Willmore surfaces in the unit n-sphere, Taiwanese J. Math. 8 (2004) 467–476
Acknowledgements
The authors would like to thank Dr. F L Yin for several helpful discussions. They are also grateful to the referee for his/her valuable comments on an earlier version of this paper. This research was supported by the NSFC, Grant No. 11261038.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicating Editor: Parameswaran Sankaran
Rights and permissions
About this article
Cite this article
XU, H., YANG, D. Rigidity theorem for Willmore surfaces in a sphere. Proc Math Sci 126, 253–260 (2016). https://doi.org/10.1007/s12044-016-0270-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12044-016-0270-y