## Tsukuba Journal of Mathematics

### A gap theorem for complete four-dimensional manifolds with $\deltaW^{+} = 0$

Takashi Okayasu

#### Abstract

Let $M^{4}$ be a complete noncompact oriented fourdimensional Riemannian manifold satisfying $\delta W^{+}=0$, where $W^{+}$ is the self-dual part of the Weyl curvature tensor. Suppose its scalar curvature is nonnegative and Sobolev's inequality holds. We show that if the $L^{2}$ norm of $W^{+}$ is sufficiently small, then $W^{+}\equiv 0$.

#### Article information

Source
Tsukuba J. Math., Volume 29, Number 2 (2005), 539-542.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496164970

Digital Object Identifier
doi:10.21099/tkbjm/1496164970

Mathematical Reviews number (MathSciNet)
MR2177026

Zentralblatt MATH identifier
1094.53028

#### Citation

Okayasu, Takashi. A gap theorem for complete four-dimensional manifolds with $\deltaW^{+} = 0$. Tsukuba J. Math. 29 (2005), no. 2, 539--542. doi:10.21099/tkbjm/1496164970. https://projecteuclid.org/euclid.tkbjm/1496164970