Power-law violations of Yau’s nodal upper bound

Proof. Begin with \widetilde h=-a_*t. Direct differentiation on the round sphere gives

At every point there is a nonzero tangent vector in \ker\dd y, because \dd y:T_xB\to\R^3 has a four-dimensional domain. A round unit vector e in this kernel satisfies \dd\widetilde h(e)=0 and (\nabla_{g_0}^2\widetilde h)(e,e)=2a_*t. Since the Hessian in (3.1) has norm at most 2a_* and the squared gradient has norm at most a_*^2, on t\ge1/4 we obtain

after choosing a_*^2\le1/8.

At a point x_* with t=1, the Hessian has eigenvalues 2a_*,2a_*,0,0, and the gradient is zero. On a small neighborhood of x_* contained in \{t>1/2\}, two Hessian eigenvalues therefore have a uniform positive lower bound and the gradient is arbitrarily small. In round normal coordinates \xi at x_*, add \varepsilon|\xi|^2/2 times a fixed smooth cutoff equal to one near \xi=0. Choose \varepsilon>0 sufficiently small that this is a C^2-small perturbation on its support. The resulting function h_0 has positive definite Hessian at x_*.

Here is why the cutoff region retains the required test. Shrinking its support and then the perturbation if necessary, the perturbed Hessian has two eigenvalues at least c a_*, its norm is at most C a_*, and |\nabla_{g_0}h_0| is sufficiently small that C a_*|\nabla_{g_0}h_0|^2<c a_*/2. The positive two-plane intersects \ker\dd h_0 in dimension at least one. A unit vector in that intersection gives J_{h_0}\ge c a_*/2. This argument also applies at a critical point. Outside the perturbation support the previous \ker\dd y test applies.

These conclusions persist in a sufficiently small common C^1 neighborhood of g_0. Indeed, the difference of the two Hessians of a fixed function is the connection difference contracted with its differential, and all metric and gradient comparisons use only C^1 bounds. Outside the perturbation support, the condition \dd y(e)=0 still implies \dd h_0(e)=0 independently of the metric. On the support, the positive two-plane and small-gradient estimates just established are uniform under a small perturbation.

Finally, positive definiteness at x_* holds on a neighborhood of that point. There is a nearby point with nonzero differential: otherwise h_0 would be constant on that neighborhood. Choose a relatively compact neighborhood U_0 of such a point where the Hessian remains positive definite and the gradient stays separated from zero. Choose U_0\Subset V\Subset\Omega\subset\{t>1/2\} in a fixed coordinate chart, and shrink \mathcal G once more. All the stated constants can then be chosen uniformly. Notice that U_0 is chosen away from the critical point x_*.□

Proof. The pattern will depend on three variables transverse to the parent gradient frozen at its center. Radial wells in these variables have two positive tangential Hessian directions; a small amplitude controls their potentially negative radial contribution to J_v. We first choose the wells and smaller concentric balls that will supply the repeatable gain. We then place this pattern in the parent and verify the Hessian test, including where the parent cutoff varies.

The periodic wells. Choose a smooth function F on [0,1], positive on (0,1), equal to v near v=0, and flat at v=1. Inside a unit cube in \R^3 take finitely many balls with pairwise disjoint closures and radii R_i>0. On a ball, at radius r from its center, set

Extend \psi by zero outside the balls and periodically to \R^3. It is smooth: near a center it is a constant plus r^2/(2R_i), and it is flat at the outer boundary. The extended value of u_0 at the center is 1/R_i. Write \sigma for the radial unit vector. Then

The first constant in these inequalities is independent of the radii, because, for v=r/R_i,

The second follows from f^2/u_0=R_i vF(v) and the bounded radii. Set f,f',u_0 equal to zero outside the balls. These coefficients are continuous, and

At a center f'=u_0, so any choice of \sigma gives the same Hessian in (3.5); outside the balls choose \sigma arbitrarily as well.

Choose a small amplitude a>0. Its smallness will use only F and the radius-independent first bound in (3.5). In particular it can be chosen before the packing. The mean gradient factor on any one well is

Put \eta_0=(\mathcal W(a)-1)/8. Choose the finite packing with volume fraction \varpi such that

Such a packing exists. To see this without requiring an infinite packing in the construction, fill a fixed positive fraction of any remaining open set by finitely many closed grid cubes lying inside it, and put a ball with strictly smaller radius than the inradius in each cube. This captures a fixed positive fraction of the remaining volume. Finitely many repetitions make the uncovered volume as small as desired, with all ball closures still disjoint.

Choose \eta>0 so small that, on the core balls r<(1-\eta)R_i, the weighted integral per period satisfies

Choose an inner cube fraction q\in(0,1) such that

Here q is a four-dimensional volume fraction. Fix a smooth cutoff on the unit parent cube which equals one on the concentric inner cube of fraction q, is supported in the parent interior, and is flat wherever it is zero. Products of one-dimensional plateau cutoffs give such a function. The wells, cores, and cutoff are now fixed. All their derivative bounds are finite, although no bound uniform in derivative order is asserted.

Normalization on a prepared parent. Take affine coordinates z=(z_1,z') centered at x_P, making the metric Euclidean at the center and \nabla h(x_P)=s\partial_{z_1}. The linear maps and their inverses have uniform bounds; the image of P is a parallelepiped of uniformly bounded distortion. The coordinate identity relating D(\dd h) to the covariant Hessian and the bounded connection gives

on P, since S\le1. Integration along segments of length O(\nu S) yields relative upper and lower bounds for the gradient and then

The constants here may depend on the preparation constants. Only the bounds already specified in the lemma enter them.

Rescale the fixed parent cutoff to P and call it \chi. Set

In the following estimates the radial coefficients are evaluated at z'/\ell. A cutoff derivative at its parent scale is denoted D_{\mathrm{sc}}\chi; thus an ordinary derivative of \chi is (\nu S)^{-1}D_{\mathrm{sc}}\chi. The bounded affine changes only alter the fixed bounds for these scaled derivatives. Since \ell/(\nu S)=\nu, Equation (3.9) gives

This proves the gradient comparison in (i) for fixed \alpha,\beta, when \nu is sufficiently small.

The new Hessian contribution. Write

where e' denotes the static transverse components in the affine coordinates. The two defining linear conditions are independent by (3.10), so \mathcal F is two-dimensional. For a metric unit vector e\in\mathcal F,

Indeed e_1=O(\nu) follows from the two linear conditions. The second estimate follows from (3.10) and G^{ij}=\delta_{ij}+O(\nu), using a denominator at least a fixed multiple of s. We may take C_0 absolute by first taking \nu small for the fixed data. These arguments are uniform even when s is arbitrarily small or large.

Let \Pi_\sigma be Euclidean projection onto \sigma^\perp in the transverse space. The leading contribution of v-h to its Hessian test on D,e, divided by s/\ell, is

By (3.11), this is at least

At the initial choice of amplitude, impose 2C_0^2a^2\sup_v vF(v)|F'(v)|\le1/4. The main term is then at least a\chi u_0/4-Ca\nu^2\chi|f'|.

The mixed cutoff terms on e,e vanish because \sigma\cdot e'=0. On D,D their size is bounded by

The pure cutoff terms have size at most C\nu^2|\psi||D_{\mathrm{sc}}^2\chi|. Finally, replacing the coordinate Hessian of v-h by its covariant Hessian gives an error of size

in the same units, because the connection is uniformly bounded.

We verify uniformity of these errors also where the favorable coefficient vanishes. Set X=\chi u_0. On a fixed compact parameter space, if continuous functions X,b\ge0 satisfy b=0 on \{X=0\}, then

For X\ge\tau the quotient is at most \nu^2\sup b/\tau; for X<\tau it is at most \sup_{X<\tau}b, which tends to zero as \tau\downarrow0. The compact parameter space here consists of cutoff position, period position, and bounded affine frames. Taking its closure is harmless. Transformed cutoff derivatives are continuous on this space and have uniform bounds.

Apply (3.12) to b=\chi|f'|, b=f|D_{\mathrm{sc}}\chi|, and b=|\psi||D_{\mathrm{sc}}^2\chi|. Their vanishing follows from (3.6) and the flatness of \chi on its zero set. The other mixed term satisfies \nu\chi f^2|D_{\mathrm{sc}}\chi|\le C\nu X. Moreover, \chi f\le C\sqrt X and \ell\le\nu^2 give

We have proved, uniformly in all prepared parents and allowed tests,

for a fixed c_a>0. No positive lower bound on a nonzero value of \chi or u_0 has been used.

Retaining one direction everywhere. The old Hessian has norm O(\nu^2) after division by s/\ell. Thus the preceding estimate compares a favorable new contribution of size X with inherited terms of size \nu^2. When X dominates \nu^2, the new favorable directions suffice. When X=O(\nu^2), the gradients are close enough to carry an old favorable direction to the new tangent space. To obtain one uniform lower bound, consider any sequence \nu\to0 of admissible configurations. If X/\nu^2\to\infty along a subsequence, the positive term in (3.13) dominates the old Hessian in every unit direction of \mathcal F.

If instead X/\nu^2 stays bounded, then \chi f\le C\sqrt X=O(\nu), so |\nabla v-\nabla h|\le\varepsilon_1|\nabla h| with \varepsilon_1=O(\nu). This also makes their normalized gradients close uniformly for unbounded slopes. In fact, for T(p)=p/\sqrt{1+|p|^2} and |q-p|\le\varepsilon_1|p|, \varepsilon_1\le1/2, the segment from p to q stays outside the ball of radius |p|/2. Therefore

There is a metric orthogonal rotation R=I+O(\varepsilon_1) taking \ker\dd h to \ker\dd v. Rotate the old good two-plane E in (3.2) by R. Inside the three-dimensional space \ker\dd v, the planes RE and \mathcal F intersect in dimension at least

For a unit vector e in their intersection, compare the old test on p/Q,R^{-1}e with the old Hessian evaluated on D,e. Equation (3.14) shows that the change costs o(\nu^2) in the normalized units. Thus the old contribution is at least \tfrac12c_{\mathrm p}\nu^2 for small \nu. The new contribution in (3.13) is bounded below by -o(\nu^2) in this bounded-ratio regime.

These alternatives prove the uniform margin c_{\mathrm p}\nu^2/8 in the normalized units for some test when \nu is sufficiently small. Indeed, a contrary sequence has a subsequence on which X/\nu^2 is bounded or tends to infinity, and both alternatives just excluded failure of that margin. Multiplication by s/\ell and (3.8) prove the Hessian assertion in (i). This is where base dimension four is used: in dimension d the corresponding intersection lower bound is d-3.

The region prepared for repetition. Inside the inner parent cube where \chi=1, keep the points for which z'/\ell lies in a core ball. On this region u_0 has a fixed positive lower bound. Equations (3.13) and (3.8) imply, for every unit e\in\mathcal F,

The leading constants depend only on the fixed pattern and the uniform metric bounds. Gradient comparison now permits fixed thresholds C_{\mathrm p}^* and c_{\mathrm p}^* such that, for C_{\mathrm p}\ge C_{\mathrm p}^* and 0<c_{\mathrm p}\le c_{\mathrm p}^*, this region satisfies (3.2) at scale \ell once \nu is small. In particular the old term O(C_{\mathrm p}\nu^2) is absorbed after C_{\mathrm p} has been fixed.

Select all closed child cubes contained in this open repeat region. Since \nu is a reciprocal integer, the parent has an exact partition into cubes of side \nu\ell. These children satisfy the preparation condition. It remains to show that their total gradient integral exceeds that of the entire old parent. To evaluate their mass, first tile the affine z coordinates by period cells of side \ell, using a dummy period in z_1. The child grid uses the original coordinates, whereas these period cells use the affine coordinates; the two grids need not align. The cells meeting the inner parent boundary have relative volume O(\ell/(\nu S))=O(\nu), uniformly over the bounded affine distortions. The points lost by retaining only whole child cubes lie in a layer of width O(\nu\ell) along the core and inner parent boundaries. In each rescaled period the core boundaries have finite area, so this gives another relative O(\nu) loss. The constants may involve the smallest radius in the finite fixed packing.

On the repeat region,

The constant affine Jacobian cancels on dividing by the parent volume. Consequently

Choose the final \nu small enough that both errors are at most \eta_0. By (3.7), the ratio of new selected mass to old entire parent mass is then greater than

Fix \zeta=\eta_0/(1+\eta_0)>0. This proves (3.3); taking \nu smaller still preserves this value of \zeta.

Derivatives and the order of choices. Differentiating (3.9) in the fixed coordinates gives (3.4). In a term with q derivatives on \chi, the scale factor is

All derivatives of the pattern and all powers of the bounded affine matrix enter the fixed constant C_j. The center slope and the affine matrix are constants in the expression being differentiated.

To summarize the choices within this proof, first fix F and its small amplitude a, then the finite packing, cores, and parent cutoff with the slack in (3.7). Choose the preparation constants next, large and small respectively, and finally choose a reciprocal \nu satisfying the local positivity, repeat preparation, and mass-error requirements. This proves the stated uniformity and completes the lemma.□

Proof. Choose C_{\mathrm p} large enough and c_{\mathrm p}>0 small enough for both Lemma 3.2 and the seed preparation following Lemma 3.1. Choose the final \nu sufficiently small for the former lemma and so that a fixed cube P_0 of side \nu has closure in U_0. All these choices are uniform for the metrics in \mathcal G. Put r=\nu^2.

Start with the prepared family \mathcal P_0=\{P_0\} for h_0. At generation i\ge1, apply Lemma 3.2 on every cube of \mathcal P_{i-1}, using parent scale S_i=r^{i-1}, and call the resulting function h_i. The selected child cubes form \mathcal P_i; their sides are \nu r^i and they are prepared at scale r^i. Every generation consists of finitely many cubes with disjoint interiors. Each perturbation is supported strictly inside its parent, so each h_i is smooth, including across all cube boundaries. Nothing outside P_0 changes.

Let

Equation (3.3) gives

The initial mass I_0 has a positive uniform lower bound by Lemma 3.1 and the fixed volume of P_0.

We give the derivative accounting for all orders with one growth constant. Let \beta\ge1 and 0<\alpha\le1 be the gradient comparison factors of Lemma 3.2. The largest gradient after i generations is at most a fixed multiple of \beta^i. On P_0 every point has gradient at least s_{\min}\alpha^i, since a point is changed at most once per generation. At generation i, the center slope of each parent is therefore at most C\beta^{i-1}. By (3.4), disjointness within that generation gives

There is no factor counting cubes in this supremum norm estimate. There is also no product of higher pattern derivative constants over generations: the increments are added, and their individual formulas involve frozen slopes and affine matrices. Passing to the fixed finite atlas only adds fixed chain-rule constants and lower-order derivatives, which satisfy the same upper bound. Summation yields

where (j-1)_+=\max\{j-1,0\} and one finite C_* works for every j\ge0.

The lower Hessian margin has the same exponential control. At a point outside P_0, the seed margin remains unchanged. At a point in P_0, consider the last generation i\le m that changes the function there. If no such generation occurs, use the seed margin. Otherwise Lemma 3.2 bounds its retained margin below by

Later perturbations vanish near that point, or all their jets there vanish at their support boundaries, so this margin persists. By increasing C_* if necessary,

This argument requires no nonzero-gradient assumption outside P_0; critical points there are covered by the seed lemma.

Fix \theta>0 so small that C_*\theta\le\delta, and for each k\ge2 take

Equations (3.18) and (3.19) give the first two estimates in (3.15). Equation (3.16) gives the third, since (1+\zeta)^m\ge(1+\zeta)^{-1}k^\gamma. Adding bounded constants affects only the zeroth-order bound, and h=-a_*t+c_k on t\le1/2 because both the seed perturbation and all corrugations lie above that set.

For clarity about subsequent uses, the constants determining C_*, \theta, and \gamma use only the fixed pattern, the seed, and common C^1 metric bounds. Higher derivatives of an individual input metric may enter constants in a later smooth coordinate change, but cannot alter these exponents. Indeed, for any fixed such coordinate change, an order-j derivative of the pullback is a finite sum of derivatives of h of orders at most j multiplied by fixed smooth coefficients. Its growth is still bounded by C_jk^{\delta(j+1)}. Each finite order may have its own constant and frequency threshold. This establishes the common exponent for arbitrarily high finite accuracy and completes the proof.□

Proof. For b>0, the first identity in (4.1) is equivalent to

The two prescribed columns are compatible with symmetry precisely when

First suppose that b\ge k^{-12\delta}, and set

Then (4.2) holds. To give the extension of D explicitly, write its blocks relative to \R e_1\oplus\R v\oplus E, where E=\{e_1,v\}^{\perp}. Denote the corresponding blocks of U by u_{11},u_{12},u_{1E},u_{2E},\ldots. Take

The identity u_{22}=-b^2u_{11} verifies both required columns. Moreover,

These imply (4.1) for large k.

Now suppose that 0\le b<k^{-12\delta}. Put K=k^{14\delta} and take

The characteristic identity follows directly. These formulas also give D=0 and Uv=0 when b=0, so no division by a small b is involved in this case.

For the domination inequality, split H-U-k^{-3\delta}I into \R v\oplus v^\perp. Its v^\perp block is at least \tfrac12KI for large k. Its scalar v block is at least

because

The Schur complement subtracts at most C|H|^2/K\le Ck^{-6\delta}. The positive k^{-2\delta} term dominates every subtraction. The matrix is therefore positive definite for large k. Finally, |U|\le Ck^{14\delta} and |D|\le Ck^{2\delta}, proving the remaining bound.□

Proof. Choose an index i with |q_i|\ge c/2, write q\cdot\partial=q_i\partial_i+D', and let I_i denote integration in X_i with zero integration constant. On \mathcal P_m, set

Each application of \mathcal K lowers the degree in the variables other than X_i. Thus \mathcal K^{m+1}=0 on \mathcal P_m. Since

the finite geometric sum gives (q\cdot\partial)R_{q,m}=I. The stated coefficient bound follows from the finite formula and the bounds on q_i^{-1} and q. The four possible choices of i have the same uniform bound.□

Proof. We divide the construction into the selection of jets, finite polynomial equations, and localization estimates.

Normal coordinates and the linear jets. For each center choose a G-orthonormal frame and use normal coordinates \xi in that frame. Put

In these coordinates the tangent and cotangent components agree at the center. The envelope bounds, with a little slack in their exponents, give

for large k.

If p\ne0, set e_1=p/|p|, and choose a favorable unit test e_2 from (3.15). If p=0, first choose such an e_2, then choose any unit e_1\perp e_2. Complete to an orthonormal basis e_1,e_2,e_3,e_4, and put

These three vectors span e_1^\perp with least spanning singular value at least c\eta_k.

Use the four pairs

For each pair define

Then p'=|p'|e_1, P\cdot P=-1, and

Changing e_2 to a v_j changes its Hessian value by at most Ck^{-5\delta}. If a=|p|, the function a^2/(1+a^2) changes by at most 2k^{-3/4} when a is increased by Qk^{-3/4}. Indeed its derivative at s\ge a is at most 2(1+a^2)^{-3/2}. The additional error in the Hessian test is at most Ck^{-3/4+4\delta}. Both errors are negligible compared with the original ck^{-\delta} margin. Consequently

for every selected pair. Apply Lemma 4.1 to obtain U,D. Multiplying its characteristic identity by Q' gives (U+iD)P=0.

Frozen charts and scaling. The choice of frame may depend on k, but each packet uses one fixed chart when spatial derivatives are taken. More explicitly, if E is its chosen frame, the chart map is

For each fixed derivative order q, the derivatives of these maps and their inverses are uniformly bounded over all centers and orthonormal frames on sufficiently small common coordinate balls. This follows from smoothness and compactness of the orthonormal frame bundle of the fixed metric. In particular, the chain rule gives

There are no spatial derivatives of the rule used to select E.

Set

For every fixed q\ge2,

All these derivatives are bounded independently of k,z. The constant may depend on q and on high derivatives of the fixed metric.

In normal coordinates write L=g^{ij}(\xi)\partial_{\xi_i\xi_j} +\beta^i(\xi)\partial_{\xi_i}. After multiplication by r_0^2, the operator L+k^2 becomes

Its coefficients have uniformly bounded derivatives of every fixed order on a fixed X-ball. The normal-coordinate identities are g^{ij}(0)=\delta_{ij} and \partial_{\xi_\ell}g^{ij}(0)=0.

Finite eikonal and transport equations. Choose an integer T\ge3, to be specified below. We construct a complex polynomial \Phi with

such that

vanishes to order T at zero. Here vanishes to order T means that all derivatives of order less than T vanish. The constant equation follows from P\cdot P=-1, and the linear equations follow from (U+iD)P=0 and the vanishing first derivatives of the metric.

At homogeneous degree m\ge2, the as-yet unspecified homogeneous term \Phi_{m+1} enters the equation through 2(P/Q)\cdot\partial\,\Phi_{m+1}. All other terms have already been determined. The Hermitian norm of P/Q lies between 1 and 3 for large k, by (4.7). Lemma 4.2 therefore solves each equation with a uniformly bounded right inverse. Solving through degree T-1 gives the desired polynomial. Its coefficients are bounded for each fixed T: the prescribed Hessian is bounded by k^{-80\delta}, the coefficient jets of \mathcal L are bounded, and only finitely many bounded algebraic operations are used.

Define

Fix an integer s\ge0. Choose polynomials a_0,\ldots,a_s such that

and each of

vanishes to order T at zero. To see this directly, solve \mathcal Ta=f by homogeneous degrees. The new amplitude term of degree m+1 enters the degree-m equation through 2(P/Q)\cdot\partial; the remaining terms use only coefficients already chosen. Lemma 4.2 again applies. First construct the entire a_0, then a_1, and so on. Thus the second derivative in \mathcal La_{j-1} causes no circular dependence. Constants at zero are free, as stipulated above. For fixed T,s, all polynomial coefficients are bounded uniformly in k,z. No bound uniform as T,s\to\infty is asserted or needed.

Choose once a real smooth cutoff \omega, equal to one on \{|X|\le1\}, supported in \{|X|<2\}, and taking values in [0,1]. Define

and extend w by zero. For large k, its support is compactly contained in the normal chart, so this extension is smooth and has the asserted support radius. On |X|\le2k^{-1/3},

where the constant depends on T,s. This follows from the chosen constant terms and \kappa\ge1.

Localization. The linear part of \operatorname{Re}\Phi-h_{\rm sc} is either zero or k^{-3/4}e_1\cdot X. Lemma 4.1, (4.6), and (4.9) imply

The third derivative bounds used here are uniform for each fixed construction order. On the packet support the cubic term can be absorbed into half the quadratic deficit, since k^{-1/3+106\delta}\to0. The maximum of the remaining linear term minus the quadratic term is O(k^{-3/2+106\delta}). Also

Consequently

for large k.

On the smaller annulus \tfrac14k^{-1/3}\le|X|\le2k^{-1/3}, the negative exponent has magnitude at least ck^{1/3-206\delta}, whereas the positive linear exponent is at most Ck^{-1/12}. Since 1/3-206\delta>1/4, a larger threshold gives

Residual and derivative estimates. For the uncut expression, direct differentiation gives

Taylor’s theorem bounds a remainder vanishing to order T, after q\le J\le T derivatives in X, by C|X|^{T-q}. On the support this is at most Ck^{-(T-q)/3}. Each ordinary derivative of such a remainder therefore has a cost at most k^{1/3+100\delta}, relative to its zeroth-order Taylor bound. A derivative falling on the exponential, amplitude, or cutoff has a cost at most a fixed constant times

Higher coordinate derivatives obey the same conservative k^2 cost per derivative, by the chain rule and the fixed normal-chart bounds.

Undoing the operator scaling costs r_0^{-2}=k^{200\delta}. Thus, for 0\le j\le J, the uncut residual divided by e^{kh} is bounded by

The commutators with the cutoff, and any prescribed derivatives of them, have polynomial factors times (4.13). They are smaller than every inverse power of k.

For example, it suffices to choose integers T,s with

These choices are made before the frequency threshold. Increasing that threshold absorbs all their finite constants and proves (4.4). The same derivative calculation without the operator factor proves (4.3) for every fixed j, with exponent 2j independent of T,s.

Real jet spanning. At the center, a(0)=1 and the cutoff is constant, so

The error is bounded by Ck^{-1+100\delta}. Before this error, the real parts contain the two vectors

Their coefficients are bounded, and their least spanning singular value on the value–e_1 plane is at least ck^{-3/4}. The imaginary parts span the orthogonal e_1^\perp slope space with least singular value at least ck^{-9\delta}. Together their least spanning singular value is at least ck^{-3/4}. Since k^{-1+100\delta}=o(k^{-3/4}), the actual center jets retain this bound, with a smaller constant. The argument applies without change at p=0.

Finally, (4.3) through order two and the first envelope derivative bound give

near the center. In differentiating this expression Q_z remains fixed, while differentiating e^{-kh(x)} costs k|D_xh|\le Ck^{1+2\delta}. A displacement of O(k^{-10}) therefore changes the jet matrix by O(k^{-7}). Uniformly bounded changes between ordinary and normal coordinates preserve these estimates. This is negligible relative to k^{-3/4}, and proves the second inequality of (4.5), after increasing the threshold.□

Proof. The bounds on the first two derivatives of h imply

in the fixed chart. A packet centered at z that meets the ball about x has

For large k, such a z, and the whole ball, belong to a compactly contained part of \Omega. Normal coordinates centered at z are defined throughout the ball. The preceding estimates give

In particular, Q_z/Q_x is bounded above and below independently of the packet. The same comparison holds for Q throughout the ball. Since |\dd h|\le CQ in the coordinate chart,

The nearest center has |X_z(x)|=O(k^{-10+100\delta}). The center formula and its nearby-jet estimate in the proof of Proposition 4.3 show that one of its packets has |w(x)|\ge c e^{kh(x)}. This proves the first line of (4.15).

Write \kappa_z=kr_0Q_z. For every fixed q\ge1, the chain rule and normal-chart bounds give

For q=1 there is also a lower bound: the matrix

has inverse bounded independently of k,x,z. In particular,

Suppose first that w\in\mathcal T_x. At every point where its cutoff or any derivative of the cutoff can contribute,

Apply (4.13). Each fixed positive Y-derivative of \kappa_z\Phi(X_z) is bounded: in a term containing m derivatives of X_z, the displayed derivative bounds leave a factor \kappa_z^{1-m}\le1. Amplitude derivatives are bounded as well. Derivatives of \omega(k^{1/3}X_z) are bounded because k^{1/3}/\kappa_z\le k^{-2/3+100\delta}\le1. Thus differentiation introduces only fixed constants at each fixed order. Together with (4.18), this proves the tail bound in (4.15). This argument uses an absolute estimate and is valid even if w(x)=0.

For w\in\mathcal N_x, the displacement estimate shows that |X_z|<k^{-1/3} throughout the ball, so its cutoff is identically one. By (4.11), the amplitude is bounded away from zero. The bounds just proved for derivatives of \kappa_z\Phi(X_z) and a(X_z) imply bounded derivatives of w_x relative to |w_x|. Integrating the bounded real logarithmic derivative along straight segments in the Y-ball gives

This proves the second line of (4.15), including j=0.

For the lower oscillation estimate, the bounded coefficients of \Phi give

By (4.7) its leading term has norm at least one. The inverse bound in (4.19) therefore gives a positive lower bound for the imaginary Y-gradient of \kappa_z\Phi(X_z). The amplitude contributes only

It cannot cancel that lower bound. This proves (4.16), including at centers with p=0 and for phases nearly isotropic in the complex bilinear form.

Packets whose supports do not meet the ball contribute zero to all the displayed sums. For the other packets, squaring (4.15), dividing by S_x^2\ge c e^{2kh(x)}, and using the cardinality bound shows that the total tail contribution is at most

The non-tail contributions are bounded by a constant times \sum_{\mathcal N_x}|w(x)|^2/S_x^2\le1. At derivative order zero the definition of S_x also shows that this latter ratio equals 1-o(1). This proves (4.17).□

Proof. We separate two deterministic regions. If W/e^{kh}>k^{50}, then \beta=1 for large k: on its transition region (5.1) instead gives W/e^{kh}\le1+e^{-cw k}. In the uncut region, the cosine formula and the angular derivative give

On \mathcal B_k, the packet value and first derivative together are bounded by Ck^{43}e^{kh}. The beam therefore dominates their sum on this region and gives a lower bound cW for the first jet. The first inequality in (5.1) puts all of t<s_1 in this region for sufficiently large k.

At a point in the remaining region there is a nearby net center. The persisting real jet span in (4.5), after converting the normalized derivative to ordinary coordinate derivatives, gives for the value and four derivatives of the random sum a covariance matrix whose least eigenvalue has square root at least

for all sufficiently large k. Only one nearby packet family is needed for this lower bound. All other families add nonnegative covariance matrices. The deterministic beam is a mean shift and does not affect the maximum of a Gaussian density. Hence, in any fixed coordinate chart and for any fixed constant C_0, the unconditional probability that the normalized jet

has norm at most C_0k^{-D} is bounded by

Indeed the density is at most Ck^{260} and the five-dimensional ball has volume Ck^{-5D}. No conditioning on \mathcal B_k is used in this estimate.

On \mathcal B_k, (5.6) through order two and the last bound in (5.3) give

for sufficiently large k. In each of finitely many coordinate patches covering B, take a grid of mesh k^{-D-48}. Its total cardinality is at most Ck^{4D+192}. Apply (5.9) at grid points in the Gaussian region; on the other grid points the deterministic beam bound already holds on \mathcal B_k. The union of the Gaussian bad events has probability at most

With D=500 this tends to zero. Choose the fixed grid-point threshold C_0k^{-D} large enough to account for the finitely many coordinate norm comparisons. The interpolation error on \mathcal B_k is at most Ck^{-D-2}, so the grid bounds imply |u|+|\dd u|_{g_0}\ge k^{-D}W everywhere. Subtracting the bad probabilities and (5.5) proves (5.8). All order-dependent constants have only increased the lower threshold on k.□

Proof. For x\in V, put

All balls below are Euclidean coordinate balls. Their radii are at most k^{-1}, so fixed enlargements lie in a common relatively compact subset of \Omega for large k. The envelope bounds give |DQ|\le Ck^{4\delta}. If two candidate balls intersect, their centers are at distance at most 2/k, and their Q values differ by at most Ck^{-1+4\delta}. Since Q\ge1, the corresponding radii are uniformly comparable.

At each fixed k the candidate radii have a positive lower bound. Choose a finite maximal disjoint family of open balls \{B(x,R_x):x\in\mathcal X_k\}, with centers in V. Uniformly bounded enlargements of these balls cover V, and Q is comparable to Q_x in each enlargement. Thus

The last inequality uses (3.15).

Fix one ball. The beam vanishes on its radius-two enlargement for large k, because \overline V\subset\{t>1/2\} and s_4<1/2. Define

In expressions differentiated with respect to Y, a packet v means its composition v(x+R_xY). Lemma 4.4 applies to the entire packet collection, including packets centered outside V that meet this ball. It gives S_x\ge ce^{kh(x)}; uniformly bounded sums of normalized squared packet derivatives through every fixed order; and, for every non-tail packet,

Here a non-tail has its cutoff identically one on the scaled ball and nonzero value at the center. For tails, the lemma supplies absolute exponential bounds, even if v(x)=0. Polynomially many tails consequently contribute o(1) to the normalized squared value and derivative sums. In particular,

All these statements are uniform over x\in\mathcal X_k.

The value U_x(0) is a standard real Gaussian. We verify that some directional derivative has a uniformly positive conditional variance given this value. Let z be the coefficient-space vector whose coordinates for packet v are (\operatorname{Re}v(x),\operatorname{Im}v(x)), and let T be the matrix whose corresponding columns are the four-dimensional real derivative vectors \dd_Y\operatorname{Re}v(0) and \dd_Y\operatorname{Im}v(0). Thus |z|^2=S_x^2. If g is the vector of independent standard Gaussian coefficients, the value and gradient are z^{\mathsf t}g/S_x and Tg/S_x. The conditional gradient covariance is

For each non-tail packet define the unit coefficient vector c_v supported on its two coordinates, with entries

These vectors are mutually orthonormal and perpendicular to z. The orthogonal projection in (5.14) therefore dominates \sum_v c_vc_v^{\mathsf t} as a nonnegative quadratic form. Moreover

Equations (5.12) and (5.13) show that the trace of (5.14) is at least a fixed c>0. There is consequently a deterministic unit direction e_x in \R^4 such that the derivative Z_x=\partial_{e_x}U_x(0), conditioned on U_x(0), has variance between fixed positive constants. Its unconditional variance is bounded above by the deterministic squared-derivative estimates. Those estimates also bound the regression coefficient \operatorname{Cov}(Z_x,U_x(0)).

For clarity, the needed uniform control of the local C^2 norm follows from finitely many derivatives. The deterministic estimates give

Integration and the Sobolev inequality on concentric balls in \R^4 yield

We choose the C^2 norm to control the Euclidean gradient norm and Hessian operator norm, changing only the fixed constant in this inequality. The Sobolev embedding used here is valid since 5>2+4/2; see [7, Sections 5.6–5.7].

Let a'=1/4 and fix 0<c'<a'/4. For every sufficiently large fixed number H, the Gaussian conditional density formula gives

Indeed, on the indicated value interval the conditional mean is bounded, and the conditional variance is bounded above and away from zero. The Gaussian density on the fixed derivative interval [1,2] then has a uniform positive lower bound. By (5.15), failure of the bound \lVert U_x\rVert_{C^2}\le H has probability at most C/H^2. Choose H once so large that the intersection of these three conditions has probability at least \pi_*=c_1/(2H)>0.

On that intersection Taylor’s formula at \pm a'e_x/H gives opposite signs, with absolute values at least a'/(2H): the linear term has magnitude at least a'/H, the center value has magnitude at most c'/H, and the quadratic error is at most a'^2/(2H). The gradient bound preserves these signs on the parallel transverse disks of radius

centered at those two points. The whole closed cylinder between the disks is compactly contained in the unit ball. The direction e_x, centers and radii are deterministic functions of the packet data. After scaling by R_x, they provide a potential certificate pair inside B(x,R_x).

Let F_k be the fraction of successful balls weighted by R_x^3. No independence among these successes is claimed or needed. Since 0\le F_k\le1 and \E F_k\ge\pi_*,

The Euclidean three-area of one successful disk is \omega_3\sigma_*^3R_x^3, where \omega_3 is the unit-ball volume in \R^3. On the event in (5.16), (5.11) therefore gives a certificate of size at least c_*k^{1+\gamma}. Its cylinders lie in disjoint open balls, as required by Definition 2.1. This proves the lemma with p_* =\pi_* /(2-\pi_*).□

Proof. Write I=g_0^{-1} and define globally

The lower bound on the first jet gives \mathcal D\ge\tfrac12 k^{-1000}W^2>0, so these are smooth. The product rule gives the exact identity

Thus \diver_\mu((A+\delta A)\dd u) +k^2(\rho+\delta\rho)u=0 identically.

We include finite derivative bounds to specify the accuracy needed in this step. For 0\le q\le R+1, the hypotheses imply

Every term in a derivative of order q of \mathcal D^{-1} is a constant times \mathcal D^{-m-1}\prod_{i=1}^m D^{q_i}\mathcal D, where q_i\ge1, \sum_iq_i=q, and m\le q; the order-zero case is included separately. Hence

By the product rule,

No derivatives of W^{-1} are required for these estimates: the displayed weights bound each differentiated factor before the factors are multiplied.

Multiplying (5.19) by derivatives of u gives |D^q(Fu)|\le C_qk^{-M+1041+1088q}. Replacing u by I\dd u gives the bound C_qk^{-M+1043+1088q}, with the derivatives of the fixed tensor I absorbed in the constant. The divergence in (5.18) costs one further derivative, and the density correction has the factor k^{-2}. These bounds give (5.17); its exponent for the density is 1043+1088(R+1)-2=2129+1088R. The cancellation of the two weights is what permits arbitrarily small coefficient changes even where the original function is exponentially small.

If \mathfrak r=0 on an open set, then F=0 there and both corrections vanish. Finally, the cone of positive pairs is open in C^0 on the compact base, so sufficiently small corrections preserve positivity.□

Proof. Take a small product coordinate patch O in the stated region on which (u,t) are the first two coordinates, and \dd u\ne0. Choose a smooth nonnegative bump \chi supported in O and strictly positive on a smaller product patch O_0. Define

extending by zero off O. This is a smooth nonnegative symmetric contravariant tensor and E\dd u=0.

Let \mathcal E be the real eigenspace at k^2 for the given pair. The nonnegative form

has kernel exactly \operatorname{span}\{u\}. Indeed, a kernel element satisfies \dd f\in\operatorname{span}\{\dd u\} on O_0. After making the product patch smaller, this means f=b(u) for a smooth one-variable function b. The round eigenfunction equations imply

The Euclidean gradient of the homogeneous harmonic polynomial defining u_* has squared norm N^2(1-t)^{N-1}, and its radial derivative is Nu. Tangential projection onto S^4 consequently gives

Differentiate (5.20) with respect to t while fixing the coordinate u. Since N>1 and t<1, it follows that b''(u)=0. Substitution back into (5.20) eliminates the constant term of b, so f=cu on O_0.

The difference f-cu satisfies a second-order elliptic equation with smooth real positive definite principal coefficients and smooth lower-order coefficients. Aronszajn’s unique continuation theorem applies: its local hypotheses allow principal coefficients of class C^2 with Lipschitz second derivatives and bounded lower-order coefficients, and smooth solutions satisfy its solution hypotheses [1, pp. 235–236]. Vanishing on an open patch gives infinite-order vanishing there. On the connected base the difference is therefore identically zero, proving the assertion about the kernel.

For \tau>0 replace A by A+\tau E. The eigenfunction u and its eigenvalue remain exact because E\dd u=0 identically. We claim that k^2 is simple for every sufficiently small positive \tau. Otherwise there would be \tau_m\downarrow0 and real eigenfunctions f_m at k^2, normalized in L^2(\rho\,\dd\mu) and orthogonal to u. Their energy identity is

so uniform ellipticity bounds them in H^1. A subsequence converges weakly in H^1 and strongly in L^2 to a normalized function f_0 orthogonal to u. Passing to the weak equation shows f_0\in\mathcal E; elliptic regularity makes it smooth. For any fixed \phi\in\mathcal E, testing the two eigenfunction equations against one another cancels the original form and gives

Weak convergence passes this fixed bilinear identity to the limit. Now choose \phi=f_0 to see that f_0 is in the kernel just identified, contradicting its normalization and orthogonality. This proves the claim. The tensor E is fixed before choosing \tau; hence \tau E can meet every prescribed finite norm tolerance, while remaining supported in O.□

Proof of Proposition 2.2. Fix the input pair, the collar, r and \eta. Choose R\ge\max\{r,1\} and a smaller positive tolerance \eta' so that total C^R movement less than \eta' implies the requested C^r movement and keeps the pair positive and in \mathcal U. Such a choice exists because the fixed input belongs to the open set \mathcal U.

Choose J=R+2 and M\ge2131+1088R, and then choose the finite packet orders giving (5.6). All these choices precede frequency selection. Lemmas 5.1 and 5.2 give, for every sufficiently large allowed k, a sample with the required first-jet bound, residual bounds and a certificate of size c_*k^{1+\gamma}. Lemma 5.3 changes the pair by at most C_Rk^{-2} in C^R while preserving that same function. Choose k large enough for this change to be less than \eta'/2.

We describe explicitly why a narrow collar only changes this frequency threshold. Packet supports avoid the region below s_1-Ck^{-1/3-100\delta}. Requiring

protects a fixed subcollar and makes any required cutoff residual C_q(w)k^Pe^{-cw k} smaller than k^{-M'}. Only finitely many orders q and finite targets M' are needed at this step. For every already fixed w>0, all inequalities in (5.21) hold for every sufficiently large k. In particular the exact corrections vanish below s_1-w/100, where the coefficients remain round and u=u_*.

In the subcollar

choose a point with \sin(N\vartheta)\ne0. Since 0<t<1, \dd t\ne0, and the nonzero angular derivative makes \dd u and \dd t independent. Lemma 5.4 applies there. After k and the sample are fixed, form its tensor E and choose \tau>0 below its simplicity threshold and so small that \lVert\tau E\rVert_{C^R}<\eta'/2. This tensor can have large finite derivatives depending on k; the subsequent scalar choice of \tau absorbs them. It preserves u, its exact eigenvalue and all current signs exactly. Its support lies in t>l_-, so the final pair is round on t<l_-.

The final pair meets every conclusion of Proposition 2.2. In particular, its certificate constant c_* was fixed after choosing the finite orders and before taking k large, and it was unaffected by either correction. The exponent \gamma, the chart and the background neighborhood were fixed upstream. Neither the collar estimates nor the arbitrary finite tolerances change them. The construction works for every sufficiently large integer N, as asserted.□

Proof. We recall the spectral and regularity facts used here. The form

is coercive on H^1(B) and defines the inverse of -L+1 in L^2(\rho\,\dd\mu). The compact embedding H^1\hookrightarrow L^2 makes this inverse compact, and it is selfadjoint. The spectral theorem and elliptic regularity give a discrete nonnegative spectrum with finite multiplicities and smooth real eigenfunctions. Its ordered eigenvalues obey the min–max principle with Rayleigh quotient

Uniform multiplicative comparisons of nearby positive tensors and densities compare (5.22) for every f and therefore imply convergence of each ordered eigenvalue under C^0 convergence of pairs. Choose a smaller interval about \lambda, with closure in I, separating it from its neighboring ordered eigenvalues. Their convergence implies that for sufficiently close pairs the corresponding ordered eigenvalue is still simple and lies in I.

We also need uniform convergence of a corresponding normalized eigenfunction, up to sign. Here only finitely many coefficient derivatives are needed. For a fixed smooth elliptic L on the closed base, the local interior estimates, a finite partition of unity and interpolation give

These are the usual interior elliptic estimates; the Sobolev and compactness statements used here are described in [7, Sections 5.6–5.7 and 6.3]. A sufficiently small finite coefficient norm makes L'-L small as an operator H^{a+2}\to H^a for each of finitely many specified a. Its contribution in (5.23) can then be absorbed on the left. Applying the resulting estimates at a=0,2, together with the eigenfunction equation, gives a uniform H^4 bound for eigenfunctions whose eigenvalues stay in the chosen bounded interval and whose L^2 norms are bounded. For example, requiring sufficiently small C^4 changes of the pair is more than enough for these finitely many operator bounds: the principal coefficients of L involve A,\rho without derivatives, and its first-order coefficients involve at most their first derivatives.

Suppose that pairs converge in such a finite norm to \mathcal P. Normalize their corresponding real eigenfunctions f_m by \int\rho_m f_m^2\,\dd\mu=1. Positivity bounds their ordinary L^2 norms, so the preceding argument bounds them in H^4. Rellich compactness and Sobolev embedding in dimension four give a subsequence converging in C^0 (for example, first converge in H^3, which embeds in C^0). Passing to the weak equation shows that the limit is a normalized eigenfunction at \lambda for \mathcal P. By simplicity it equals the prescribed normalized eigenfunction or its negative.

The finite collection of strict signs on compact disks has a positive minimum absolute margin after this normalization. Hence all sufficiently close such eigenfunctions have the same opposite signs on the same disks, allowing an overall change of sign. If no sufficiently small finite-norm neighborhood had this property, a sequence converging in that norm would contradict the preceding compactness argument. Finally, decrease its radius to put a closed finite-norm ball inside the neighborhood. This proves the lemma, including the assertion for varying densities.□

Proof of Theorem 1.1. Use monotone fixed coordinate norms for pairs. Choose b>0 so that the closed C^1 ball of radius b about \mathcal P_0=(g_0^{-1},1) consists of positive pairs in \mathcal U. Fix

We construct pairs \mathcal P_j=(A_j,\rho_j), frequencies k_j, and certificates \mathcal C_j inductively. Every output is round on t<l^{(j)}.

After producing \mathcal P_i, apply Lemma 5.5 to its simple eigenvalue k_i^2, its certificate, and the interval (k_i^2/2,2k_i^2). Denote a resulting finite norm order and closed-ball radius by s_i and \varepsilon_i>0. At stage j, these numbers are already fixed for i<j. Put

For j=1, the inner minimum is omitted. These are finite requirements at each stage. Apply Proposition 2.2 to \mathcal P_{j-1}, with l_+=l^{(j-1)}, l_-=l^{(j)}, order r_j and tolerance \eta_j. Use the certificates from its construction in Lemma 5.2, with balls centered in V and radii at most k_j^{-1}. The proposition gives a positive constant c_j and all sufficiently large allowed frequencies. Choose one with k_j>2k_{j-1} when j>1, and also so large that

The output certificate then satisfies

Only after this output has been constructed do we select its own s_j,\varepsilon_j by Lemma 5.5. They constrain subsequent stages, so no current frequency choice depends circularly on its own future persistence radius.

The increments are summable in every fixed C^r norm: for all large j, r_j\ge r and their C^{r_j} norms are less than 2^{-j}. Therefore the pairs converge to one smooth pair \mathcal P_\infty=(A_\infty,\rho_\infty). Their total C^1 movement from the round pair is less than b\sum_{j\ge1}2^{-j-1}=b/2, so the limit remains positive and belongs to \mathcal U. For each fixed i, the same budgets give

The limit lies in every selected persistence neighborhood. Hence there are real smooth eigenfunctions v_i for the limiting base operator, with the same certificate disks \mathcal C_i, and

The frequency separation makes these eigenvalue intervals disjoint and tending to infinity. The functions v_i are nonzero by their normalization, or just by their strict signs.

Set G_\infty^{-1}=A_\infty/\rho_\infty and let

This is a smooth positive function, the ratio of two smooth positive base densities. We use the standard warped-product realization of a weighted Laplacian [2, Section 4, equation (2)], with a circle as the fiber. On the fixed manifold M=B\times S^1 define

where \dd\varphi^2 is the standard unit-circle metric. The manifold is smooth, closed and connected, of dimension five, and

For a function f pulled back from B, the coordinate divergence formula gives

Here \mu in the coordinate expression denotes its positive coordinate density. Thus the pullbacks u_i of v_i are exact real Laplace–Beltrami eigenfunctions with eigenvalues \lambda_i.

It remains to convert their certificates into the claimed measure bound. Fix a closed interval I_0 of positive length in a circle coordinate chart. All certificate cylinders are in one fixed compact subset of \Omega: their centers lie in V\Subset\Omega and their radii tend uniformly to zero, with the initial frequencies chosen large enough for the same inclusion. On the product of this compact set with I_0, coordinate and Riemannian distances have fixed Lipschitz comparison constants. One may see this directly by extending the coordinate functions smoothly with cutoffs to the compact manifold; their gradients are then uniformly bounded in the single fixed metric g.

For one certificate cylinder C_i^{(a)}, let D_i^{(a)} be one of its transverse disks. The opposite endpoint signs imply that every longitudinal segment through the disk contains a zero of v_i. Therefore the subset

projects onto D_i^{(a)}\times I_0 by coordinate orthogonal projection in the base and the circle coordinate. The Lipschitz constant of every such projection is bounded by one fixed K, independent of the disk’s radius and orientation. The defining covering inequality for Hausdorff measure under a Lipschitz map gives

The closed cylinders lie in disjoint open balls. Their indicated nodal subsets are consequently pairwise disjoint Borel sets, so their Hausdorff measures add. Using (5.25), we obtain one constant c>0 belonging to the final metric such that

This argument uses continuity and signs only, without any regularity assumption on the nodal set.

Finally, (5.26) and \epsilon_0=\gamma/4 imply

Combining this with (5.29) yields

Both g and \epsilon_0 are fixed throughout this sequence, which proves Theorem 1.1.□

Proof. Take the product of (5.27) with any fixed smooth closed connected manifold of dimension n-5, and pull back the same eigenfunctions. The product Laplacian preserves their eigenvalues. Multiply each of the projection certificates above by a fixed compact coordinate box of positive (n-5)-volume in the extra factor. The same Lipschitz projection argument, now in dimension n-1, gives a fixed positive multiple of the preceding certificate lower bound. The denominator and its fixed exponent are unchanged, so the ratio still tends to infinity.□