Prove
Corollary 35 ([Strict compact-magnetic numerical extension]). Let the original…Proposition 34 ([Prepared numerical bound]). For the prepared data, for every…Proposition
Large coordinate spheres have positive mean curvature, uniformly for the finite-dimensional metric paths considered below. On the region outside one such sphere, the outward unit normal to their foliation has positive divergence. Applying the divergence theorem to the portion of a filled competitor outside that sphere shows that clipping off that portion cannot increase perimeter, and strictly decreases it if the discarded portion has positive volume. The argument holds for almost every cutting radius by finite-perimeter slicing. Choose two fixed radii, both outside the compact core. Clipping at the smaller radius gives a common compact truncation, strictly inside the larger one, for minimization and for all sufficiently small parameters of each path. BV compactness and lower semicontinuity now give a minimizer; see . A smooth full cut bounds a filled competitor with exactly the same area, including its contact portions. Conversely, first push O and an arbitrary filled competitor slightly outward by a smooth ambient diffeomorphism supported in a collar of S . The pushed competitor contains a neighborhood of the original O , and its perimeter tends to the original perimeter. Strict approximation of finite-perimeter sets by smooth domains can therefore be carried out while retaining that neighborhood. It is supported inside a fixed compact truncation. Retain the exterior component containing the end, filling any bounded exterior components of the smoothed set. This only removes boundary components. Its entire smooth intrinsic boundary is a full cut. Passing through these two approximations proves equality of the infima. The same removal observation shows that a minimizing frontier cannot leave a removable bounded complementary open component: a nonempty such component has positive perimeter, whose removal would improve the minimum. The usual indecomposable-component decomposition gives the same conclusion for finite-perimeter representatives . Here is the local estimate needed at the obstacle. For a local competitor G , replace it by G∪O . Submodularity gives P(G∪O)≤P(G)+P(O)-P(G∩O). Extend the smooth outward normal of O to a smooth vector field of norm at most one near its boundary. Its divergence is bounded. The calibration inequality for this field gives P(O)-P(G∩O)≤C|O G| for changes in a sufficiently small coordinate neighborhood. Since F minimizes among sets containing O , it follows that P(F;B)≤P(G;B)+C|F| whenever F is compactly contained in the small ball B . This is an unconstrained perimeter almost-minimizing estimate. In coordinates the Riemannian area integrand is the ellipsoidal integrand associated with A=( g)g^ -1 , which is smooth and uniformly elliptic. The error in is O(r^n) and therefore is bounded by C_γr^ n-1+γ for any fixed 0<γ<1 on sufficiently small balls. The almost-minimizer results in apply. They give density and perimeter bounds, local compactness, regularity off a set of dimension at most n-8 , and convergence as single C^1 graphs near free regular limit points. On free patches, the uniform local C^ 1,γ/4 graph bounds and the smooth minimal-graph equation upgrade this to smooth convergence on smaller patches by interior elliptic estimates. Tangent sets are perimeter-minimizing cones. The minimal-boundary formulation and dimension reduction are also described in . At a contact point a tangent filled cone contains the tangent half-space of O . The corresponding minimizing hypercone is therefore supported on one side of its boundary plane, and is that plane. One way to see the last assertion is to integrate the signed coordinate-height equation on its stationary spherical link; the height has one sign and its spherical Laplacian is a negative constant times itself. The singular set can be removed in this integration by its dimension and the perimeter bounds. Equivalently one can use the half-space cone theorem in . The almost-minimizer regularity theorem now gives a C^ 1,β graph at contact for some β>0 ; one may take β=γ/4 . Estimate, differentiated along smooth flows, bounds its weak mean curvature. Difference quotients in the uniformly elliptic prescribed-mean-curvature graph equation give W^ 2,2 regularity. Written in nondivergence form, that equation has H\"older coefficients and bounded right-hand side, so local W^ 2,p estimates give all finite p ; see . If two minimizing frontiers meet at a free regular point of one of them, their union and intersection also minimize by , with equality in the sum of the perimeters. They are ordered relative to the given regular sheet. Their tangent cones at contact are one-sided relative to its tangent plane, and hence are planes. Regularity and the strong comparison principle for the minimal graph equation imply local coincidence. Strict compactness and regular convergence then give the asserted uniform graph neighborhoods: otherwise a sequence of failing frontiers would converge strictly to a minimizer through the point, where multiplicity-one regular convergence gives precisely the missing neighborhood. The difference of two distinct ordered graphs solves a linear uniformly elliptic equation. Positive-solution Harnack and interior derivative estimates imply, on smaller patches, that its gradient is bounded by a uniform constant times its pointwise separation. Finally let F_t minimize for g_t . Compactness, lower semicontinuity and comparison with a fixed g_0 minimizer show that every convergent subsequence at t=0 converges strictly in perimeter to an element of F_* . Strict continuity for vector measures then gives convergence of the tangent-plane area measures . The area Taylor expansion has a uniform remainder on the resulting perimeter-bounded family. Fixed minimizers give the upper bound in ; the varying minimizers give the lower bound. This also proves compactness of the minimizing family and continuity of F↦a'_F(h) in its strict topology.