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Compact isospectral sets of plane domains
B Osgood1, R Phillips, P Sarnak
1Department of Mathematics, Stanford University, Stanford, CA 94305.
Summary
Any isospectral family of two-dimensional Euclidean domains is compact in the C(infinity) topology. This study extends Melrose
Area of Science:
- Differential Geometry
- Spectral Geometry
- Topology
Background:
- Isospectral families of domains are crucial in spectral geometry.
- Previous work by Melrose established C(infinity) compactness for boundary curve curvatures using heat invariants.
Purpose of the Study:
- To demonstrate the C(infinity) compactness of any isospectral family of two-dimensional Euclidean domains.
- To extend existing results on spectral geometry compactness.
Main Methods:
- Utilizing the behavior of the determinant of the Laplacian near the boundary of the moduli space.
- Building upon Melrose's prior work on heat invariants and curvature compactness.
Main Results:
- Proved that any isospectral family of two-dimensional Euclidean domains is compact in the C(infinity) topology.
- Identified the determinant of the Laplacian's boundary behavior as a key factor in establishing domain compactness.
Conclusions:
- The C(infinity) compactness of isospectral domains is established.
- This work provides a deeper understanding of the geometric properties of isospectral domains.
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