## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 32, Number 3 (2009), 432-441.

*L*^{2} harmonic 1-forms on complete submanifolds in Euclidean space

#### Abstract

Let *M*^{n} (*n* ≥ 3) be an *n*-dimensional complete noncompact oriented submanifold in an (*n*+*p*)-dimensional Euclidean space **R**^{n+p} with finite total mean curvature, i.e, ∫_{M}|*H*|^{n} < ∞, where *H* is the mean curvature vector of *M*. Then we prove that each end of *M* must be non-parabolic. Denote by φ the traceless second fundamental form of *M*. We also prove that if ∫_{M}|φ|^{n} < *C* (*n*), where *C* (*n*) is an an explicit positive constant, then there are no nontrivial *L*^{2} harmonic 1-forms on *M* and the first de Rham's cohomology group with compact support of *M* is trivial. As corollaries, such a submanifold has only one end. This implies that such a minimal submanifold is plane.

#### Article information

**Source**

Kodai Math. J., Volume 32, Number 3 (2009), 432-441.

**Dates**

First available in Project Euclid: 11 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1257948888

**Digital Object Identifier**

doi:10.2996/kmj/1257948888

**Mathematical Reviews number (MathSciNet)**

MR2582010

#### Citation

Fu, Hai-Ping; Li, Zhen-Qi. L 2 harmonic 1-forms on complete submanifolds in Euclidean space. Kodai Math. J. 32 (2009), no. 3, 432--441. doi:10.2996/kmj/1257948888. https://projecteuclid.org/euclid.kmj/1257948888