A Global Curvature Pinching Result of the First Eigenvalue of the Laplacian on Riemannian Manifolds
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 : let be an -dimensional compact Riemannian manifold without boundary with , then the first eigenvalue of Laplacian on will satisfy that , and the inequality becomes equality if and only if .
The above result implies that the first eigenvalue of the Laplacian will have a lower bound less than if the Ricci curvature of manifolds involved has a lower bound except on a small part where the Ricci curvature satisfied that . Now a natural question arises: what is the lower bound of the first eigenvalue of Laplacian on such a manifold? In , Petersen and Sprouse gave a lower bound under the assumption that the bad part of the manifolds is small in the sense of -norm, where is a constant larger than half of the dimension of the manifold. In this paper, we are interested in the lower bound of the first eigenvalue under the global pinching of the Ricci curvature and we obtain a universal estimate of this lower bound on a certain class of manifolds.
2. A Sobolev Constant on the Geodesic Ball
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.
Definition 1. Let be a geodesic ball with radius ; we define the Sobolev constant on it to be the infimum among all the constant such that the inequality holds for all .
Definition 2. Let be a geodesic ball with radius ; we define the isoperimetric constant on it to be the supremum among all the constant such that the inequality holds for all with smooth boundary.
For any fixed point and radius , Croke proves that the equality holds , but one expects the constant to be independent on the location of the point , under some assumptions. In what follows, we will give an upper bound to independent of the point .
Let be an -dimensional Riemannian manifold, is the unit tangent bundle of , and is the canonical projective map. is the normalized geodesic from with the initial velocity . We define some notations as follows:
is the arc length from to the cut locus point along . Consider where is the standard surface measure of the unit sphere, is denoted to be the area of the unit sphere .
Definition 3. Using the Notation above, is called the visibility angle of .
If the manifold has which ensures that any minimal geodesic starting from any point in will reach the boundary before it reaches its cut locus, then the visibility angle of for any point which we denote by satisfies .
Lemma 4. Let be a closed Riemannian manifold with , then for any , the following Sobolev inequality holds on : , where and .
Proof. Croke proved the following inequality :
where , and is just the visibility angle of the domain .
As discussed above, we will have if ; then according to Croke’s inequality, we obtain . The relation between and tells us that , where is a constant only depending on the dimension .
Proposition 5. Let be a closed -dimensional Riemannian manifold with then for all , where is a constant only depending on the dimension .
Proof. Also take the inequality of Croke then the result can easily be derived from the fact that and after we integrate both sides of the inequality.
3. The First Eigenfunction and Eigenvalue
Let be a closed -dimensional Riemannian manifold; suppose that is the first eigenvalue of the Laplacian and is the first eigenfunction. In other words, they will satisfy that . By linearity, we can assume that and for the linearity. For the convenience, we call it the normalized eigenfunction. Next we will study some properties of the normalized eigenfunction and the eigenvalue.
Lemma 6. Let be a closed -dimensional Riemannian manifold with and . Then, a constant can be found such that .
Proof. One of the theorems of Yau and Schoen  shows that if , where is the diameter of the manifold and is a constant depending only on .
We will now introduce some notation. Let denote the lowest eigenvalue of the Ricci curvature tensor at . For a function on , we denote . Notice that a Riemannian manifold satisfies if and only if .
The well-known Myers theorem shows that a closed manifold with would have a bounded diameter . In other words, one can deduce that if one has . We will show next a result analogous to the one in  which we will use in our estimation of the eigenvalue. The proof follows identically; so it will be omitted (the reader can refer to the aforementioned article).
Lemma 7. Let be a closed -dimensional Riemannian manifold with , then for any , there exists such that if then the diameter will satisfy . In particular, there exists such that if then the diameter will satisfy . This fact, together with the volume comparison theorem, implies that , where is also a constant only dependent of .
Now, we can get a rough lower bound for the first eigenvalue.
Lemma 8. For , let as above and suppose that is a closed manifold with then there exists a constant such that .
Proof. The proof mainly belongs to Li and Yau . Let be the normalized eigenfunction of , set where . Then, we can easily get that
Denote that , and we then have by the Ricci identity on manifolds with : For the term , we have and for the term , we have Therefore, assume to be the maximum of ; then at , we have Therefore,
Denote to be the minimizing unit speed geodesic joining the maximum and minimum points of ; then integrating along , one will get:
Let ; then for any , we have .
Considering the maximum of the right hand and the upper bound of the diameter derived in Lemma 7, we can deduce that a positive constant can be found such that where is the diameter of the manifold.
Corollary 9. If the manifold one discussed satisfies all the conditions in Lemma 8 and its injectivity radius satisfies and if one let to be the normalized eigenfunction, then there exists a constant such that .
Proof. Set in the (13) from above. Then applying Lemma 6, one obtains
Proposition 10. Let be a closed -dimensional Riemannian manifold, the first eigenfunction of the Laplacian, and the corresponding eigenvalue, then holds in the sense of distribution. Moreover, if is compact with boundary, then the same conclusion holds for its Neumann boundary value problem.
Proof. From the definition, we know that holds on . Denote
According to the maximum principle of elliptic equation and the discussion about nodal set and nodal regions in , we can conclude that is a smooth manifold with dimension .
For all , integrating by parts we then have where and denote the outward normal direction with respect to the boundaries of and , respectively. Note that on and on for the definition of and . This completes the proof.
When has boundary, we can apply the same reasoning, except that the test function will require . This gives the proof.
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 -norm of first normalized eigenvalue of Laplacian from below by the geometric quantities. In this sense, we have the following result.
Theorem 11. Let be a closed -dimensional Riemannian manifold with and . If is the normalized eigenfunction of the Laplacian, then there exists a constant such that .
Proof. We use Moser iteration to get the result. From Proposition 10, we know that holds on in the sense of distribution. Set and take the point such that .
For , denote ; is a cut-off function on , then we have by integrating by parts: However, using the identity we have therefore, using the Sobolev inequality in Lemma 4,
Putting into the inequality above and we then have by splitting the integral into three parts and using the values of on each of them: where we denote only for emphasizing the integral domain.
And putting into (25), we can derive after iteration that
Let , then
The product can be estimated as follows:
The right hand will converge to a fixed number by using the fact that and the fact is finite for some . From , we can find a positive constant such that
4. The Lower Bound of the First Eigenvalue
Using the same notation as above, we can state the following result.
Theorem 12. For , , there is an such that any closed manifold with and will satisfy that .
Proof. Assume that is the normalized eigenfunction of Laplacian on , let , direct computation shows that
Integrating both sides on , we have therefore,
if we suppose that If is the one obtained in Lemma 7, then one has:
Finally, if one chooses then as long as , and this proves the theorem.
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.
S. T. Yau and R. Schoen, Lectures in Differential Geometry, Scientific Press, 1988.
P. Petersen and C. Sprouse, “Integral curvature bounds, distance estimates and applications,” Journal of Differential Geometry, vol. 50, no. 2, pp. 269–298, 1998.View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Yang, “Convergence of Riemannian manifolds with integral bounds on curvature. I,” Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, vol. 25, no. 1, pp. 77–105, 1992.View at: Google Scholar | Zentralblatt MATH | MathSciNet
I. Chavel, Riemannian Geometry: A Mordern Introduction, Cambridge University Press, Cambridge, UK, 2000.View at: Publisher Site | MathSciNet
C. Sprouse, “Integral curvature bounds and bounded diameter,” Communications in Analysis and Geometry, vol. 8, no. 3, pp. 531–543, 2000.View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Li and S. T. Yau, “Estimates of eigenvalues of a compact Riemannian manifold,” in AMS Proceedings of Symposia in Pure Mathematics, pp. 205–239, American Mathematical Society, Providence, RI, USA, 1980.View at: Google Scholar | Zentralblatt MATH | MathSciNet