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The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lower bound for the first eigenvalue of the Laplacian on manifolds with small negative curvature. The derivation involves Moser iteration.

The Laplacian is one of the most important operator on Riemannian manifolds, and the study of its first eigenvalue is also an interesting subject in the field of geometric analysis. In general, people would like to estimate the first eigenvalue of Laplacian in terms of geometric quantities of the manifolds such as curvature, volume, diameter, and injectivity radius. In this sense, the first interesting result is that of Lichnerowicz and Obata, which proved the following result in [

The above result implies that the first eigenvalue of the Laplacian will have a lower bound less than

The Sobolev inequality is one of the most important tools in geometric analysis, and the Sobolev constant plays an important part in the study of this field. In this section, we will obtain a general Sobolev constant only depending on the dimension of the manifold on the geodesic ball with small radius.

Let

Let

For any fixed point

Let

Using the Notation above,

If the manifold has

Let

Croke proved the following inequality [

As discussed above, we will have

Let

Also take the inequality of Croke

Let

Let

One of the theorems of Yau and Schoen [

We will now introduce some notation. Let

The well-known Myers theorem shows that a closed manifold with

Let

Now, we can get a rough lower bound for the first eigenvalue.

For

The proof mainly belongs to Li and Yau [

Denote that

Denote

Let

Considering the maximum of the right hand and the upper bound of the diameter derived in Lemma

If the manifold one discussed satisfies all the conditions in Lemma

Set

therefore,

Let

From the definition, we know that

For all

When

As long as the given manifold is compact, one knows that the first normalized eigenfunction is then determined. This indicates that the first normalized eigenfunction of the Laplacian has a close relation with the geometry of the manifold. In particular, one would hope to bound the

Let

We use Moser iteration to get the result. From Proposition

For

Let

Putting

Set

And putting

Let

The product can be estimated as follows:

The right hand will converge to a fixed number by using the fact that

Therefore,

Using the same notation as above, we can state the following result.

For

Assume that

Integrating both sides on

if we suppose that

Finally, if one chooses

The authors owe great thanks to the referees for their careful efforts to make the paper clearer. Research of the first author was supported by STPF of University (no. J11LA05), NSFC (no. ZR2012AM010), the Postdoctoral Fund (no. 201203030) of Shandong Province, and Postdoctoral Fund (no. 2012M521302) of China. Part of this work was done while the first author was staying at his postdoctoral mobile research station of QFNU.