The Scientific World Journal

Volume 2014 (2014), Article ID 697132, 8 pages

http://dx.doi.org/10.1155/2014/697132

Research Article

Gap Phenomenon of an Abstract Willmore Type Functional of Hypersurface in Unit Sphere

Yanqi Zhu, Jin Liu, and Guohua Wu

College of Information System and Management, National University of Defense Technology, Changsha, Hunan 410073, China

Received 18 April 2014; Accepted 26 May 2014; Published 5 June 2014

Academic Editor: Luc Vrancken

Copyright © 2014 Yanqi Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called -Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the -Willmore hypersurface, for which the variational equation and Simons’ type integral equalities are obtained. Moreover, we construct a few examples of -Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.

1. Introduction

Let be an -dimensional compact without boundary hypersurface in unit sphere. Choose an orthonormal frames field along such that are tangent to and is normal to . Their dual frames are and , respectively; obviously, when it is restricted over . Let denote the second fundamental form of the immersion . Write

and define

One calls , , to be mean curvature, square of the length, and th power polynomial of second fundamental form, respectively. It is well known that hypersurface is minimal when and is totally geodesic when . For invariant , it is obvious that for all , if and only if is an umbilical point of , and for all and some positive constant since is compact.

In differential geometry, there is a famous classical Willmore functional of hypersurface which is defined as

See [1–4] for the details. It is well known that this functional is invariant under conformal transformation. Its critical point is called classical Willmore or -Willmore hypersurface. The famous Willmore conjecture says that holds for all immersed tori ; recently, it has been completely solved by Marques and Neves in article [5] using Min-Max theory for two dimensional case, but higher dimensional case is still an open problem. When , the functional was studied extensively in [6–11].

Due to the importance of Willmore conjecture, many geometric experts generalized the classic Willmore functional to a wide range and some interesting results have been obtained.

In [12], Cai studied the -Willmore functional for surfaces in -dimensional unit sphere :

where and denote the principal curvatures of and . Under some proper conditions, Cai obtained some interesting inequalities.

In [13], Guo and Li considered the -Willmore functional for submanifolds in unit sphere :

They calculated the first variation and obtained Simons’ type inequalities and finally classified the point-wise gap phenomenon. In [14], Xu and Yang derived the global but not point-wise gap phenomenon for the same functional.

In [15], Wu researched the -Willmore functional for submanifolds in unit sphere :

The first variation formula, Simons’ type inequalities, and point-wise gap phenomenon are obtained.

Motivated by the work mentioned above, in this paper, we study an abstract Willmore type called -Willmore functional:

where we always assume that the abstract function satisfies

Obviously, functionals ,,, and are special cases of , so the construction and investigation of are meaningful. As usual, the critical point of functional is called -Willmore hypersurface. In Section 3, we calculate the st variation formula of functional .

Theorem 1. Let be an -dimensional hypersurface in an -dimensional unit sphere ; then is a -Willmore hypersurface if and only if

In particular, if is an isoparametric (all principal curvatures are constant) hypersurface, then is a -Willmore hypersurface if and only if

Remark 2. When , Theorem 1 coincides with corresponding results in [6, 13, 15], respectively.

It is well known that Simons’ integral inequality plays an important role in the study of minimal hypersurface. It says that if is an -dimensional compact minimal hypersurface in -dimensional unit sphere , then

For classic Willmore hypersurface, Li [6] proved a Simons’ type integral result which says if is an -dimensional compact classical Willmore hypersurface in unit sphere , then

Similarly, for functionals and , [13, 15] obtained integral inequalities, respectively. A natural question is that whether a Simons’ type inequality or equality can be established for -Willmore functional. We give a confirm answer in this study.

Theorem 3. Let be an -dimensional compact -Willmore hypersurface in sphere ; then we have Simons’ type equality and can give a discussion according to the sign of , , and :

When , there holds

When , there holds

When , , there holds

When , , , there holds

When , , , there holds

Remark 4. When , Theorem 3 coincides with corresponding results in [6, 13, 15], respectively. Similar equalities also hold for the other sign cases of , , and .

Using the integral equalities in Theorem 3, together with two famous Chern et al. results in [16], we can obtain some conclusions of gap phenomenon.

Theorem 5. Let be an -dimensional closed -Willmore hypersurface in unit sphere , we have the following.

(1)When , over , if , then or .

(a)For , then , , , and is totally geodesic;(b)For , then , , , and for some with .

(2)When ,, over , if , then or .

(a)For , then is totally umbilical;(b)For , then , , , , and .

(3)When , , over , if , then , , , or .

(a)For , is totally umbilical;(b)For , for some with .

(4)When , , over , if , , then , or , .

(a)For , , then is totally geodesic;(b)For , , then , , , , .

Remark 6. When , Theorem 5 coincides with corresponding results in [6, 13, 15], respectively. Similar conclusions also hold for the other sign cases of , , and . The examples appeared in Theorem 5 would be constructed in Section 3.

2. Preliminaries

Let be an -dimensional closed hypersurface in an -dimensional unit sphere and let be a variation of ; it means that

is still an isometric immersion with .

Let and be the orthonormal local frames of and , respectively; then is the orthonormal local frames of the pullback vector bundle over , such that are tangent to and is normal to due to

Use to denote the connection form over ; by the pullback operation, we have the following decomposition:

Thus, one can derive that are the orthonormal frames of and when it is restricted over , are the variation vector fields of , is the connection form of , and is the second fundamental form. In particular, we have

Use and to denote curvature forms of and , respectively and write their components:

From any one standard differential geometry book, we know that the curvature of unit sphere is

Using connection form , the covariant derivatives of , , , , , and can be defined as

We use , and to denote the differential operators on , , and , respectively. By calculating directly and comparing both sides of the following equations:

we would derive out the lemmas below; also see article [10]. In fact, the right hand side of (27) is

while the left hand side of (27) is

comparing (29) with (30), we obtain

The equation of (28) is

Comparing both sides, we obtain

From (31) and (33), we can easily conclude the following lemmas which are useful in the variation calculation and establishment of Simons’ type integral equalities.

Lemma 7. Let be a hypersurface; one has structure equations:

Lemma 8. Let be a hypersurface; one has Ricci identity:

Lemma 9. Let be a hypersurface and let be a variation vector field; one has

Lemma 10. With the same notations as above, one has

3. Variation Calculation and Examples

To calculate the first variation of -Willmore functional, we need two lemmas.

Lemma 11. Let be a hypersurface and let be a variation vector field. Suppose that denotes the volume element; one has

Proof. By Lemma 9, we have . Hence,

With this, we complete the proof of Lemma 11.

Lemma 12. Let be a hypersurface and let be a variation vector field; one has

Proof. By the definition of , and , one has

It follows Lemmas 9 and 10 that

By Lemmas 9 and 10 again, one has

Hence,

With this, we complete the proof of Lemma 12.

Proof of Theorem 1. By Lemmas 11 and 12, we can calculate

With this, we complete the proof of Theorem 1.

From Theorem 1, we can find a few examples in unit sphere , in particular, the isoparametric hypersurfaces. Since all principal curvatures are constant, then , , are constant and , , . Thus, -Willmore hypersurface equation becomes

Example 13. Obviously, totally geodesic hypersurfaces are -Willmore for any abstract function , totally umbilical but not totally geodesic hypersurfaces are -Willmore if and only if .

Example 14. For a particular hypersurface with ,

All principal curvatures are

We can derive the quantities ,,,, respectively, ,,,. Obviously always is a -Willmore hypersurface of for any function .

Example 15. For a family hypersurfaces with parameters ,,, , .

Obviously, all principal curvatures are

Then the quantities , , , are, respectively,

Setting , -Willmore hypersurface equation becomes

Example 16 (Clifford torus [16]; ). Consider that

are minimal hypersurfaces with ,,.

Example 17 (classic Willmore torus [6]; ). Consider that

are -Willmore hypersurfaces with . When some is minimal, then , .

4. Simons’ Type Equalities and Gap Phenomenon

Some lemmas are needed for the establishment of Simons’ type integral equalities and the discussion of gap phenomenon.

Lemma 18. For , one has

where .

Proof. By the definition of and Laplacian operator, we have

for term ; by Lemmas 7 and 8, easily we obtain

for term ; by Lemma 8, we have

Substituting (58) into (57), we have

Substituting (59) into (56), we have

With this, we complete the proof of Lemma 18.

Lemma 19 (Huisken’s estimate [17]). Consider that

and if and only if .

Proof. We decompose the tensor as follows:

where

Through a computation, we obtain

Finally, by the triangle inequality, we have

If , then all inequalities become equalities, , , and ; thus, . With this, we complete the proof of Lemma 19.

Lemma 20. For , one has

Proof. We know that

By Lemmas 18 and 19, we can easily obtain the identity.

Proof of Theorem 3. Integrating the equality in Lemma 20 over , we have

By Theorem 1, we know that

Using Stokes theorem, we obtain

We complete the proof of Theorem 3.

In order to prove Theorem 5, we need two very important conclusions which are treated as a lemma and a main Theorem in Chern et al.’s article [16]. For a hypersurface, we choose frame fields in such a way that for all,.

Lemma 21 (Lemma 3 of [16]). Let be a compact hypersurface with ; then there are two cases.

(1), and is a totally umbilic () or totally geodesic ();(2), , and is a Riemannian product of , where ,.

Lemma 22 (main Theorem of [16]). Clifford torus ≤ ≤ are the only compact minimal () hypersurfaces of dimension in unit sphere satisfying .

Proof of Theorem 5. Obviously, case of Theorem 5 is a corollary of Lemma 22; thus, we just focus cases and ; the rest can be proved by similar argument.

When ,, over , by the case of Theorem 3, we have

If , the right hand side of identity (71) is nonpositive, while the left hand side of (71) is nonnegative; thus, we can conclude that or . For , is totally umbilical; for , substituting it into the above equality together with Lemma 19, we can derive , , , and , by Lemma 22, we know that ; moreover, by -Willmore equation, we conclude that , , , , and .

When , , , we know that , . By the case of Theorem 3, we have

Moreover, if , then or . For , is totally umbilical; for , substituting it into the above equality together with Lemma 19, we obtain ; by Lemma 22, we know that is a torus; then, by Example 17, we can conclude that is a classic Willmore torus for some .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The second author would like to express thanks to Professor H. Z. Li for his encouragement and help.

References

B. Y. Chen, “Some conformal invariants of submanifolds and their applications,” Bollettino della Unione Matematica Italiana, vol. 10, pp. 380–385, 1974.

F. J. Pedit and T. J. Willmore, “Conformal geometry,” Atti del Seminario Matematico e Fisico dell'Università di Modena, vol. 36, pp. 237–245, 1988.

F. J. Willmore, Total Curvature in Riemannian Geometry, Ellis Horwood, 1982.

T. J. Willmore, Riemannian Geometry, Oxford Science Publications, Clarendon Press, Oxford, UK, 1993.

F. C. Marques and A. Neves, “Min-max theory and the Willmore conjecture,” Annals of Mathematics, vol. 179, no. 2, pp. 683–782, 2014.

H. Z. Li, “Willmore hypersurfaces in a sphere,” Asian Journal of Mathematics, vol. 5, pp. 365–378, 2001.

H. Z. Li, “Willmore surfaces in Sn,” Annals of Global Analysis and Geometry, vol. 21, no. 2, pp. 203–213, 2002.

H. Z. Li, “Willmore submanifolds in a sphere,” Mathematical Research Letters, vol. 9, no. 5-6, pp. 771–790, 2002.

Z. Guo, “Willmore submanifolds in the unit sphere,” Collectanea Mathematica, vol. 55, no. 3, pp. 279–287, 2004.

Z. J. Hu and H. Z. Li, “Willmore submanifolds in Riemannian manifolds,” in Proceedings of the Workshop on Contemporary Geometry and Related Topics, pp. 251–275, World Scientific, Hackensack, NJ, USA, 2005.

Z. Guo, H. Z. Li, and C. P. Wang, “The second variation of formula for Willmore submanifolds in Sn,” Results in Mathematics, vol. 40, pp. 205–225, 2001.

M. Cai, “ willmore functionals,” Proceedings of the American Mathematical Society, vol. 127, no. 2, pp. 569–575, 1999.

Z. Guo and H. Z. Li, “A variational problem for submanifolds in a sphere,” Monatshefte für Mathematik, vol. 152, no. 4, pp. 295–302, 2007.

H. W. Xu and D. Y. Yang, “The gap phenomenon for extremal submanifolds in a sphere,” Differential Geometry and its Application, vol. 29, no. 1, pp. 26–34, 2011.

L. Wu, “A class of variational problems for submanifolds in a space form,” Houston Journal of Mathematics, vol. 35, no. 2, pp. 435–450, 2009.

S. S. Chern, M. do Carmo, and S. Kobayashi, “Minimal submanifolds of a sphere with second fundamental form of constant length,” in Functional Analysis and Related Fields, pp. 59–75, Springer, Berlin, Germany, 1970.

G. Huisken, “Flow by mean curvature of convex surfaces in to shperes,” Journal of Differential Geometry, vol. 20, pp. 237–266, 1984.