| Title:
|
The gap theorems for some extremal submanifolds in a unit sphere (English) |
| Author:
|
Wu, Xi Guo and Lan |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
23 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
85-93 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $M$ be an $n$-dimensional submanifold in the unit sphere $S^{n+p}$, we call $M$ a $k$-extremal submanifold if it is a critical point of the functional $\int _M\rho ^{2k}\,\mathrm{d}v $. In this paper, we can study gap phenomenon for these submanifolds. (English) |
| Keyword:
|
Extremal functional |
| Keyword:
|
Mean curvature |
| Keyword:
|
Totally umbilical |
| MSC:
|
53C24 |
| MSC:
|
53C40 |
| idZBL:
|
Zbl 1342.53077 |
| idMR:
|
MR3394079 |
| . |
| Date available:
|
2015-08-25T14:01:07Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144360 |
| . |
| Reference:
|
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| Reference:
|
[2] Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds.Comm. Pure. Appl. Math., 27, 1974, 715-727, Zbl 0295.53025, MR 0365424, 10.1002/cpa.3160270601 |
| Reference:
|
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| Reference:
|
[4] Li, A.-M., Li., J.-M.: An intrinsic rigidity theorem for minimal submanifolds in a sphere.Arch. Math., 58, 1992, 582-594, MR 1161925, 10.1007/BF01193528 |
| Reference:
|
[5] Li., H.: Willmore hypersurfaces in a sphere.Asian. J. Math., 5, 2001, 365-378, MR 1868938, 10.4310/AJM.2001.v5.n2.a4 |
| Reference:
|
[6] Li., H.: Willmore submanifolds in a sphere.Math. Res. Letters, 9, 2002, 771-790, MR 1906077, 10.4310/MRL.2002.v9.n6.a6 |
| Reference:
|
[7] Simons., J.: Minimal varieties in Riemannian manifolds.Ann. of Math., 88, 1968, 62-105, MR 0233295, 10.2307/1970556 |
| Reference:
|
[8] Xu, H.-W., Yang., D.: The gap phenomenon for extremal submanifolds in a Sphere.Differential Geom and its Applications, 29, 2011, 26-34, MR 2784286 |
| Reference:
|
[9] Wu., L.: A class of variational problems for submanifolds in a space form.Houston J. Math., 35, 2009, 435-450, MR 2519540 |
| . |
