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Metric tensor estimates, geometric convergence, and inverse boundary problems
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## Metric tensor estimates, geometric convergence, and inverse boundary problems

### Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas, and Michael Taylor

**Abstract.**
Three themes are treated in the results announced here. The first is the regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is the geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts.

*Copyright 2003 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**09** (2003), pp. 69-79
- Publisher Identifier: S 1079-6762(03)00113-6
- 2000
*Mathematics Subject Classification*. Primary 35J25, 47A52, 53C21
*Key words and phrases*. Ricci tensor, harmonic coordinates, geometric convergence, inverse problems, conditional stability
- Received by editors December 17, 2002
- Posted on September 2, 2003
- Communicated by Tobias Colding
- Comments (When Available)

**Michael Anderson**

Mathematics Department, State University of New York, Stony Brook, NY 11794

*E-mail address:* `anderson@math.sunysb.edu`

**Atsushi Katsuda**

Mathematics Department, Okayama University, Tsushima-naka, Okayama, 700-8530, Japan

*E-mail address:* `katsuda@math.okayama-u.ac.jp`

**Yaroslav Kurylev**

Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

*E-mail address:* `Y.V.Kurylev@lboro.ac.uk`

**Matti Lassas**

Rolf Nevanlinna Institute, University of Helsinki, FIN-00014, Finland

*E-mail address:* `lassas@cc.helsinki.fi`

**Michael Taylor**

Mathematics Deptartment, University of North Carolina, Chapel Hill, NC 27599

*E-mail address:* `met@math.unc.edu`

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