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ERRATUM TO: "UNIQUE CONTINUATION RESULTS FOR RICCI CURVATURE AND APPLICATIONS"
 

Summary: ERRATUM TO: "UNIQUE CONTINUATION RESULTS FOR RICCI
CURVATURE AND APPLICATIONS"
MICHAEL T. ANDERSON AND MARC HERZLICH
Abstract. Corrections are given to some of the proofs of the paper above
In this note, we point out and correct some errors in the proofs of the main results in the paper
[1]. The main results themselves are correct as stated, but the proofs need modification.
To begin, the proof of Lemma 3.3 in [1] is incorrect. Thus, h in [1, (3.12)] is the linearization of a
mapping C C, and so has n degrees of freedom, not (n + 1) as indicated there. Moreover, the
information given on solving the PDE in (3.12) is insufficient. In addition, the proof of Lemma 3.2
also requires the Nash-Moser implicit function theorem; the linearization L of H is not surjective
since one has no gain of regularity in the direction.
In Lemma 1 below, we state a slightly more general version of Lemmas 3.2 and 3.3 of [1] and then
proceed with the proof. We recall that the main point of these results is to construct a foliation
of prescribed mean curvature with harmonic coordinates along the leaves, such that the lapse and
shift are prescribed at the boundary. The rest of the work in [1, 3] then proceeds as before.
We begin by describing the initial set-up of the issue. Let C0 be the unit ball Bn(1) in Cartesian
coordinates xi, 1 i n and let D0 = C0 [0, 1] be the vertical cylinder over C0 in coordinates
x = (, xi). Let 2 = (xi)2 - 1 be the Euclidean distance (squared) to C0 and view the graph
of the function = a-1 as a cone with boundary C0. Let Da be the interior solid cone, where
a-1. We will assume a >> 1, so that the cone is almost flat.

  

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

 

Collections: Mathematics