Proposition

Lemma 6 ([Rotation approximation and coefficient continuity]). Every q∈ T_ τ,α is a limit, in \|·\|_β , of finite angular type tensors q_ν∈ T_ τ,α . The approximants may be chosen as averages of rotated copies of q . Their weighted norms …

Lemma 6 ([Rotation approximation and coefficient continuity]). Every q∈ Tτ,α is a limit, in \|·\|β , of finite angular type tensors qν∈ Tτ,α . The approximants may be chosen as averages of rotated copies of q . Their weighted norms are uniformly bounded, and the construction preserves a nonnegative boundary value qss |S when that condition holds. The maps q↦u1[q] and q↦u2[q] defined for finite angular type seeds extend uniquely by these approximations to continuous, respectively linear and quadratic, maps on Tτ,α with values in the weighted uniform norm. The corresponding pairs (Fj,Bj) in extend continuously in the norms \|·\|2β and \|·\|C0(S) .

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Lemma 5 ([Green representation of the mass]). For every sufficiently small…Lemma 4 ([Finite coefficient asymptotics]). For every smooth finite angular…Proposition
Let U(R)q denote the pullback by a rotation R∈SO(3) , and let R be normalized invariant volume. Define k_ν(R)=c_ν((1+ R)/(4))^ν, ∫_ SO(3) k_ν\, R=1, q_ν=∫_ SO(3) k_ν(R)U(R)q\, R. The expression raised to the power ν lies in [0,1] and attains one only at the identity. The normalized densities therefore concentrate there. To verify this directly, outside any fixed neighborhood their base is at most 1-δ , whereas on a smaller neighborhood of positive volume it is at least 1-δ/2 ; the ratio of the corresponding integrals tends to zero. The map R↦U(R)q is continuous in \|·\|_β . On a compact part of M this follows from smoothness. On the tail, the bound Ce^ -(τ-β)s is uniform in R , so the strict weight margin controls it. Concentration gives \|q_ν-q\|_β→0 . Rotations preserve g_0 , the end background, and s . They commute with trace and divergence and preserve the weighted regularity. Consequently each average is smooth and TT, with the asserted uniform bounds. They also preserve the radial normal, so positivity of q_ ss |_S is retained by the nonnegative average. For fixed ν , the density is a polynomial in matrix entries. After a change of integration variable, the rotation orbit of q_ν depends on translates of this polynomial. These translates lie in a finite-dimensional polynomial space; their coefficients multiply finitely many fixed averaged tensors. Thus q_ν has finite angular type. It remains to check that the coefficient limits are independent of the approximation. On a bounded set in \|q\|_β , comparison applied to gives, for finite type seeds q, q , \|u_1[q]-u_1[ q]\|_β , ≤C\|q- q\|_β,; \|F_2[q]-F_2[ q]\|_ 2β +\|B_2[q]-B_2[ q]\|_ C^0(S) , ≤C\|q- q\|_β,; \|u_2[q]-u_2[ q]\|_β , ≤C\|q- q\|_β. Here the elementary product estimate is \||q|^2-| q|^2\|_ 2β ≤(\|q\|_β+\| q\|_β)\|q- q\|_β. The comparison constant has no angular-cutoff dependence. These estimates prove existence, uniqueness, and continuity of the limits in the complete weighted uniform space. Formula preserves their linear and quadratic dependence. No derivative convergence of the limiting coefficients is needed in this construction or in the mass functional.
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