A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold

Proof. The regular set is relatively open in \spt\mu, so the singular set is Borel. The compact-subset theorem for Hausdorff measure [16, Theorem 8.13] gives a compact subset of \Sing(V) with finite positive \Hh^s measure. We recall the two measure-theoretic refinements that we need [10, 16]. On a set of finite positive \Hh^s measure, the upper s-density is positive and finite almost everywhere. By restricting to a positive-measure subset where its upper bound is uniform, and then taking a compact subset, we obtain a compact set K such that

An upper bound initially obtained for centers in K extends to arbitrary centers by doubling the radius. Enlarging C handles the remaining large radii. Lusin’s theorem permits a further compact restriction of positive measure on which \Theta_V is continuous. The upper bound in (2.4) is preserved.

For \Hh^s-almost every p\in K, positive upper density supplies radii r_i\downarrow0 and a constant c_p>0 with

Pass to a subsequence on which V_{p,r_i} converges to a stationary integral tangent cone C_p. Define measures on the closed unit ball by

They have uniformly bounded mass, mass bounded away from zero, and the upper ball bound in (2.4). A further subsequence converges to a nonzero measure \lambda with the same upper ball bound. Its support lies in \spt C_p.

Set q_p=\Theta_V(p); at this point q_p is not assumed to be an integer. If z\in\spt\lambda, there are points p_i\in K with (p_i-p)/r_i\to z. Continuity on K and (2.1) imply

On the other hand, the density at infinity of C_p about z equals q_p: sandwich \mathbf B_R(z) between the balls about zero of radii R-|z| and R+|z|, and use conicality about zero. Monotonicity about z consequently has equality at every radius.

We spell out the geometric consequence of this equality. The radial displacement X-z lies in the tangent plane of C_p almost everywhere. Testing stationarity with \varphi(X)(X-z) gives

Differentiating dilated scalar integrals shows that the weight is m-homogeneous about z. Since a rectifiable weight determines its tangent plane almost everywhere, the varifold is conical about z as well. Composing dilations about z and about zero now gives translation invariance along the whole line \R z.

The support of \lambda spans at least m linearly independent directions. Otherwise it lies in a linear space of dimension at most m-1; covering its bounded support by O(\varepsilon^{-(m-1)}) balls of radius \varepsilon would give \lambda(\R^{m+n})\le C\varepsilon^{s-m+1}\to0. Choose m independent such directions and let P be their span. The cone is invariant under every translation in P. If its support contained a point outside P, subtracting its P-component would give w\in P^\perp\setminus\{0\} in the support. Conicality and translation invariance would then place every disk tw+(P\cap\mathbf B_1), 1\le t\le2, in the support. These are disjoint disks of equal positive m-dimensional measure, all in a bounded region, contradicting local finiteness of \Hh^m(\spt C_p). Thus its support is precisely P. Tangential stationarity on this plane gives constant multiplicity, and integrality gives

We have proved that \Theta_V is integer-valued at \Hh^s-almost every point of K. One of its countably many integer level sets therefore has positive \Hh^s measure. Choose a compact positive-measure subset K_Q of that level set. Repeating the preceding argument with K_Q, choose a fixed point p\in K_Q and radii L_i\downarrow0 on which its normalized s-mass is bounded below. The resulting tangent cone is a Q-plane. Translate p to zero, denote that plane by \Pi, and set

Compactness of K_Q ensures that the support consists entirely of points from that integer-density level set. Dilation preserves both density and singularity. The asserted convergence and bounds follow.□

Proof. If t(w)>0 and \rho(w)<r, choose a failure (x,\ell) for which \ell-|w-x|>t(w)/2. Since |w|\le r, either |x|\le16\ell or 16\ell<r+\ell. Thus

Equation (4.3) gives \delta\le Cr^{k-1}. The projected cylinder is contained in an ambient ball of comparable radius, so monotonicity gives its normalized mass bound.

If \delta=0, the support lies on S in a neighborhood of ambient 0: its projection lies inside \{\rho<r\} because its distance from the radial center is O(m_0b^k). Tangential stationarity then makes the multiplicity, extended by zero on the disk, constant. It is positive because 0 belongs to the support. Normal stationarity and elliptic regularity make S smooth and minimal there. This contradicts the singularity of 0.

We next locate the height that pays for a witness. The argument is the weighted construction of [19, Proposition 7.4], with (4.9) replacing its typical-point estimate. If the witness is near the kernel boundary, move inward by \ell along the radial geodesic and take a ball of radius a sufficiently small fixed multiple of \ell; otherwise take a nearby such ball. This ball lies within the kernel support, and its center remains within B_{30\ell}(x) in the fitting coordinates. On it \phi and \psi dominate, up to fixed constants, their suprema on a small cell containing the original point, and \psi\ge c(\ell/r)^{p-1}. Lemma 4.1 supplies separated entries above a set of base volume c\ell^m. Two entries separated in a nearly vertical direction cannot both lie within a sufficiently small multiple of their separation from a small-slope graph. After transferring to S, at least one entry therefore has height at least cm_0\ell^k and h\ge3/4. Small slopes and the height bounds make changes of horizontal argument smaller than the available subball margin. The projection Jacobian bound converts base volume into the same lower bound for mass.

In particular \delta r\ge cm_0\ell^k. At a point approaching the height supremum, the witness also satisfies \delta r\le Cm_0\ell^k, by (4.3). Hence

where the last inequality uses r\le1. This proves the lower bound in (4.6), for example with J_0\ge2+(m+p-1)/k; its upper bound follows from the mass bound. The comparison \delta r\ge cm_0\ell^k, together with k>4, proves (4.7).

To prove (4.8), group witnesses by dyadic scale and use each covering grid cell once at its scale. By (4.3), the curvature-square integral on a cell of size \ell is at most Cm_0^4\ell^{4k-8+m}\sup\phi. Its paying ball contributes at least cm_0^2\ell^{2k+m}\sup\phi to the height-square integral. At each scale the paying balls have bounded overlap, since they are moved by only O(\ell). Summing the factors Cm_0^2\ell^{2k-8} proves the assertion. The cells may be used as an overcover, so no measurable choice of witnesses is necessary.

Finally choose a smooth decreasing radial mass test on a small fixed fraction of r. Compare it with the corresponding ambient radial test at 0. The displacement of the center is O(m_0b^k), the support height is O(r^k), and chord and geodesic radii on S differ by O(r^3). The radius replacement has error O(r^2), so the normalized averages differ by O(r), including their reference normalizations. All support encountered by the ambient test lies in the cylinder and has projected radius below r/2. The ambient average is close to Q by the density-ratio requirement at the start. Decreasing the fixed parameters gives the stated 1/8 bound.□

Proof. We verify the transfer of the calculation in [19, Lemmas 8.1–8.2]; this also identifies the precise single-radius inputs. The definitions and positivity of B give

For the first bound use 1-h\le Ce; for the second split at r/2 and use 1-h\le Cb_z on the outer part. The last bound also uses (4.8), with physical units equal to the current start units.

Here are the details of the mass comparison, where preserving the coefficient one in (4.11) is essential. For the fixed 0<\beta_1<1/4 put \gamma_1=\delta^{\beta_1}. Away from the outer collar we use signed excess; on the collar we use a radial first-variation test. Choose a radial cutoff \chi equal to one at relative depth 1-\rho/r\le\gamma_1^2 and zero at depth at least 2\gamma_1^2, and put

Cover the support of \phi_{\rm bulk} by cells centered at x_j of radius \lambda_1=c_1\gamma_1^2r, with a smooth partition of unity \vartheta_j and bounded overlap of fixed enlargements. Choose c_1 small enough that these enlargements stay inside \{\rho<r\} and \phi is comparable there to \phi_j=\phi(x_j). Their mass is O(\lambda_1^m) because \delta r=o(\lambda_1). Testing normal stationarity with a cutoff times z and using (4.7) yields

This is the local estimate in [19, Equation (8.6)]. 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 \epsilon=C\delta^{2-4\beta_1}. Partition the smooth mass test from Lemma 4.2 over the cells. Each piece has the scaled derivative norm in Equation (3.6) bounded by Cr^{-m}. That homogeneous estimate bounds its error relative to q by C\epsilon\lambda_1^m r^{-m}, without dividing by the piece’s possibly small integral. Bounded overlap gives \sum\lambda_1^m\le Cr^m, so the total error is O(\epsilon). Since the test average is within 1/8 of Q, integrality gives q=Q.

The scalar estimate now bounds |M|(B_{4\lambda_1}(x_j)) by C(D+\Lambda)(B_{8\lambda_1}(x_j)). Summing with comparable weights gives the absolute bulk estimate

In the last inequality a small multiple of E_* is absorbed using (4.15).

For signed excess, use fresh charts on the cells and set

The local bounds and the floor give c|\log\delta|\le T_j\le(2J_0+4)|\log\delta|. The physical smallness hypothesis follows from

These are precisely the fixed parameter choices in Equation (3.12); the required derivative orders were chosen after J_0 and \kappa, and before the fitting exponent. The partitioned tests \vartheta_j\phi_{\rm bulk}/(C\phi_j) have size and Lipschitz constant at most one in cell units. Applying signed excess to these tests and restoring the measure factors gives

The weighted budgets satisfy

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^2\delta^{2J_0+4}\le C\delta^{J_0+4}H by (4.6). The signed-excess remainder is therefore at most Cg(E_*+H). The separate C_Q\Lambda term is bounded by Cr^2H+Cr^2\delta^2E_*; since r^2\delta^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 \sup(|a|+r|\nabla_Sa|)\le1. The radial primitive constructed in [19, Lemma 8.1, Equation (8.11)], cut off at relative depths between \gamma_1 and 2\gamma_1, gives a compactly supported tangent field X with

Here q_c is supported on that thicker cutoff collar and |q_c|\le C\gamma_1^{p+1}\phi. Its mass contribution is controlled by the absolute bulk estimate (4.16). In horizontal stationarity, the principal derivative of X pairs with radial tilt B[u,u]; the remaining derivative has the small factor \gamma_1^2. The resulting bound is

For the last inequality use, on the thin collar, \phi_{\rm thin}\le C\gamma_1^2(\rho/r)^2\psi and B[u,u]\le2b_z+C|z|^2e. The errors use only horizontal stationarity, fixed geodesic geometry, and (4.15). Combining the absolute bulk and thin estimates proves (4.10), 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 \delta is o(g); the radial error remains \eta R_*. This proves (4.11).

Normal stationarity gives the flux estimate

Projection algebra gives the exact inequality |(P_T)_{vh}v_z|^2\le b_zh; Cauchy–Schwarz proves (4.12). Horizontal stationarity with the lift of \rho u gives (4.13), as in [19, Lemma 8.2]. 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

Then DY=P_h+O(\rho+|z|), where P_h is horizontal projection at the nearest point of S. To see this, the pulled-back radial field \rho u has derivative P_h+O(\rho+|z|). Its correction in Y is \rho times a function of the tube coordinates and u that is O(|z|), including its angular derivatives. Angular differentiation therefore costs \rho\cdot O(|z|)/\rho=O(|z|), and the remaining derivative costs O(\rho). Testing with \psi|z|^2Y gives the following identity, whose integrand is extended by zero on the central fiber:

This is [19, Equation (8.18)]. First omit the central fiber using a radial cutoff. The transition term tends to zero because Y=O(\rho) 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 \rho. The error in DY is O(r) on the support, and (4.15) and the normal flux estimate give the upper bound in (4.14). The lower bound follows by direct differentiation of the nonnegative weight r^{-m}\psi. All parameter choices have been fixed independently of the start.□

Proof. Equations (4.10) and (4.12) imply |P|\le C(\sqrt{q_*}+r^2)+C\eta q_*. For |P| large, either the square-root term or the last term controls a fixed fraction of |P|, proving the first assertion of (4.18). Equation (4.13) then gives its second assertion, after increasing K and decreasing r_*.

For positive P, quotient differentiation and the upper height bound give the principal expression 2P-n_*+q_*-2Pn_*, with the errors displayed in Lemma 4.3. If q_* is larger than a sufficiently large fixed multiple of P^2, it absorbs all negative terms, since n_*\le C\sqrt{q_*}+Cr^2 and \eta T^4=o(1). Otherwise n_*\le CP and

For large K the principal terms factor as (n_*-1)(n_*-2P), bounded below by -CP(gP+r^2+\eta P^2). Since \eta T^4\le g at small radii, the term \eta P^3 is absorbed by gP^2. This proves (4.19).

If P\le-K, the lower height bound gives \dot P=v_0-P\dot H/H\ge cq_*-m|P|>0. Thus P decreases in inward time -\log r whenever it is below -K. This proves the finite-interval lower bound from the inner endpoint. On an infinite interval, such an excursion would also make A<0 decrease inward and remain below a fixed negative value. That is impossible because A\to0: its normalized mass integral is bounded and it has a factor r^2.

To prove the upper bound, work on a component where P\ge K before any exit through P=T^4. In inward time, (4.19) gives

The integral of this right-hand side from a sufficiently small radius to zero is less than one, since T\ge2|\log r|-C. Moreover Z=T-\log P=\log(r^2/A) satisfies Z'=v_0/P-2>0 by (4.18). If the component starts at r_0 with P_0\le T_0^2, then P\le\exp(1)P_0 and T\ge T_0-\log P_0+\log K\ge T_0/2. For large T_0 these inequalities exclude P=T^4. Components starting later at P=K satisfy the same argument, giving (4.20).

On an infinite interval, suppose no sufficiently small radius satisfied P\le T^2. Then eventually P>T^2, and (4.18) implies Z'\ge cT^2-2. The lower bound for T makes Z positive eventually. Since Z\le T in this regime, one then has Z'\ge c'Z^2, which forces blow-up in finite inward time. The continuous positive quantities A,H are defined at every finite time in this regime, a contradiction.

Finally, for |P|\le C_0, use the upper height derivative bound if P\ge0 and the lower bound if P<0. Absorbing \eta q_* gives

Integration bounds \int q_*\,\dd\log r on every bounded-ratio interval. Equation (4.12) and Cauchy–Schwarz bound \int n_*\,\dd\log r, and (4.14) gives both height comparisons in (4.21).□

Proof. Coverage of the targets. The top of the interval belonging to R is (r_*/N_*)R. Successive tops decrease to zero unless the construction stops, since the next start is at most FbR<R/4. Suppose a target L_i lies below one top and above the next, but misses the first interval. It must then lie below its lower endpoint, so L_i<B_*bR and FbR>10N_*L_i/r_*. A dyadic scale between 2N_*L_i/r_* and 4N_*L_i/r_* is therefore eligible for the next start. For all sufficiently late i it is good, by convergence at L_i to a plane. Maximality of the next start would make its top larger than L_i, a contradiction. Hence all sufficiently late targets are covered.

A fixed interval containing targets tending to zero necessarily has b=0. On that interval Lemma 4.4 first supplies a lower bound and a sufficiently small radius with P\le T^2, then an upper bound and height doubling inward. This proves the assertion in the fixed-interval case. After taking a subsequence, we may therefore restrict to targets whose covering interval indices tend to infinity. There are then infinitely many starts, all with positive failure parameter b.

Geometry of the late starts. We first show that the b values of these covering intervals tend to zero. Otherwise their start scales would be comparable to their target scales, since B_*b\le L_i/R\le r_*/N_*. The start varifolds would then converge to planes, contradicting failed-fit separation at a scale bounded away from zero.

In fact the start varifolds converge to multiplicity-Q planes even without comparability to the targets. Any subsequential limit is a stationary tangent cone at 0, of density Q. Pass also to limits of the start coordinates and their uniformly controlled center graphs. Because t(w)\le b+|w| and b\to0, the limiting cone lies within C|w|^k of a C^1 graph through 0. Every conical ray is therefore in the tangent plane of that graph at 0. Constancy gives a multiplicity-Q plane. Thus, if a minus sign denotes the preceding interval, eventually

Indeed the right-hand side is dyadic. If it exceeded the chosen R, then 2R would still be eligible; planar convergence of V_R on every fixed ball makes 2R good, contradicting maximality. It follows also that b_-\to0. If not, the preceding starts would be comparable to the current planar starts, again contradicting separation.

The outer endpoint: the preceding failure. We now work entirely in the current start’s units. The preceding failure will give an upper support-height bound and a matching lower height integral on fixed room. Together these bounds will control P(r_*). Define the preceding failure height in current units by

In every required fixed bounded region of the current start, all support points are within Ch_0 of the preceding center, expressed in current units: in preceding units the region has radius O(Fb_-), and (4.3) applies with t\le b_-+|w|. The preceding and current start planes make a small angle. To verify this, take a subsequence: the preceding centers in current units have small slope and pass within Ch_0 of 0, and the current planar support limit lies in their limiting graphs. The current chosen planes converge to this support plane. This proves the asserted angle bound, allowing a fixed enlargement of the small slope threshold.

The current center consequently satisfies

on the region containing \{\rho<4r_*\}. Indeed a current witness of scale \ell supplies two entries separated by cm_0\ell^k in a direction nearly vertical also for the preceding plane. Both lie within Ch_0 of its small-slope center, so cm_0\ell^k\le Ch_0. Equation (4.3) now proves (4.26). By Lemma 4.2 at radius 4r_*, the mean curvature and its first normal derivative are at most Ch_0 there as well. Every comparison is internal to both cylinders: the current cylinder has fixed room, whereas its required region in preceding units shrinks with Fb_-\to0.

A preceding failure attaining b_- supplies the complementary bound

Its center is within 16b_- of horizontal 0, so the failed-fit lemma may be used on a subball centered near horizontal 0. The entries lie within ambient radius Cb_- in preceding units: their heights over the fit are O(m_0b_-^k), and the glued center has the same bound at 0. In current units they are well inside \{\rho<r_*\} by (4.22); their mass is bounded below by a constant times F^{-m}. Their tilt remains small relative to the current center. At least one entry of each separated pair has height at least ch_0 over that center, proving (4.27).

The cutoff normal test with z on fixed room around r_*, using (4.26) and its curvature bound, gives E_*(r_*)+R_*(r_*)\le Ch_0^2. Hence (4.10) and (4.27) bound |P(r_*)| uniformly, while T(r_*)\to\infty. In particular P(r_*)\le T(r_*)^2 for all sufficiently late covering intervals, as required for the inward upper bound.

The inner endpoint: the current failure. For b>0 we use the same height-versus-tilt comparison at radius B_*b, now supplied by the current failure. On a fixed enlargement, |z|\le Cm_0b^k follows from t\le b+|w|; the curvature bounds of Lemma 4.2 apply there. The cutoff tilt estimate on room of scale b gives

The inverse-square scale in that estimate is canceled by the factors in (4.5). A failure attaining b, tested on a subball of radius 10^{-2}b near horizontal 0, supplies low-tilt entries well inside the kernel and proves H(B_*b)\ge cm_0^2b^{2k}. Thus |P(B_*b)| is bounded as well.

The outer bound controls P from above as the radius decreases. The inner bound controls it from below: once P<-K, it decreases inward, so a value below the inner-endpoint bound could not recover before that endpoint. Lemma 4.4 therefore gives (4.24). Its final smallness requirement on \eta is uniform on late intervals: use \delta\le Cr on the sufficiently small radii and (4.26) on the remaining radii. This proves the uniform height comparison.□

Proof. We prove successively the integral normalization, negligible curvature, uniform support height, and first-moment centering. In all intermediate estimates we retain larger fixed interior regions than those in the statement.

Integral normalization and convergence of the centers

For the interval covering L_i, let R be its start, use its center S, and put

The notation R and S here may depend on i. If r_i\to0, then a_i\to0 by (4.6) and \delta\le Cr_i^{k-1}. If r_i stays bounded away from zero, we are in the multiple-interval case, and (4.26), with h_0\to0, gives the same conclusion. These cases also show that \delta(cr_i)\to0 for every fixed enlargement used below. Subsequences suffice to handle them.

Proposition 4.5 yields

where these expressions remain in start units. Indeed (4.21) supplies a radius in [400r_i,600r_i] at which q_*, and hence n_*, is bounded. These radii lie below r_* because r_i\le r_*/1000. The weights at that radius are bounded below on \{\rho<200r_i\}, so (4.15) gives the height bound, and the definition of E_* gives the tilt bound. There is also a converse height lower bound ca_i^2r_i^2 already on \{\rho<2r_i\}, directly from H(r_i)=a_i^2r_i^2.

Dilate the center into target units and denote it by S_i. Its graphs have uniform high derivative bounds on fixed balls: their start derivatives are uniformly bounded and the dilation introduces factors r_i^{j-1} in derivative order j. Their heights at horizontal 0 before dilation are at most Cm_0b^k, and their slopes are small. Since b\le r_i/B_*, the center positions remain bounded after dilation. The sup height estimates show that support points on each needed fixed region are at distance tending to zero from these graphs in target units. For example, support points in an ambient ball of radius 60 have projected geodesic radius below 180, by the fixed projection bounds and the position of the radial center. Support convergence to Q|\Pi| therefore places a corresponding patch of \Pi in every subsequential graph limit. A small-slope graph containing this patch must equal the plane there. Compactness with extra derivative orders gives the asserted smooth-enough convergence. Passing to a subsequence also fixes the limiting rotations of the start coordinates. We may now regraph over \Pi on smaller fixed room.

Curvature below the height scale

We claim that, on a fixed region larger than the one in the statement,

Temporarily return to start units. At a point with horizontal argument of norm at most 40r_i and t>0, choose a witness (x,\ell). The argument for (4.9) gives \ell\le\max(b,40r_i/15)\le3r_i. Apply failed-fit separation on a small subball near that point; its center lies within the allowed 30\ell of x. The subball, and the changes between its fitting and normal projections, remain inside the upper-bound region in (5.4). Separation and small slopes give height at least cm_0\ell^k on mass at least c\ell^m. Consequently

All such witness scales tend to zero uniformly. This is immediate when r_i\to0, and otherwise follows from a_i\to0 in (5.6).

The pointwise start-unit curvature estimate is Cm_0^2\ell^{2k-4}, by (4.3) and t\le2\ell. Mean curvature and its first normal derivative acquire factors r_i and r_i^2, respectively, in target units. Thus, for mean curvature, (5.6) gives

The derivative has one additional factor r_i\le1. The exponent of \ell is positive by (4.1). At t=0 both curvature quantities vanish by (4.3). This proves (5.5).

A uniform bound for support height

Integral control alone would not suffice for all later expansions. We strengthen it to

on fixed inner room. The normal-displacement trace formula [19, Equation (7.9)] is

Since \nabla|z_i|^2=2z_i, the nonnegative ambient function G=|z_i|^2+a_i^2 satisfies, on the support,

Here the positive 2e_i term absorbs its small multiple in (5.8), and (5.5) bounds the term linear in height by C(|z_i|^2+a_i^2).

For completeness, (5.9) has the requisite mean-value consequence on a stationary varifold. Fix a support point X_*, put d=|X-X_*|, and take a nonnegative decreasing smooth radial kernel f(d/l), supported in d<l and positive at d=0. Define

Radial stationarity gives

The discarded term is nonnegative because f is decreasing. Use the primitive F_l(d)=\int_d^\infty t f(t/l)\,\dd t, whose gradient is -f(d/l)(X-X_*) and which satisfies F_l\le Cl^2f(d/l). Stationarity and (5.9) bound the right-hand side below by -Cl^2I(l). Integration of this differential inequality down to zero, using density at least one, bounds G(X_*) by a fixed-radius integral of G around X_*. The integral is O(a_i^2) by (5.4) and the mass bound, proving (5.7).

These tests may be made for all points with horizontal arguments in a fixed ball of radius 25 in target units, using small fixed ambient balls. Their heights already tend to zero by the earlier sup height estimate. To check that the normal coordinates include all relevant support, in start units use t(w)\le b+|w| and (4.3): on these arguments the heights are small compared with r_i, and changes from horizontal to projected arguments are smaller still by the small slope. Decreasing r_* once fixes all needed tube and chart margins. Regraphing over \Pi preserves smaller fixed interior regions.

Vanishing of the first height moment

It remains to prove

We first use horizontal tests and vertical graph-height differences in the start coordinates. Take an arbitrary fixed smooth test of w/r_i, supported in horizontal B_{12r_i}, so that its derivatives are bounded at that scale. We do not yet pull tests back through the changing centers. By convergence of their coordinate changes, this region contains the supports of target tests on B_8\subset\Pi for all large i; coordinate conversion will be made after proving the fixed-coordinate limit. Cover \{t>0\} there by balls of radii a small fixed multiple of t at their centers, selecting a disjoint smaller family and a bounded-overlap enlargement. The Lipschitz property of t gives a partition of unity whose derivatives are bounded at the corresponding cell scales. Height is zero off \{t>0\}.

On a cell choose a constructed fit of dyadic scale L comparable to t and sufficiently large compared with it, as in the gluing construction. Cells meeting the test support have centers in an O(r_i) region. More explicitly, choose the enlarged cell radius at most ct(w) with c<1/10. A cell meeting B_{12r_i} has

because b\le r_i/100. In particular L\le Cr_i and these centers lie strictly inside the 40r_i region used for (5.6). Applying the first-moment estimate in Lemma 4.1 to the partitioned test, whose required scaled derivatives are bounded, gives an error at most

This controls the height over the local fit. Its difference from the glued center has height Cm_0^2L^{2k-1}, so the same bound controls that remaining contribution, using the cell mass bound.

To compare (5.11) with the target height scale, choose a witness at the cell center. It satisfies \ell\ge t/2, hence L\le C\ell, though no converse comparison is required. Equation (5.6) implies

uniformly over the cells. Positivity of the exponent follows from (4.1), and the witness scales tend uniformly to zero as proved above. Independently, bounded overlap of the cells and L\asymp t give \sum L^m\le Cr_i^m. Summing (5.11) therefore gives an error o(a_i r_i^{m+1}). This is exactly the vanishing first moment after mass and height are expressed in target units.

Finally convert from vertical graph-height differences to normal displacements. At first order this conversion is the normal projection at the graph point, with uniformly bounded quadratic error. By (5.7) the vertical differences also have size O(a_i). The quadratic error is therefore O(a_i^2), and test arguments change by O(a_i), which has the same harmless effect. The coefficient matrices converge uniformly on the fixed chart. The normalized first-moment measures have bounded total variation, so their convergence against fixed smooth tests extends to these converging tests and matrices. This proves (5.10).

We can now collect the conclusions in target units. The tilt estimate in (5.4) gives D_i=O(a_i^2) on the required room. The scalar estimate, Lemma 3.1, with (5.5) and (5.7), gives |M_i|=O(a_i^2) there as well. Its integer is Q, identified by smooth averages and convergence to Q|\Pi|. If a fixed small physical radius is needed for that estimate, a fixed dilation changes only constants. The lower height bound after (5.4), originally in geodesic radius 2r_i, lies in graph-coordinate B_3 for all large i, because the centers converge to \Pi and their radial origins tend to 0. This proves (5.2). All calculations retained enough room to select the fixed cylinder over B_8, with a vertical cutoff in a support-free gap. Hence they concern the full local varifold measure required in the statement, and the proposition follows.□

Proof. All estimates are on compact subdisks with a fixed enlargement. Graph volume factors are included when measures are transported to \Omega, and tensor components are taken in smoothly converging orthonormal frames. The assumed mass bounds give weak compactness. The projection identity

shows both that the flux measures have bounded mass and that the block matrix before passage to the limit is positive semidefinite. This proves (6.2); the lower second-moment bound gives \nu\ne0.

Vanishing geometric errors

We record the horizontal first variation carefully, since the height normalization may be much smaller than the curvature of the reference graph. Suppress i temporarily. For a tangent field Y on S, let J_z=I-\mathcal S_z, where \mathcal S_z is the shape operator, and lift Y to the ambient field J_zY. The curved first-variation calculation in [19, Lemma 2.2] gives

with the following sharper error in the present setting. Choose \varepsilon_i\to0 dominating the fixed geometric derivatives of S_i that tend to zero and such that |\mathcal H_{S_i}|+|\nabla^\perp\mathcal H_{S_i}|\le\varepsilon_i a_i. Then

Indeed, in frames whose tangent and normal connections vanish at the point of calculation, the horizontal derivative has blocks

Conjugation preserves the trace of \nabla^SY, and its pairing with B has error O(\varepsilon_i|z|e|\nabla^SY|). Codazzi gives \tr\mathcal K_z=\langle\nabla_Y^\perp\mathcal H_S,z\rangle. The remaining horizontal terms are bounded by C\varepsilon_i(a_i|z|+|z|^2+e|z|)|Y|. The symmetric sum of the off-diagonal blocks is, when paired with a tangent vector a and a normal vector v,

It is O(\varepsilon_i^2|z||a||v||Y|); pairing with (P_T)_{vh} costs at most C\sqrt e. This proves (6.6), including in arbitrary codimension.

The corresponding normal formula, obtained from D\pi=J_z^{-1}P_h, is

This is the normal trace computation used in the minimal-fitting argument of [19, Equation (7.9)]. At normalization a_i^{-2}, every error in (6.6) and (6.7) tends to zero in mass: use |z_i|\le Ca_i, \int e_i\,\dd\mu_i=O(a_i^2), and Cauchy–Schwarz for \int |z_i|\sqrt{e_i}\,\dd\mu_i.

Stationarity tested with J_{z_i}Y now yields the last identity in (6.3); the reference-volume integral of a divergence is zero. Testing instead with |z_i|^2J_{z_i}Y gives \nabla\nu=2f. Terms containing both B_i and |z_i|^2 disappear, and the differentiated height contributes 2z_i\cdot(P_T)_{vh}J_{z_i}Y; replacing J_{z_i} by the identity has vanishing normalized error. Finally, the normal test \varphi z_i, with \varphi a smooth base function, gives \int\varphi\,\dd d_0+\int\nabla\varphi\cdot\dd f=0 by (6.7). Thus all equations in (6.3) hold.

Applying signed excess at the height normalization

It remains to prove (6.4). There are two requirements for applying Theorem 3.2 at this normalization. First, its geometric error \Lambda_i must be o(a_i^2); in target units its height-square term is only O(a_i^2). We achieve this by changing physical units. Second, the physical chart radius and support height must be small relative to the local excess budget. We meet that condition by dividing the base into cells whose radii are powers of a_i. These choices do not require a rate relating a_i to the original target scales.

Fix a compact test support and an interior enlargement. Dilate physical space by d_i'\downarrow0 sufficiently slowly that all required finite geometric derivatives remain bounded and

Such a choice exists because each numerator divided by a_i tends to zero, and all the finitely many graph derivatives tend to their flat values. To obtain a uniform normal tube after dilation, write the graph as g_i and extend it using a fixed smooth cutoff \chi equal to one on the enlargement. The new physical graph and its derivatives are

The cutoff graph tends to zero through all the required finite orders, including order zero. Choosing d_i' still more slowly keeps these derivatives bounded and the second derivatives small, which provides the fixed tube. The mean-curvature bounds are needed only on the unchanged inner region. The cylinder restriction has a support gap at its artificial vertical boundary, so this operation introduces no first-variation term.

In the new physical units, the geometric measure \Lambda_i charges

Expressed with mass returned to target units, it therefore satisfies \Lambda_i=o(a_i^2). The measures M_i,D_i in target normalization are unchanged. Thus the geometric error will disappear after division by a_i^2; we now arrange the local hypotheses for signed excess.

Use \zeta and \kappa fixed in Equation (3.12), and cover the test support by cells of target radius \lambda_i=a_i^\zeta, with bounded overlap of their fixed enlargements. In particular 3\kappa<\zeta. On each enlarged cell use a normalized budget b_{i,j} for |M_i|+D_i+\Lambda_i, enlarged by fixed constants and with floor a_i^3. Thus, uniformly in the cells,

Fresh normal charts on these cells have physical radius O(d_i'\lambda_i) and physical support height O(d_i'a_i). Since 3\kappa<\zeta<1, these are at most b_{i,j}^{\kappa} for all large i, with the fixed factors absorbed by strictness of the inequality. All physical hypotheses of signed excess are satisfied.

Partition a nonnegative smooth test by a smooth partition of unity at scale \lambda_i. After the fixed normalization of its size and Lipschitz constant, the signed-excess error on a cell is o(1)b_{i,j}, uniformly because b_{i,j}\to0 uniformly. Multiplying by \lambda_i^m, summing (6.8), and dividing by a_i^2 proves 2\mathfrak m\le d_0.

For the absolute bound, apply Lemma 3.1, from [19, Lemma 2.5], in the same physical units on any fixed interior ball and its enlargement. Smooth mass averages identify its integer as Q for all large i. Since \Lambda_i=o(a_i^2), passage to the limit gives, after harmless fixed changes of radii,

This inflated-ball estimate implies domination as measures, without a doubling assumption. For a compact K\subset O\Subset\Omega, cover K by sufficiently fine equal-radius balls whose triple enlargements lie in O and have bounded overlap. Summing gives |\mathfrak m|(K)\le C d_0(O). Outer regularity as O decreases to K, followed by inner regularity, proves |\mathfrak m|\le C d_0. This completes the proof.□

Proof. Weak compactness and the stated conclusions about \sigma follow from the mass and ball bounds and from convergence of the varifold supports to \Pi. We prove (7.2) for each fixed 0<l<l_*, where l_* is chosen so all the following constructions have common interior room. All estimates below are uniform over X_0\in\spt\sigma_i.

A radial test adapted to the reference graph

Write w=X_0-\pi_iX_0 and translate the reference graph to S=S_i+w, which passes through X_0. By the height bound, |w|\le Ca_i. In this part of the proof, \pi,z,B,e,J_z refer to this translated graph, and \rho is geodesic distance on S from X_0, pulled back by \pi. Translation preserves its geometric estimates. Changing the reference tangent plane by this translation and the nearby projection changes squared tilt by at most

Indeed the two projections differ by O(a_i) and the second fundamental forms are uniformly bounded; squared tilt is comparable to the squared distance of the corresponding orthogonal projections.

Define

These fields are smooth at the central fiber, and Y\cdot G=d^2. Let W be the inverse volume density in geodesic normal coordinates on S centered at X_0, again pulled back by projection. Thus integration of W\vol_S is ordinary integration in those coordinates. Enlarging the sequence \varepsilon_i\to0 from (6.6) if necessary, geodesic coordinate estimates give

The last identity is exact: U is the coordinate radial field and W is the reciprocal volume density.

Apply (6.5) to U and add the normal trace (6.7). Their leading tilt terms cancel. The identity in (7.4) then cancels the reference-volume error. The result is

For completeness, pairing \nabla^SU-\Id with B costs C\varepsilon_i\rho^2e. Differentiating W in the horizontal part gives U\cdot\nabla_SW and another error of this size. Its vertical pairing costs C\varepsilon_i\rho\sqrt e\,|z|. The remaining terms are exactly those bounded in (6.6) and (6.7).

Put

Differentiate in \log t and test stationarity with W\Phi_t(d)Y. Since Y\cdot G=d^2, this gives

The test is compactly supported within the normal tube; the cutoff power p\ge4 gives more regularity than this calculation requires.

Here the estimate must hold down to arbitrarily small t, with no comparison between t and a_i. On d\le t\le l_*\le1,

The first inequality follows from |Y-G|\le C\varepsilon_i|z|\rho. For the second, take the vertical component of G and use |P_vP_T|\le C\sqrt e and |G|\le Ct. Set b'=|(I-P_T)G|/t. Then both |z|\le t and |z|\le t(b'+C\sqrt e) hold. In particular,

Using \Phi_t\le C\Psi_t, every adverse term in (7.6) is therefore at most C\varepsilon_i t\Psi_t((b')^2+e+a_i^2). Absorb the (b')^2 term into the positive term from (7.7). Equation (7.3) and comparison of d with ambient distance yield

Averaging over the centers

Let c_\Phi=\int_{\R^m}(1-|y|^2)_+^p\,\dd y. At the center X_0, the squared radius d^2 agrees with the squared ambient distance to leading order and W(X_0)=1. Since X_0 has density Q,

Take any bounded nonnegative continuous center weight g. Average (7.8) against g\,\dd\sigma_i and integrate in logarithmic radius. Fubini’s theorem and the upper ball bound for \sigma_i give

The first integral is over fixed interior room and is O(a_i^2). The second is finite precisely because s>m-1. To justify the lower endpoint, first integrate between positive radii, then use Fatou’s lemma for the nonnegative functions gI_i and the displayed density limit. This avoids any uniform rate of density convergence in X_0.

The averaged inequality now gives a lower bound with error o(a_i^2). To obtain the asserted comparison, we need an upper bound for the same integrand in terms of the original mass defect and squared height. Translating the reference graph introduces a cross term and an additional negative square in this upper expansion. The vanishing first moment removes the cross term, and the negative square can be discarded.

Expansion at a fixed radius

Keep l>0 fixed. Write y=\pi_iX and z_i=X-y. After translating the new base back to S_i, its projection and residual are

All relevant support points satisfy |z_i|+|z'|\le Ca_i in common interior room. Functions such as q(y)=W(y+w)\Phi_l(\rho(y+w)), and the analogous function with \Psi_l, have uniform C^2 bounds and compact interior support. We claim that, uniformly in the centers,

Indeed the nearest-projection formula and smooth convergence to a plane give

For the first estimate, D\pi_i(X)=J_{z_i}^{-1}P_h(y) differs from P_h(y) by O(\varepsilon_i a_i), and the second derivative of \pi_i is o(1). For the second, w is normal to S_i at the center and all tangent planes tend uniformly to \Pi. Taylor expansion thus leaves only -\int\nabla^{S_i}q\cdot P_hw\,\dd\mu_i, up to o(a_i^2). Replacing the projected measure by Q\vol_{S_i} costs o(a_i^2) because |M_i|=O(a_i^2). Integration by parts then uses

to prove (7.10). Compact support eliminates all boundary terms.

Taylor expansion in the squared normal height gives, uniformly on the test region,

The normalized reference-volume integral of W\Phi_l(\rho) is exactly c_\Phi, by its geodesic-coordinate definition. Hence (7.10) expresses the zeroth-order term of I_i-Qc_\Phi as an integral against M_i, with error o(a_i^2).

For the height term, z'=z_i-w+o(a_i) uniformly. Put q_\Psi(y)=W(y+w)\Psi_l(\rho(y+w))\ge0. Moving its argument from y' to y costs o(a_i^2) in the height-square integral, and

This integral enters I_i with the negative coefficient -l^{-m}/(2l^2), so discarding the |w|^2 term gives an upper bound. The mixed term is o(a_i^2) by the centering assumption. We spell out its uniformity, since the centers move with i. The vector measures a_i^{-1}(\pi_i)_\#(z_i\mu_i) have uniformly bounded total variation and converge weakly to zero. At fixed l, the above center-dependent tests and their smooth coefficient factors form a relatively compact family in the uniform norm on common interior room. A finite approximation of this family therefore makes the weak convergence uniform over all its members. Multiplication by the bounded vectors w/a_i preserves this conclusion; these vectors need not converge.

It follows that, after division by a_i^2, the limiting upper bound for I_i(X_0,l)-Qc_\Phi at a center tending to x\in\Pi is

More explicitly, let \operatorname{pr}_{\Pi} denote orthogonal projection onto the limiting plane. The same compact-test argument applied to the bounded measure sequences in (6.1) gives

The function F_l is continuous. Combining this uniform upper bound with (7.9), and passing to the limit against any nonnegative continuous g, gives \int gF_l\,\dd\sigma\ge0. Continuity implies F_l\ge0 on \spt\sigma, which is (7.2). Since l\in(0,l_*) was arbitrary, the comparison holds for every such radius.□

Proof. Suppose that \nu vanishes on a nonempty open subset. Because \nu\ne0 and \Omega is connected, choose a point x outside \spt\nu so close to its boundary that

Then H(l)=0 for l\le l_0 and H(l)>0 for l_0<l\le l_1, where l_1>l_0 is fixed and interior. Put t=\log l, t_j=\log l_j, and on (t_0,t_1] define

By (8.7),

For P\ge1 the right-hand side is nonnegative. Therefore, going inward from t_1, P(t)\le\max\{1,P(t_1)\}.

On B_{l_0}(x), the distributional identities give f=0 and d_0=0, and (8.4) gives \mathfrak m=0. The compact kernels converge uniformly and vanish on the boundary sphere, so A(t)\to0 as t\downarrow t_0, even if \mathfrak m charges that sphere. Moreover,

because q-n\ge n^2-n\ge-1/4. Since \dot H=2D\ge0, integration from t_0 yields

Thus P is bounded on this finite interval. Using q\ge n^2 in the exact equation for \dot P now gives

Integrating from t_0+\varepsilon to t_1 bounds \int_{t_0+\varepsilon}^{t_1}n^2\,\dd t independently of \varepsilon. In particular n is integrable down to t_0. But (\log H)^{\displaystyle\cdot}=2n, which prevents H from tending to zero there. This contradiction excludes every nonempty open zero-height region. Since the kernel is positive in its open ball, the stated conclusion follows.□

Proof of Proposition 8.1. The preceding lemma makes P(x,l)=A(x,l)/H(x,l) well defined at every interior ball. In the harmonic-map model from Section 6, this quotient is half the weighted Dirichlet frequency

For a nonzero harmonic map homogeneous of degree d about x, the flux identity in (8.7) makes this frequency equal to d, so P=d/2. For the measure data, constant frequency at the retained centers, together with the lower bound 2P\ge1 supplied by their density comparison, will force the radial measure identity used below.

At a center in Z, the additional hypothesis (8.5) gives 2P\ge1 for small radii. Hence (8.8) improves to

In particular the finite limit P(x,0+) exists and is at least 1/2.

Suppose that \dim_{\Hh}Z>m-1, and choose m-1<s'<\dim_{\Hh}Z. We may first restrict to centers in a compact interior region for which (8.5) holds on one common interval 0<l<l_*. To justify this restriction, exhaust \Omega by compact sets with positive boundary distance and impose the comparison for all rational radii below 1/k. The resulting sets are Borel by continuity of the kernels, their union is Z, and continuity in radius supplies the comparison at every radius. One of them still has dimension greater than s'.

The compact-subset theorem and Egoroff’s theorem now give a compact set K of positive finite \Hh^{s'} measure on which

Indeed first apply Egoroff at dyadic radii; monotonicity supplies the same conclusion between them. Since each function P(\cdot,l) is continuous, its uniform limit is continuous on K. A further compact restriction gives

Choose x\in K with positive upper s'-density, and radii l_j\downarrow0 such that \Hh^{s'}(K\cap B_{l_j}(x))\ge c l_j^{s'}.

We next normalize the measure tuple at these scales. Fix an open neighborhood \Omega'\Subset\Omega of x, so that the constant in (8.4) is fixed on \Omega'. Put T_j(y)=(y-x)/l_j and N_j=l_j^m H(x,l_j). On the expanding domains T_j(\Omega) define

The identities and inequalities (8.2)–(8.4) are invariant under this normalization, and the new height at center zero and radius one equals one.

Here are the local mass bounds needed to pass to a limit. At the fixed center x, P is bounded on 0<l<l_*/2. The exact quotient equation and n\le\sqrt q therefore give

On every interval of radii with bounded ratio, this bounds \int q\,\dd\log l. Since \dot H/H=2n\le2\sqrt q, Cauchy–Schwarz gives two-sided comparison of H on the same interval.

To see the resulting mass estimates explicitly, fix R>0 and discard finitely many j so that all balls used below lie in \Omega'. Comparison on the interval from \min\{1,R\}l_j to \max\{1,4R\}l_j gives H(x,\tau l_j)\le C_RH(x,l_j) for R\le\tau\le4R. The kernel \psi_{x,2Rl_j} is at least c_Rl_j^{-m} on B_{Rl_j}(x); hence

The integrated bound for q on [2Rl_j,4Rl_j] supplies \tau_j\in[2R,4R] with n(x,\tau_jl_j)\le C_R. The kernel \phi_{x,\tau_jl_j} is also at least c_Rl_j^{-m} on B_{Rl_j}(x), so the definition of D=nH gives

Positivity bounds \beta_j by its trace d_j, and Cauchy–Schwarz for (8.2) bounds f_j. Finally, |\mathfrak m_j|\le C_{\Omega'}d_j on every fixed ball for all sufficiently large j. In particular the same constant is available on all fixed balls of the eventual limit.

Pass to a diagonal subsequence with local weak limits on \R^m, and denote them by

They satisfy the same measure identities and inequalities. The normalization gives H_\infty(0,1)=1, so \nu_\infty\ne0 and Lemma 8.2 applies on all of \R^m.

The rescaled center measures

have a nonzero weak limit \lambda on \overline B_1, with the upper ball bound (8.11). Consequently \spt\lambda spans \R^m. Otherwise its bounded support lies in an (m-1)-dimensional linear space and can be covered by O(\varepsilon^{-(m-1)}) balls of radius \varepsilon. The ball bound would give \lambda(\R^m)\le C\varepsilon^{s'-m+1}\to0, contradicting its nonzero mass.

We have now retained enough centers in the second blow-up. Its remaining feature is that they all have the same constant frequency. If z\in\spt\lambda, choose y_j\in K with T_j(y_j)\to z. At every fixed radius r>0,

This follows from (8.10) and continuity of the limiting frequency on K. Local mass bounds and continuous compact kernels permit passage to the weak limits even with these moving centers. Since the limiting height is positive,

The comparison inequality also passes to these centers.

Equality at independent centers

Fix z\in\spt\lambda\cup\{0\}. In the limit, (8.9) and (8.13) give

As n\ge2P_0\ge1, this forces n=2P_0=:n_0, including the case P_0=1/2. The exact quotient equation then gives q=n_0^2, or equivalently D=n_0H and R=n_0^2H.

Let \mathcal M_\infty be the positive matrix measure in (8.2) for the limiting data. Contract it with (-n_0,y-z) and integrate against \psi_{z,r}. The result is

With respect to the trace measure, a positive semidefinite matrix whose quadratic form vanishes on a vector annihilates that vector. Since \psi_{z,r}>0 on B_r(z), the first row therefore gives

The radius is arbitrary, so this is an identity on \R^m. Subtracting the identity for center zero yields z\cdot f_\infty=0. The support of \lambda spans \R^m, and hence f_\infty=0. The identity at zero then gives n_0\nu_\infty=0, contradicting n_0\ge1 and H_\infty(0,1)=1. This proves the proposition.□

Proof of Theorem 1.1. Suppose that \dim_{\Hh}\Sing(V)>m-1, and choose m-1<s<\dim_{\Hh}\Sing(V). Proposition 2.1 gives the fixed singular point, flat target scales, and center measures satisfying (2.3). Proposition 5.1 supplies the centered, nonzero height normalization at these scales. Proposition 6.1 then gives on the connected base ball B_4\subset\Pi the nonzero measure tuple satisfying (6.2), (6.3), and (6.4). These are precisely the hypotheses (8.2)–(8.4).

Pass to a weak limit \sigma of the center measures. It is nonzero, is supported in \Pi\cap\overline{\mathbf B}_1, and retains the upper ball bound with exponent s. The mass distribution principle therefore gives \dim_{\Hh}\spt\sigma\ge s. Proposition 7.1 supplies (7.2) at every point of \spt\sigma, which is (8.5). Applying Proposition 8.1 with Z=\spt\sigma gives

a contradiction.

For sharpness, take two distinct m-planes intersecting in an (m-1)-plane, each with multiplicity one. Their sum is stationary by additivity of first variation. Away from the intersection it is locally a single smooth plane, whereas at the intersection its support is not a smooth embedded m-submanifold. Its singular set therefore has dimension exactly m-1.□