Estimates for small eigenvalues of the Laplacian and conformal Laplacian on closed manifolds

Thesis

A central and well-established theme in geometry is eigenvalue estimates for geometric operators on manifolds. In this thesis we obtain new estimates for small eigenvalues of the Laplacian and conformal Laplacian respectively, in two distinct geometric contexts.

In the first part of this thesis we consider eigenvalues of the Laplacian on closed hyperbolic surfaces. It is known that for such surfaces degenerating by the collapse of a single simple closed geodesic, the first eigenvalue to highest order depends only on the length of that geodesic and the topology of the limiting surface. We consider the behaviour of the eigenvalues in the more general case of surfaces degenerating by the collapse of multiple simple closed geodesics. We exploit the fact that the geometry of a closed hyperbolic surface of fixed genus depends on a finite number of parameters (up to pullback by diffeomorphisms), the Fenchel-Nielsen coordinates. We prove new local energy estimates for the eigenfunctions of small eigenvalues and use these to obtain estimates for the derivatives of the eigenvalues with respect to the Fenchel-Nielsen coordinates. These are then used to prove that in the case of multiple collapsing geodesics, the small eigenvalues of the Laplacian again to first order depend only on the lengths of the collapsing disconnecting geodesics nd the topology.

In the second part of the thesis, we consider the behaviour of the first eigenvalue of the conformal Laplacian for particular families of metrics on closed manifolds of dimension at least three. It is known that the volume-normalised first positive eigenvalue of the conformal Laplacian is not bounded on any conformal class of metrics, contrasting with the behaviour of the Laplacian eigenvalues. We initiate an exploration of this difference in behaviour by proving sharp estimates for the first eigenvalue of the conformal Laplacian for particular families of conformal metrics, which we call asymptotically conical. We also obtain sharp estimates for the behaviour of the volume- and diameter-normalised eigenvalues for these metrics, giving new examples of conformal families of metrics for which the volume-normalised eigenvalues are unbounded.

Authors: Chaudhary, A

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: spectral geometry; hyperbolic geometry; Riemannian geometry; eigenvalue estimates; differential geometry; conformal Laplacian
• Subjects: Mathematics; Geometry