证明
Lemma 33 ([A freely chosen late plateau]). For all sufficiently large N one can…Theorem 2.1 ([Neutral numerical input]). Let d≥3 and let (U^d,h,L) be a…Lemma 7 ([Increasing the ADM energy]). For purely electric data with the weak…观点
Keep s , a smooth profile, and ε fixed. Take N sufficiently large for all preceding constructions. Sard's theorem gives a regular value c_h∈(b_3/2,b_3) of |h| . Let D_N be the closure of the component of |h|<c_h containing the entire far end. By, this component cannot cross S into the filling. Its entire boundary is a compact smooth embedded two-sided hypersurface in the original interior. It is nonempty: a path in the original connected exterior from the far end to S must cross the chosen level. All boundary components are retained. The mean curvature of this fixed compact boundary in γ is finite. We may now choose the plateau M in Lemma 33 ([A freely chosen late plateau]) so large that, with normal toward the end, H_ ∂ D_N +tr_ ∂ D_N K_ new =H_ ∂ D_N -k a<0 every component. This late choice does not require repeating either scalar solve or increasing N , by. Restricted to D_N , the new tensor has compact support, since h→0 and H vanishes near zero. We check the neutral input's hypotheses explicitly. The exterior D_N is connected and oriented, smooth up to its nonempty compact boundary, and has exactly the original coordinate end with compact complement. It is complete with boundary included: it is closed in the original complete exterior, and bounds its metric below by a positive constant times g . The metric is smooth and has all-order differentiated decay of order 2-n in normalized end coordinates. Thus the metric and tensor satisfy the stronger falloffs in Theorem 2.1 ([Neutral numerical input]), with any exponent strictly between (n-2)/2 and n-2 . The neutral DEC follows from Lemma 33 ([A freely chosen late plateau]); its matter densities are integrable because the tensor is compactly supported and the scalar curvature is integrable at infinity. The ADM momentum is zero. The ADM energy exists and is given by. Finally the new boundary is strictly future outer trapped by. Every full cut of D_N , regarded as a subset of the original exterior, is also a full cut there. This includes any portions coincident with ∂ D_N , and counts every component. Therefore its full-cut infimum a_N satisfies a_N≥e^ ky_0 a_*>0. Indeed the metric comparison acts on all k tangent directions, and the class of available cuts has only been restricted. No minimizing boundary is assumed to be smooth or to exist. It remains to verify the timelike energy hypothesis of the neutral input for these auxiliary data. Write e_N= E(γ) . If e_N>0 , their zero momentum makes that hypothesis immediate. If e_N≤0 , apply the conformal end increase of Lemma 7 ([Increasing the ADM energy]), taking its positive function φ constant on a core containing the boundary and the new tensor's support. For U=1+λφ , transform the data by γ_λ=U^ 4/(n-2) γ, K_λ=U^ 2/(n-2) K_ new . The neutral DEC and trapping signs persist, the stronger asymptotic derivatives persist, momentum remains zero, and full-cut areas do not decrease. Its energy is e_N+2λ . For every arbitrarily small e>0 choose λ=(e-e_N)/2≥0 . These data satisfy all hypotheses of Theorem 2.1 ([Neutral numerical input]), which would give e≥12(a_N/ω)^ (k-1)/k ≥12e^ (k-1)y_0 X_*>0. Letting e0 is a contradiction. Thus e_N>0 , and the same neutral input applied directly gives e^ (k-1)ε ( E_*+o_N(1)) -H_0(0)k|Q| ≥12 e^ (k-1)y_0 X_* =(1-s^2)/(2)X_*. The parameter order is as follows. For the fixed prepared data, fix s∈(0,1) , a profile tolerance, and ε>0 first. Complete the two scalar solves and the regular cut for each sufficiently large N ; the tensor plateau is chosen last for that N . Now let N→∞ in , then let ε0 , and then let the profile tolerance decrease to zero. Equations and give . Only fixed-parameter estimates were required in the second scalar solve. The electromagnetic input used in this proof was precisely and its lower bound by I_g(w) , which also holds for the indicated compact magnetic form.