证明
Lemma 4 ([Weighted bounds]). On, δ>0 , δ≤Cr^ k-1 , and normalized mass is…Lemma 3 ([Local fitting inputs]). For every x∈B_5 , there are smooth minimal…观点
We verify the transfer of the calculation in ; this also identifies the precise single-radius inputs. The definitions and positivity of B give r^ -m ∫φ|z|^2\,μ, ≤C(H+δ^2E_*),; r^ -m ∫ψ|z|^2\,μ , ≤C H+δ^2(E_*+R_*) ,; r^ -m ∫φ\,Λ, ≤C(H+δ^2E_*). For the first bound use 1-h≤Ce ; for the second split at r/2 and use 1-h≤Cb_z on the outer part. The last bound also uses , with physical units equal to the current start units. Here are the details of the mass comparison, where preserving the coefficient one in is essential. For the fixed 0<β_1<1/4 put γ_1=δ^ β_1 . Away from the outer collar we use signed excess; on the collar we use a radial first-variation test. Choose a radial cutoff χ equal to one at relative depth 1-ρ/r≤γ_1^2 and zero at depth at least 2γ_1^2 , and put φ_ thin =χφ, φ_ bulk =(1-χ)φ. Cover the support of φ_ bulk by cells centered at x_j of radius λ_1=c_1γ_1^2r , with a smooth partition of unity ϑ_j and bounded overlap of fixed enlargements. Choose c_1 small enough that these enlargements stay inside ρ<r and φ is comparable there to φ_j=φ(x_j) . Their mass is O(λ_1^m) because δr=o(λ_1) . Testing normal stationarity with a cutoff times z and using yields λ_1^ -m (D+Λ)(B_ cλ_1 ) ≤Cδ^ 2-4β_1 . This is the local estimate in . The scalar estimate selects an integer on each cell; the small exceptional projection sets cannot fill the prescribed overlaps, so the integers agree throughout the connected bulk; call their value q . Put ε=Cδ^ 2-4β_1 . Partition the smooth mass test from Lemma 4 ([Weighted bounds]) over the cells. Each piece has the scaled derivative norm in Equation bounded by Cr^ -m . That homogeneous estimate bounds its error relative to q by Cελ_1^m r^ -m , without dividing by the piece's possibly small integral. Bounded overlap gives ∑λ_1^m≤Cr^m , so the total error is O(ε) . Since the test average is within 1/8 of Q , integrality gives q=Q . The scalar estimate now bounds |M|(B_ 4λ_1 (x_j)) by C(D+Λ)(B_ 8λ_1 (x_j)) . Summing with comparable weights gives the absolute bulk estimate r^ 2-m ∫φ_ bulk \,|M| ≤Cr^ 2-m ∫φ\,(D+Λ) ≤C(E_*+r^2H). In the last inequality a small multiple of E_* is absorbed using . For signed excess, use fresh charts on the cells and set d_j= Cλ_1^ -m τ(B_ 4λ_1 (x_j)), δ^ 2J_0+4 , T_j=- d_j, τ=|M|+D+Λ. The local bounds and the floor give c|δ|≤T_j≤(2J_0+4)|δ| . The physical smallness hypothesis follows from λ_1+|z|≤Cδ^ 2β_1 , κ(2J_0+4)<2β_1. These are precisely the fixed parameter choices in Equation; the required derivative orders were chosen after J_0 and κ , and before the fitting exponent. The partitioned tests ϑ_jφ_ bulk /(Cφ_j) have size and Lipschitz constant at most one in cell units. Applying signed excess to these tests and restoring the measure factors gives 2r^ 2-m ∫φ_ bulk \, M , ≤r^ 2-m ∫φ_ bulk \, D +C_Qr^ 2-m ∫φ_ bulk \,Λ; , +Cr^ 2-m ∑_jφ_jλ_1^m d_j e^ -F_Q(T_j) . The weighted budgets satisfy ∑_jφ_jλ_1^m d_j ≤C∫φ\,(D+Λ)+Cr^mδ^ 2J_0+4 , by scalar control on the enlarged cells and bounded overlap. The exponential factor is smaller than any fixed negative power of T , and the floor costs r^2δ^ 2J_0+4 ≤Cδ^ J_0+4 H by . The signed-excess remainder is therefore at most Cg(E_*+H) . The separate C_QΛ term is bounded by Cr^2H+Cr^2δ^2E_* ; since r^2δ^2=o(g) , it contributes only Cr^2H+CgE_* . This proves the bulk signed comparison with leading tilt coefficient one. For the thin part, let a be a multiplier with (|a|+r|∇_Sa|)≤1 . The radial primitive constructed in , cut off at relative depths between γ_1 and 2γ_1 , gives a compactly supported tangent field X with _SX=aφ_ thin +q_c, |X|/r+|∇^SX-aφ_ thin u⊗u| ≤Cγ_1^2φ. Here q_c is supported on that thicker cutoff collar and |q_c|≤Cγ_1^ p+1 φ . Its mass contribution is controlled by the absolute bulk estimate. In horizontal stationarity, the principal derivative of X pairs with radial tilt B[u,u] ; the remaining derivative has the small factor γ_1^2 . The resulting bound is r^ 2-m |∫aφ_ thin \, M| , ≤Cr^ 2-m ∫φ_ thin B[u,u]\,μ +Cδ^αE_*+Cr^2H ; , ≤Cδ^ α' (E_*+R_*)+Cr^2H. For the last inequality use, on the thin collar, φ_ thin ≤Cγ_1^2(ρ/r)^2ψ and B[u,u]≤2b_z+C|z|^2e . The errors use only horizontal stationarity, fixed geodesic geometry, and. Combining the absolute bulk and thin estimates proves , including its version with any bounded scaled C^1 multiplier. For the signed estimate take a=1 and combine with the signed bulk bound. The small tilt error is absorbed in CgE_* , since every positive power of δ is o(g) ; the radial error remains ηR_* . This proves. Normal stationarity gives the flux estimate |E_*-r^ -m ∫ρψ\, z·(P_T)_ vh v_z\,μ| ≤Cr^2H+ηE_*. Projection algebra gives the exact inequality |(P_T)_ vh v_z|^2≤b_zh ; Cauchy--Schwarz proves . Horizontal stationarity with the lift of ρu gives, as in . Its geodesic volume-distortion multiplier has bounded scaled C^1 norm, so the preceding absolute estimate applies with a uniform constant. One modification is needed for the height identity because our center need not pass through ambient 0 . Off the central fiber put Y=(ρK_zu)/(|K_zu|^2). Then DY=P_h+O(ρ+|z|) , where P_h is horizontal projection at the nearest point of S . To see this, the pulled-back radial field ρu has derivative P_h+O(ρ+|z|) . Its correction in Y is ρ times a function of the tube coordinates and u that is O(|z|) , including its angular derivatives. Angular differentiation therefore costs ρ·O(|z|)/ρ=O(|z|) , and the remaining derivative costs O(ρ) . Testing with ψ|z|^2Y gives the following identity, whose integrand is extended by zero on the central fiber: H=r^ -m ∫ψ |z|^2((P_TDY)-mh) + 2ρz·(P_T)_ vh K_zu |K_zu|^2 \,μ. This is . First omit the central fiber using a radial cutoff. The transition term tends to zero because Y=O(ρ) and the mass of the open shrinking transition annulus tends to zero. On the fiber h=0 almost everywhere, so the left-hand side is unchanged; the flux term has a factor ρ . The error in DY is O(r) on the support, and and the normal flux estimate give the upper bound in. The lower bound follows by direct differentiation of the nonnegative weight r^ -m ψ . All parameter choices have been fixed independently of the start.