The Spectrum and Isometric Embeddings of Surfaces of Revolution

The Spectrum and Isometric Embeddings of Surfaces of Revolution For Gus and Sonia

A sharp upper bound on the first S invariant eigenvalue of the Laplacian for S invariant metrics on S is used to find obstructions to the existence of S equivariant isometric embeddings of such metrics in (R, can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in (R, can). This leads to ge...

A Note on Isometric Embeddings of Surfaces of Revolution

Covers and the Curve Complex

We propose a program of studying the coarse geometry of combinatorial moduli spaces of surfaces by classifying the quasi-isometric embeddings between them. We provide the first non-trivial examples of quasi-isometric embeddings between curve complexes. These are induced either via orbifold coverings or by puncturing a closed surface. As a corollary, we give new quasiisometric embeddings between...

The inverse spectral problem for surfaces of revolution

We prove that isospectral simple analytic surfaces of revolution are isometric. 0 Introduction This article is concerned with the inverse spectral problem for metrics of revolution on S. We will assume that our metrics are real analytic and belong to a class R∗ of rotationally invariant metrics which are of ‘simple type’ and which satisfy some generic non-degeneracy conditions (see Definition (...

Local Rigidity of Surfaces in Space Forms

We prove that isometric embeddings of closed, embedded surfaces in R are locally rigid, i.e. they admit no non-trivial local isometric deformations, answering a question in classical differential geometry. The same result holds for abstract surfaces embedded in constant curvature 3-manifolds, provided a mild condition on the fundamental group holds.