Erratum To: “unique Continuation Results for Ricci Curvature and Applications”

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 τ 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 B (1) in Cartesian coordinates x, 1 ≤ i ≤ n and let D0 = C0 × [0, 1] be the vertical cylinder over C0 in coordinates x = (τ, x). Let ρ2 = ∑ (xi)2 − 1 be the Euclidean distance (squared) to ∂C0 and view the graph of the function τ = a−1ρ as a cone with boundary ∂C0. Let Da be the interior solid cone, where τ ≥ a−1ρ. We will assume a >> 1, so that the cone is almost flat. Next consider mappings φ from (a neighborhood of) Da, with range into the space with coordinates (t, y), which are diffeomorphisms onto their image, which are close to the identity, and for which φ = id on C0. The horizontal level surfaces Στ = τ ×B n of τ in Da are mapped under φ to a foliation Sτ of the image domain. Given a metric g, close to the standard flat metric in the (t, y ) coordinates, the pullback φg is defined on Da. We will only consider metrics which equal a given fixed metric g0 on C0, for which the coordinates y i are harmonic on (C0, g0). We seek foliations such that x are harmonic on each leaf Στ with respect to φ g, i.e. ∆ ∗g Στ x = 0, and such that H ∗g Στ (τ, x) = ψ(x), where ψ is a given function, close to 0. In particular, this implies ∂τH φ∗g Στ = 0. The unit normal vector of the foliation Στ in the metric φ g is given by Nφ∗g = u (∂τ−σ) where (u, σ) are the lapse and shift of the Στ foliation. As in [1], one has (φ ∗g)00 = u−2, (φ∗g)0i = −u−2σi. Prescribing the lapse and shift of the τ -foliation Στ in the (τ, x ) coordinates at the boundary ∂Da The first author is partially supported by NSF Grant DMS 0604735/0905159; the second author is partially supported by ANR project GeomEinstein 06-BLAN-0154. MSC Classification: 58J32, 53C21, 35J60.