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Boundary Regularity for the Ricci Equation, Geometric Convergence, and Gel'fand's Inverse Boundary Problem
 

Summary: Boundary Regularity for the Ricci Equation,
Geometric Convergence, and Gel'fand's Inverse Boundary Problem
Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev,
Matti Lassas, and Michael Taylor
Abstract
This paper explores and ties together three themes. The rst is to establish regular-
ity 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 to establish 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 bound-
ary injectivity radius, making use of the rst 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 rst two parts.
1. Introduction
The goals of this paper are to establish regularity, up to the boundary, of the met-
ric tensor of a Riemannian manifold with boundary, under Ricci curvature bounds
and control of the boundary's mean curvature; to apply this to results on Gromov
compactness and geometric convergence in the category of manifolds with bound-
ary; and then to apply these results to the study of an inverse boundary spectral

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics