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Set E = ∂ek+1 ∩ (∂e1 ∪ · · · ∪ ∂ek ). If E = ∅, we easily connect u0,k to some u0,k+1 which is Lip near ∂e1 ∪ · · · ∪ ∂ek+1 . If E = ∅, recall that u0,k is Lip in the −neighborhood of E for some > 0. The value of will be taken small enough so that the following arguments can go through. Let B be the ball of radius in Rn−[p] centered at the origin, and let ϕ : B × S[p] → M be a diffeomorphism such that for any (x, y) ∈ ∂B × S[p] , {ϕ(sx, y) ; 0 < s < 1} is a geodesic parameterized by arclength s; moreover, ∂ek+1 ⊂ ϕ(B 4 × S[p] ), ϕ({0} × S[p] ) ⊂ ek+1 .

We point out one modification, since the others are more obvious. 2. Let ρ ∈ C ∞ (A), 0 ≤ ρ ≤ 1, ρ(a) = 1 for a ∈ A2 , ρ(a) = 0 for a ∈ A \ A . 2. 1, set, for 0 < t ≤ 1, f, tρ(a) )2x, a), ut (x, a) := u((1 − |x| u(x, a), |x| ≤ tρ(a), a ∈ A, tρ(a) < |x| < 2tρ(a), a ∈ A, 2tρ(a) ≤ |x| ≤ 4, a ∈ A, and u0 = u. Then ut ∈ C([0, 1], W 1,p(B4 × A)). 1 and its proof). We leave the details to the reader. 5. Let n = dim M. 2. 2. So we only need to consider the case n≥3 and 2 ≤ p < n. As always, we discuss only the case where ∂M = ∅.

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