The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 924–950

# Heat Invariants of the Steklov Problem

• Iosif Polterovich
• David A. Sher
Article

## Abstract

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley calculus to the Dirichlet-to-Neumann operator, whose spectrum coincides with the Steklov eigenvalues. As an application, it is proved that a three-dimensional ball is uniquely defined by its Steklov spectrum among all Euclidean domains with smooth connected boundary.

## Keywords

Steklov problem Heat trace Dirichlet-to-Neumann operator Riemannian manifold Spectral rigidity

## Mathematics Subject Classification

58J50 58J35 58J40

## Notes

### Acknowledgements

The authors would like to thank P. Hislop and P. Perry for useful discussions, and the anonymous referee for helpful remarks. Research of I.P. was supported in part by NSERC, FQRNT, and the Canada Research Chairs Program. Research of D.S. was supported in part by the CRM-ISM postdoctoral fellowship.

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