One-sided negative sectional curvature and the holomorphic Liouville property

Proof. Topology and the metric. The map

is a smooth diffeomorphism from \mathbb C^{n+1}\times\mathbb R\times(0,\infty) onto \mathcal T_\ell. The claimed topology follows, for example by using \log b as the last real coordinate.

Let \pi(z,w)=z. Differentiating the potential gives, as Hermitian forms,

Here d^cv=-du, and consequently

In real notation Equation (2.3) is

The pullback in the first term is implicit when evaluating forms from the base. In particular, the sign before d^c\ell is positive with the convention above. Formula (2.3) is positive definite: its first summand controls the base component, and on a vector with zero base component its second summand controls the remaining complex w component. Being the complex Hessian of a smooth potential, it is Kähler.

Completeness. Let \gamma be a locally piecewise smooth curve of finite g_{\mathcal T}-length, with its parameter tending to an endpoint. Equation (2.4) gives

Thus \log b has finite total variation. In particular, along this curve there are constants 0<b_-\le b\le b_+<\infty, and b has a positive limit. The same metric identity implies

The base projection therefore has finite length and converges by completeness of g_\ell. Its image has compact closure. On that compact set |d^c\ell|_{g_\ell} is bounded by some constant C, so

Hence u also converges. Since v=b+\ell(z), the curve has a limit in \mathcal T_\ell. This finite-length criterion proves metric completeness. Indeed, a Cauchy sequence without a limit would have a subsequence whose successive distances are less than 2^{-j}; joining successive points by paths of length less than 2^{1-j} would produce a finite-length curve without a limit.

The curvature comparison. For a Hermitian matrix g=(g_{a\bar b}) arising from a Kähler potential, denote its Hermitian curvature components by

We record explicitly how these components determine the real sectional numerator in our normalization. If \xi^a=dz^a(X) and \eta^a=dz^a(Y), set

Expansion of X=\sum_a(\xi^a\partial_{z^a} +\bar\xi^a\partial_{\bar z^a}) and similarly of Y, using the Kähler curvature symmetries, gives

The factor 1/2 reflects that the complexification of the real metric satisfies g(\partial_{z^a},\partial_{\bar z^b})=g_{a\bar b}/2.

Fix a point of the tube over a base point z_0. Choose holomorphic Kähler normal coordinates on the base, centered at z_0, so that (\ell_{a\bar b}) is the identity and its first derivatives vanish there. Such coordinates follow by a linear normalization followed by a quadratic holomorphic change of coordinates; the Kähler symmetry makes the quadratic coefficients symmetric in their two holomorphic indices. The mixed terms of total degree three in the potential then vanish. Subtracting the real part of a holomorphic polynomial of degree at most three removes its remaining nonconstant pure terms. This subtraction is absorbed by a holomorphic translation of w: if \ell=\ell_0+\operatorname{Re}H, replace w by w-iH. Thus in the adapted coordinates the base potential has three-jet

Compare the tube potential at this point with

The two metrics and their first derivatives agree at the point. Their cubic contractions in Equation (2.5) therefore agree. In the fourth potential derivatives the only difference is the term b^{-1}\ell_{a\bar b c\bar d} with all four indices in the base. The base cubic contractions vanish in the normal coordinates, so

Applying Equation (2.6) gives the corresponding identity of real sectional numerators:

In particular, the extra term is nonpositive for every real pair X,Y. This remains true when their base projections are linearly dependent, in which case that term is zero. The comparison metrics agree at the point, so the sectional-curvature denominators also agree.

The model normalization. Put \widetilde w=w-ic. The Cayley coordinates

satisfy

Consequently the potential in Equation (2.7) equals -\log(1-|\mathcal Z|^2-|\mathcal W|^2) up to a pluriharmonic summand and a constant. Our comparison point has z=0. A real translation of \widetilde w and a dilation (z,\widetilde w)\mapsto(z/\sqrt r,\widetilde w/r), r>0, are model isometries and place it at (0,i), which the Cayley map sends to the ball origin.

Writing Z for all the ball coordinates, expansion of -\log(1-|Z|^2) at zero gives

For real orthonormal X,Y at the origin, their complex components \xi,\eta have norm one and satisfy \operatorname{Re}\langle\xi,\eta\rangle=0. Write \langle\xi,\eta\rangle=i\beta, where |\beta|\le1. Equation (2.6) now gives

Combining Equations (2.9) and (2.10) proves \operatorname{Sec}_{g_{\mathcal T}}\le-1 on all real planes.□

Proof. Let X,Y be an orthonormal basis of such a base plane \sigma. At a point above it with height b, the distribution

is J-invariant, and d\pi identifies its metric with b^{-1}g_\ell. If \widehat X,\widehat Y are the horizontal lifts of X,Y, then \sqrt b\,\widehat X,\sqrt b\,\widehat Y are g_{\mathcal T}-orthonormal. Equation (2.9) and the model computation give the exact formula

Letting b\to\infty proves the assertion.□

Proof. We use the standard unitary-invariant Gaussian ensemble of homogeneous polynomials; its covariance and conditional zero-density framework appear in Bleher et al. [1, Sections 2.2 and 2.4]. The proof below supplies the needed formulas and all additional quantitative grid properties. Randomness is used only to establish existence. A standard circular complex Gaussian has density \pi^{-1}e^{-|z|^2} on \mathbb C. Choose the components of Q_N independently, with

Here all g_{a,\beta} are independent standard circular complex Gaussians. The covariance identity is the multinomial formula. In particular, at a unit vector the components of Q_N are independent standard complex Gaussians, and the law is invariant under unitary changes of coordinates.

We first prove (3.1) with probability tending to one. If A=\sup_{|z|=1}|Q_N(z)|, homogeneity gives \sup_{|z|\le1+1/N}|Q_N(z)|\le eA. Cauchy’s inequality on complex lines therefore bounds the first derivative on the unit sphere by CNA, with an absolute constant C. A sphere net of mesh c/N, with a sufficiently small absolute c, consequently contains a point where the norm is at least A/2. Such a net has O_n(N^{2n-1}) points. Gaussian tails show

On the event A\le N, the Lipschitz constant on the sphere is at most CN^2. Use a second net of mesh cN^{-C_n-2}. If the norm somewhere on the sphere is less than N^{-C_n}, it is less than 2N^{-C_n} at a point of this net. A standard complex Gaussian vector in \mathbb C^n has probability at most C_n' r^{2n} of lying in a ball of radius r. The union bound is therefore at most

Choose, for example, C_n=8n. This proves both sphere bounds with probability tending to one.

Put d=n-1. Almost surely the projective zeros of the d-component system H_N are transverse. To see this directly, let \mathcal A be its finite-dimensional complex coefficient space. Evaluation at any point of \mathbb P^d maps \mathcal A onto the fiber of \mathcal O(N)^{\oplus d}. Hence the universal zero incidence manifold is smooth, and a coefficient vector is a regular value of its projection to \mathcal A precisely when the corresponding section has only transverse zeros. Sard’s theorem gives a Lebesgue-null exceptional set; the Gaussian law is absolutely continuous. Transversality and compactness imply that the zero set is finite. The projective Bézout theorem for a zero-dimensional intersection of d hypersurfaces of degrees N,\ldots,N gives total intersection number N^d; every transverse intersection has multiplicity one [4, Theorem III-71]. Thus the number of zeros is the deterministic quantity M_N=N^d almost surely.

We next justify the counting formulas needed for equidistribution. They are the one- and two-point, zero-dimensional cases of the correlation formula in Bleher et al. [1, Theorem 2.1]. Our subsequent off-diagonal argument proves the particular convergence needed here; the more general simultaneous-zero convergence theorem of Shiffman [10, Corollary 1.3] supplies a broader context. In a holomorphic coordinate chart of \mathbb P^d, regard the system as a holomorphic map H_a(z) depending linearly on its coefficients. Locally split those coefficients as (a,b) so that

This is possible because evaluation is surjective. The equation H_{a,b}(z)=0 is equivalent to a=A(z,b):=-E(z)^{-1}R(z,b). At a zero, with J=D_zH_{a,b}(z),

For fixed b, the change-of-variables formula with multiplicity for z\mapsto A(z,b) thus introduces the real Jacobian |\det E|^{-2}|\det J|^2. Integrating in b against the smooth Gaussian coefficient density yields the one-point zero density

The same argument works with a bounded measurable mark multiplying the zero count. It first applies to nonnegative marks, and the general bounded case follows by decomposition.

For two distinct projective points, joint evaluation onto the two fibers is surjective. Indeed, powers of linear forms that vanish at one point and not the other give independent prescribed values for each component. Solve now for 2d coefficient variables and apply the same change of variables in the two point coordinates. The point Jacobian is block diagonal, giving

Partitions of unity give these formulas on projective space and on the complement of its diagonal, respectively. This derivation uses only the coefficient density and ordinary change of variables; no independence of zeros has been assumed.

At a point represented by a unit vector u, take affine tangent coordinates u+\sum_{j=1}^d z_j e_j with (e_j) orthonormal in u^\perp. Differentiating the covariance shows that H_N(u) is independent of N^{-1/2}DH_N(u)|_{u^\perp}, and that the latter is a d\times d matrix of independent standard complex Gaussians. At two distinct projective points, the cross-covariances of these normalized value and first-derivative data tend to zero locally uniformly off the diagonal. More explicitly, on a compact set of pairs with |u\cdot\bar v|\le\rho<1, they are bounded by a fixed power of N times \rho^{N-2}. This follows by differentiating (u\cdot\bar v)^N at most once at each point. The joint Gaussian covariance consequently converges uniformly there to the direct sum of its one-point covariances. Conditional Gaussian covariances given the values zero converge as well. Gaussian moments of the fixed polynomials |\det J|^2 and their products therefore converge. Equations (3.4) and (3.5) imply that the two-point density, divided by M_N^2, converges locally uniformly off the diagonal to the product of the normalized one-point densities.

The exact total number of zeros lets us pass from off-diagonal decorrelation to weak convergence without estimating the two-point density near the diagonal. Let

Unitary invariance and the deterministic count imply \mathbb E\mu_N=\sigma. The positive measure \mathbb E(\mu_N\otimes\mu_N) has total mass one. Off the diagonal, its density converges to that of \sigma\otimes\sigma, by the preceding calculation. This implies convergence on the whole product: given \varepsilon>0, choose a compact subset of the off-diagonal region having (\sigma\otimes\sigma)-mass greater than 1-\varepsilon. Its mass under \mathbb E(\mu_N\otimes\mu_N) is greater than 1-2\varepsilon for all sufficiently large N. At most 2\varepsilon of the total mass can remain elsewhere. Since the diagonal has zero (\sigma\otimes\sigma)-mass, there is no additional limiting mass there. Thus

For every continuous test function, its integral against \mu_N therefore has variance tending to zero. This proves weak convergence to \sigma in probability.

It remains to discard the bad zeros. Use (3.4) with marks. If G is a d\times d standard complex Gaussian matrix, then at a zero the derivative condition in (3.2) fails only when \operatorname{sing}_{\min}(G)<N^{-3/2}. Its expected discarded fraction is

Indeed a Gaussian square matrix is nonsingular almost surely, |\det G|^2 is integrable, and dominated convergence applies. The independent polynomial p_N has a standard complex Gaussian value at every unit representative. Its two forbidden tails have probability

Markov’s inequality shows that the total discarded fraction tends to zero in probability. Removing that fraction does not alter weak convergence. The modulus and singular-value conditions are unchanged on multiplying a unit representative by a phase, so they hold for every such representative.

We may now select deterministic systems. Take a countable uniformly dense family of continuous functions on projective space, and impose successively accurate conditions on successively longer finite subfamilies. Convergence in probability, the vanishing bad fraction, and the probability-one transversality statement show that these conditions and (3.1) have a nonempty intersection for every sufficiently large degree. Choose one system from that intersection. A diagonal choice gives all the asserted limits and works equally well along any prescribed sequence of degrees.

We have obtained the projective sampling properties. It remains to provide disjoint neighborhoods in which Section 8 can localize its shear coefficients. Fix a good direction and a lift v with p_N(v)=1. Homogeneity and (3.2) give N^{-1/N}\le |v|\le N^{1/N}. In the splitting of the domain into the radial complex line and its Hermitian orthogonal complement, DH_N(v) vanishes on the radial line, while Dp_N(v) has radial derivative N/|v|. The transverse derivative of H_N has least singular value at least N^{-4} for large N: the scaling from a unit representative costs at most one additional factor N, so this is a deliberately weaker bound than necessary. Homogeneity, the sphere upper bound, and Cauchy’s inequality at scale c/N give first- and second-derivative bounds O_n(N^5) in the stated neighborhood. At the lift itself, the transverse first derivatives are O_n(N^4). Block inversion therefore gives

for all sufficiently large N. Taylor’s formula, uniformly for |z-v|\le4N^{-30}, now yields

This proves (3.3). It also excludes every other zero of (H_N,p_N-1) within distance 4N^{-30}, proving disjointness of the balls about all good lifts.□

Proof. Dimension gives

for all sufficiently large N. Indeed its limit is the sum in (3.6).

There is a number t_h>0 such that, on every sufficiently accurate grid, every polynomial with \|P\|=1 satisfies

For completeness, a nonzero homogeneous polynomial has a projective zero set of \sigma-measure zero. In affine charts this is the elementary fact that the zero set of a nonzero complex polynomial has Lebesgue measure zero, obtained by induction and Fubini’s theorem. Thus, for each norm-one polynomial P, some positive threshold has a superlevel set of measure greater than 0.9. Uniform continuity in the coefficients makes the same superlevel estimate, with a smaller threshold, valid in a neighborhood of P. The coefficient unit sphere is compact, so finitely many such neighborhoods suffice and their thresholds have a positive common lower bound. Weak convergence of the grid measures applied to these finitely many open superlevel sets gives the estimate for unit representatives, with a fixed margin in both threshold and counting proportion. Since |v_\xi|^h\to1 uniformly, it gives (3.8). This proof requires only finitely many fixed approximation conditions for the given h.

The projection has 0\le(\Pi_N)_{\xi\xi}\le1 and diagonal sum less than 0.02M_N. Consequently at most 0.04M_N diagonal entries exceed 1/2. Choose the b_\xi temporarily as independent uniform phases. Writing Q_N'=1-\Pi_N, independence gives

when \|P\|=1. The last inequality uses the intersection of the 0.8M_N indices in (3.8) with the complement of the at most 0.04M_N high-leverage indices.

We need a single phase choice for all polynomials, rather than a choice depending on P. Let (e_a) be a fixed orthonormal basis of the degree-h coefficient space, and put q_a(\xi)=e_a(v_\xi). These evaluations have a common bound depending only on n,h and the chosen norm. The Gram matrix of the random linear map in (3.7) has entries

Its diagonal-index terms are deterministic. For distinct indices, the products \overline{b_\xi}b_\eta are orthonormal in L^2 of the phase probability space: two such ordered products have nonzero inner product only when their ordered pairs agree. In particular, reversal of the pair gives expectation \mathbb E(\overline{b_\xi}^{\,2}b_\eta^2)=0. It follows that

The Gram matrix has fixed finite size. Chebyshev’s inequality therefore implies that its operator-norm distance from its mean tends to zero in probability. Its mean is bounded below by 0.38t_h^2 times the identity. A phase choice with least eigenvalue at least 0.19t_h^2 exists for every sufficiently large grid. Taking \delta_h=\sqrt{0.19}\,t_h proves the assertion. All estimates used only absolute values of lift evaluations, so the threshold and lower bound are uniform over their arbitrary representative phases.□

Proof. Cauchy’s inequality on the complex line through v gives

The homogeneous expansion converges absolutely at the evaluation points. The finite Fourier average annihilates every degree not congruent to h modulo N, giving the identity. Put \theta=R|v|/S. The first tail is bounded by

The case k=1 gives the other estimate. The denominator is bounded away from zero for c\ge1.□

Proof. For each stage choose the phases supplied by Lemma 3.2. Assign its phase to all N_j lifts of the same direction. We prove the assertion for a holomorphic function with |f|\le1, which suffices by constant rescaling.

Fix a stage and temporarily omit its subscript. For each w with \operatorname{Im}w=D, put

Both are holomorphic on a neighborhood of the closed spatial ball |s|\le S, by the strict height margin and compactness. The maximum principle and (3.12) give |A|\le e^{p_0c} on the ball. The unit w-disk centered at any (s,\zeta,w) in this ball lies in \mathcal T, by (3.11); Cauchy’s inequality gives |I|\le1. On a sheared disk, the derivative at t=0 is A(Rv)+\alpha b_\xi I(Rv). Thus

Every estimate so far is uniform in the real part of w.

For each good direction take the Fourier average in Lemma 3.4 over its cycle. Since b_\xi is constant on that cycle, it factors out of the averaged second term of (3.14). The lift radius bound gives

Let a and i' be the resulting vectors of averaged A and I values, respectively. There is a vector a_0\in V_N such that

If P is the homogeneous degree-h Taylor term of the function s\mapsto I(Rs), then

The averaged node inequalities still have supremum norm at most e^{-m_0c}. Apply 1-\Pi_N and use a_0\in V_N. Since the number of good directions is at most M_N, conversion from the supremum norm to M_N^{-1/2} times Euclidean norm costs at most one. Lemma 3.2 consequently gives

In particular, with the fixed positive number

we have

The constants are independent of the stage once h is fixed. This is exactly the uniformity in Lemma 3.2; the Fourier tail denominator is 1-e^{-0.99c}, not 1-e^{-0.99c/N}. Since every degree-h coefficient of I(Rs) is R^h times the corresponding coefficient of I(s) and R\ge1, (3.15) bounds each degree-h coefficient of f_w(s,\zeta,w) by the same exponential rate.

We have obtained exponential decay for every coefficient of the prescribed degree, uniformly along the entire horizontal w-line. It remains to bring that estimate down from the growing testing heights to a fixed open part of the tube.

Fix now a scheduled center \zeta and a multiindex \beta of length h. The coefficient

is holomorphic on the whole connected half-plane \operatorname{Im}w>\ell(0,\zeta). At every point of this half-plane a spatial neighborhood is available, so the derivative is well-defined and holomorphic there. Choose once a small closed spatial polydisk about zero and a number t_0 exceeding the supremum of \ell(s,\zeta) on that polydisk by more than two. Cauchy’s inequalities in s and on a unit w-disk give

where B_\beta is independent of every testing stage and of \operatorname{Re}w. Along the stages assigned to (\zeta,h), (3.15) gives

For any fixed t>t_0, apply Hadamard’s three-lines theorem [12, Theorem 7.13] to the horizontal strip between heights t_0 and D_j, once D_j>t. Writing \theta_j=(t-t_0)/(D_j-t_0), it yields

If B_\beta=0 there is nothing to prove. Otherwise, the logarithm of the right side is

by (3.10). Hence g_\beta vanishes in the upper half-plane and, by the identity theorem, throughout \operatorname{Im}w>\ell(0,\zeta). Repeating this argument for each order proves that all spatial Taylor coefficients of f_w at zero vanish at every scheduled center. Dependence of the constants and of the preliminary height t_0 on the order causes no difficulty: each order has its own infinite sequence, and analytic continuation gives the same full half-plane afterwards.

To pass from these statements to a common open set, choose a spatial ball about zero and an open subdisk of the center disk, both with compact closures. Choose L larger than the supremum of \ell on their product. On the product with \operatorname{Im}w>L, the function f_w is jointly holomorphic. At each scheduled center in the subdisk, all of its spatial Taylor coefficients vanish for every such w. Taylor uniqueness and the identity theorem on the spatial ball give vanishing on that ball. Continuity in y and density of the scheduled centers then give vanishing on the whole product. The tube is connected: the coordinates

identify it with \mathbb C^{n+1}\times\mathbb R\times(0,\infty). The several-variable identity theorem therefore gives f_w=0 everywhere on \mathcal T.

For every fixed (s,y) the w-fiber is a connected half-plane, so f is constant along that fiber. It descends to a bounded function F on \mathbb C^{n+1}. This descended function is holomorphic: near any base point choose a single constant w_0 whose imaginary part exceeds the supremum of \ell on a small base neighborhood, and write F(s,y)=f(s,y,w_0) there. These local holomorphic definitions agree on overlaps by fiber constancy. Liouville’s theorem, applied on complex lines in \mathbb C^{n+1}, makes F, and therefore f, constant.□

Proof. Use the holomorphic coordinate z_0=\log(y-\zeta), so that \rho=z_0+\bar z_0. Differentiating U_\tau+\rho=0 gives

Differentiation of \ell=U+\tau\rho now gives

which is (4.6). For regularity, write F=\log\tau+f(\tau,s). The equation f_\tau=(q^{-1}-1)/\tau, with f(0,s)=0, shows that f is smooth. Equation (4.3) becomes \tau e^{f(\tau,s)}=e^\psi|y-\zeta|^2. Its derivative with respect to \tau at zero is one, so the smooth implicit-function theorem applies. In particular, \tau=e^\psi|y-\zeta|^2+O(|y-\zeta|^4). Substitution in (4.4) proves the stated extension and axis metric.□

Proof. In addition to the tilt, first choose spatial coordinates normal for h at the point. On functions of (s,\tau), physical coordinate derivatives act as

and their conjugates. At the point the complete list of cubic entries needed below is

For example, the identity P^2\partial_\tau U_{\tau i}=P_i, obtained by spatial differentiation of U_{\tau\tau}=-1/P, and its conjugate give the second and third entries. Directly differentiating the metric entries before evaluating gives

For completeness, the fourth derivative giving the spin block has a cancellation which is useful later. Put a_i=U_{\tau i}. Before evaluation,

At the point a_i=a_{ik}=0, while

Applying D_{\bar0}=P\partial_\tau therefore gives P_{ik}: the two positive terms P_iP_k/P cancel the negative doubled term.

Subtract the cubic contractions in (4.5). For A the vertical contraction is PP_\tau^2 and the spatial one is |\partial P|_h^2. For A_i they are P_iP_\tau and P(\mathcal Nh^{-1}\partial P)_i. For W they are P_iP_{\bar j}/P and P^2(\mathcal Nh^{-1}\mathcal N)_{i\bar j}. The first combines with P_{i\bar j} to give P\partial_i\partial_{\bar j}\log P. For the purely spatial block the contraction is the one defining C^h. These calculations prove all formulas in normal spatial coordinates. Without spatial normalization, the last entry of (4.13) is \partial_i h_{k\bar j}; its contractions replace P_{ik} by \nabla^h_i\partial_kP and \partial_i\mathcal N_{k\bar j} by \nabla_i^h\mathcal N_{k\bar j}. This also verifies the displayed tensorial formulas in general spatial coordinates.□

Proof. For real vectors with (1,0) parts v,w, set H=i(v\otimes\bar w-w\otimes\bar v). This is Hermitian of rank at most two. Kähler symmetry gives

Expand R(v+\bar v,w+\bar w,w+\bar w,v+\bar v). The only four nonzero terms are the two alternating type patterns and their conjugates. In the convention in which a real vector with complex components v has squared length g_{i\bar j}v^i\bar v^j, the complex bilinear extension of the real metric on a holomorphic and an antiholomorphic vector is one half the Hermitian pairing. Accordingly the real curvature numerator is -\tfrac12 C(H,H). This proves the sign assertion. Expanding a general block matrix H gives (4.15)–(4.16); each mixed conjugate pair accounts for its factor two or four. A principal block T has rank at most two.

For the quantitative assertion use 2t(W:T)\ge-.98At^2-|W:T|^2/(.98A). The remaining diagonal coefficient is .02A. The spin term costs at most W/2. The other two terms obey

Since 1.5-408\eta^2\ge1 when \eta\le1/100, (4.21) follows.□

Proof. Set Q=\Delta H-\Gamma\Delta g. Then H+\Delta H=\Gamma(g+\Delta g)+Q. Multiply out the corresponding cubic contraction and subtract \Gamma g\Gamma^*. The two cross terms and \Gamma\Delta g\Gamma^* combine to give the first three terms in (4.22). Taking operator norms gives (4.23). The inverse comparison is the elementary spectral estimate for I+g^{-1/2}\Delta g g^{-1/2}; the final statement uses (4.14) twice.□

Proof. Write a=(1+\tau)^{-1/16}, z=e^Eu, and q_b=K_0^{-1}\log(1+\tau). The logarithmic slopes of the two summands of q are

They satisfy

For the second inequality use \log(1+\tau)\ge2\tau/(2+\tau), obtained by differentiation from equality at zero. The derivative of a weighted mean of logarithmic slopes is the weighted mean of their derivatives plus their nonnegative variance. Hence r_x\ge-r/2, proving the first bound for D. Also (q_a)_{xx}/(q_a)_x=p+p_x/p\ge-1/16 and (q_b)_{xx}/(q_b)_x=(1+\tau)^{-1}>0. Taking the weighted mean with weights (q_a)_x,(q_b)_x gives the other slope bound.

Every positive E derivative of q_a of order at most four is bounded by Cq_a\min(1,z). Differentiating q^{-1} thus gives

Likewise (q_a)_x and its E derivative have ratio bounded in absolute value by a profile constant, so |r_E|\le Cr. Since a\ge2^{-1/16} for \tau\le1, comparison with p gives e_0,q_{EE}/q\le Cr there. The remaining bounds in (5.4) follow directly from the displayed formulas.

Differentiate the normalized integral for F. For x\le0, use d x=d\log z/a, where a\ge2^{-1/16}, in (5.11); the resulting absolute integral is at most

For x\ge0, put M=e^{b_0E}. The positive lower and upper bounds on u for \tau\ge1 imply cM\le q_a\le CM and q\ge cM+x/K_0. The bound in (5.11) is therefore at most CM/(cM+x/K_0)^2, whose integral over [0,\infty) is uniformly bounded. This proves (5.5), also for the average by integration.

Integration by parts gives, including for j=0,

If \tau\ge e^{-E/4}, split at t=e^{-E/4}. On the first part the integrand is uniformly bounded. On the second part \tau t\ge e^{-E/2}, and u(v)\ge c\min(v,1) implies e^E u(\tau t)\ge c e^{E/2}. There the integrand, also for j=0, is at most C e^{-b_0E/2}. This proves the first bound in (5.6). For \tau\ge2, split instead at t=\tau^{-1/2}. Monotonicity and the positive finite limit of u, together with \log(1+\sqrt\tau)\ge\tfrac12\log(1+\tau), give q(\log\sqrt\tau,E)\ge c q(\log\tau,E). Thus the integral is at most C(\tau^{-1/2}+q^{-1}). The estimates q\le C(M+\log\tau) and q_x\ge c(1+M\tau^{-1/16}) show that q\tau^{-1/2}\le Cq_x and 1\le Cq_x; hence this last bound is at most Cr. On [e^{-E/4},2] the explicit formula for p gives a positive lower bound on r. This completes (5.6).

The estimates (5.8) follow from the same two summands of q. For 1\le\tau\le e^{3b_0E}, the logarithmic summand divided by q_a tends uniformly to zero as the lower threshold on E increases; also z/(1+z)\to1 uniformly. This proves (5.9). On [e^{-E/4},e^E] one has z\ge c e^{3E/4} and a\ge c e^{-E/16}, so

Finally, on (0,2e^{-E}], the quantity z is comparable to e^E\tau and remains bounded by a profile constant. The formulas for the first two x derivatives of q then give (5.10). Between 2e^{-E} and 1 the same formulas give q_x\ge c; for \tau\ge1, use (5.8). These statements also prove the two final consequences.□

Proof. Differentiating U_\tau=\psi-F gives

The bar denotes averaging from 0 to \tau. The bounded derivatives in (5.5) give (1-C\mu)G\le\mathcal N,\bar{\mathcal N}\le(1+C\mu)G. Choose \mu small enough that C\mu<1/10. This proves positivity and (5.13).

We will repeatedly use finite-jet estimates of the following precise kind. Differentiating (5.15) at most twice spatially gives sums with at most four derivatives of E and at most four E derivatives of F or \bar F. Each summand is a product of a fixed number of the assumed jets. The formulas for the inverse, connection, and curvature add a fixed number of inverse metric factors. Consequently there is an absolute finite constant C_*, independent of dimension, such that these estimates cost at most C e^{C_*\epsilon E}, with the factors 1+\tau or (1+\tau)^{-1} displayed explicitly below. This conclusion follows by the product rule and h^{-1}\le C e^{\epsilon E}(1+\tau)^{-1}I; it uses operator norms, not sums of tensor entries. The same observation applies after normalizing spatial axes by \mathcal N, because e^{-\epsilon E}I/C\le\mathcal N\le C e^{\epsilon E}I. Enlarge C_* once to cover these finitely many products, and choose \epsilon>0 so small that C_*\epsilon<1/100. Any profile-dependent multiplicative constants are absorbed by raising E_{\min}.

In \mathcal N-unit axes define

The displayed formula indicates the normalization; equivalently it is I-\tau h^{-1} after \mathcal N=I has been imposed in the fixed axes at the point. We claim

Always \|\mathbf Y\|\le1+C\mu, because h\ge\tau\bar{\mathcal N} and \bar{\mathcal N} is close to \mathcal N. On [2e^{-E},1], the explicit slope p in the scalar lemma yields r\ge .35b_0 for all sufficiently large E: indeed u\ge e^{-1/16}\tau, z\ge2e^{-1/16}, a\ge2^{-1/16}, and q_a/q\ge(1+\log2/K_0)^{-1}\ge(1+\log2)^{-1}. The product of these three lower bounds in p(q_a/q)=b_0a[z/(1+z)](q_a/q) exceeds .35b_0. This proves the first bound on that interval. For \tau\ge1, (5.6) gives \|\bar{\mathcal N}-\mathcal N\|_{\mathcal N}\le C\mu r. Set h_0=V+\tau\mathcal N. The inverse identity and h,h_0\ge c\tau\mathcal N show that replacing h by h_0 in (5.16) costs at most C\mu r. In the normalized axes V\le\gamma I, where \gamma=(1-C\mu)^{-1}, and hence

This is at most r/40 on [1,e^{3b_0E}], by (5.9) and b_0\ge1000. For \tau>e^{3b_0E}, use r\ge c/q and q/\tau\le C(e^{b_0E}+\log(1+\tau))/\tau; the supremum of the last expression on this range tends to zero with the lower threshold on E. Thus the same estimate holds there. Decreasing \mu pays the inverse-identity error. Finally, if \tau\le2e^{-E}, then h\ge V and the raw size bounds give \tau\|\mathcal N^{1/2}h^{-1}\mathcal N^{1/2}\| \le C\tau e^{C_*\epsilon E}, proving the second bound.

Since P_\tau=q(1+r) and P^2P_{\tau\tau}=Pq^2D, the curvature blocks in Lemma 4.2 become

The final term in (5.18) and the spin form divided by Pq have norm at most C\mu r when \tau\ge2e^{-E}, and at most C\mu on the remaining interval. This follows from (5.4), the three smallness assumptions on E, and q^{-1}\le Cr away from the exceptional interval. After decreasing \mu,

and the spin inequality (4.18) holds.

For the fiber diagonal term,

Here h\ge c\tau G; the final scalar quotient is bounded using e_0\le Cr for \tau\le1, and qD\ge cqr=cq_x\ge c for \tau\ge1. Thus

In the same normalized axes the cancellation in (4.8) gives

The last inequality uses e_0\le Cr on the exceptional interval. Combining with (5.20) and (5.21), the squared ratio to the dual norm determined by AW is at most C\mu. Consequently (4.19) holds with any fixed desired small \eta>0 after decreasing \mu.

We next pay the trace coupling. Outside the exceptional interval, (5.17) and (5.18) imply

By D\ge r^2, its cost against .98A is at most

For 0\le s\le\sqrt2, (s+\sqrt2/16)^2/(1+s^2)<.8; this can also be checked by differentiating, since its maximum on this interval occurs at the right endpoint. Because T has rank at most two, decreasing \mu therefore makes (5.22) at most .85P((\operatorname{tr}_{\mathcal N}T)^2+|T|_{\mathcal N}^2). These are exactly the positive symmetrized terms in (4.12). On \tau\le2e^{-E}, the bounds D\ge c e^E\tau, q,r\le C, and (5.20) instead give a trace cost at most Ce^{-E}|T|_{\mathcal N}^2.

To estimate the remaining C^h, note from (5.15) that the difference h-V=\tau\bar{\mathcal N} and its first two jets are at most C\tau e^{C_*\epsilon E}. On \tau\le e^{-E/4}, the curvature formula or Lemma 4.4 consequently gives

In the complementary range, the inverse bound and the jet bounds give

This is arbitrarily small compared with P|T|_{\mathcal N}^2: indeed, for \tau\ge e^{-E/4}, \tau q_a/(1+\tau)\ge e^{2E} after increasing the threshold on E, by the explicit power b_0\ge1000; normalization of T costs only e^{C_*\epsilon E}. On the exceptional interval, its trace cost Ce^{-E}|T|_{\mathcal N}^2 is likewise negligible compared with e^{-\epsilon E}|T|^2. We have thus obtained (4.17) with

before paying the K coupling.

For that coupling split K into P\nabla^h\mathcal N and the two terms containing P_i=Pe_0E_i. The latter have normalized size at most CP|e_0|\sqrt\mu; their squared dual W norm after pairing with T is at most C\mu P|T|_{\mathcal N}^2. To see this directly outside the exceptional interval, the scalar quotient to bound is e_0^2/(qr); it is bounded by (5.4) and (5.8). On the exceptional interval replace r in that denominator by 1+r, which is easier. For P\nabla^h\mathcal N use three ranges:

1. 1.

   If \tau\le e^{-E/4}, then W\ge cPq\mathcal N, and the raw jet bounds give squared dual W cost at most C\tau e^{C_*\epsilon E}|T|^2, negligible compared with the second term of (5.23).

2. 2.

   If e^{-E/4}\le\tau\le e^E, use \|\nabla^h\mathcal N\|_{\mathcal N}\le Ce^{C_*\epsilon E}. The squared dual cost is at most CPe^{C_*\epsilon E}(qr)^{-1}|T|_{\mathcal N}^2. By (5.7) this is negligible compared with the first term of (5.23).

3. 3.

   If \tau>e^E, the identity \nabla^h h=0 permits

   \nabla^h\mathcal N =\nabla^h(\mathcal N-\bar{\mathcal N}-V/\tau).

   Equation (5.6), differentiated spatially once, and the raw bounds give a normalized norm at most C e^{-E/4+C_*\epsilon E}. In making this estimate the connection has raw norm at most Ce^{C_*\epsilon E}, because h has first jets at most C(1+\tau)e^{C_*\epsilon E} and inverse norm at most Ce^{C_*\epsilon E}/(1+\tau). Since qr=q_x\ge c, the squared dual W cost is again negligible compared with P|T|_{\mathcal N}^2.

Fix \eta\le1/100 in Lemma 4.3. Choose \mu small enough for the terms proportional to \mu, and then E_{\min} large enough for the exponentially small terms, so that (4.20) holds. That lemma proves positivity of every rank-two contraction with the retained margins (5.14), and hence nonpositive real sectional curvature. Smoothness and positivity at the axis follow from Lemma 4.1; curvature extends by continuity.□

Proof. Where the coefficient is locally constant the added potential is pluriharmonic, so it changes no metric. It remains to estimate the derivative collar in (5.24). Since E is constant, P_i=0, \mathcal N=G, and h=V+\tau G. The holomorphic tilt used in Lemma 4.2 has linear and quadratic coefficients bounded by Ce^{\epsilon E}, because U_\tau=\psi-F. Its constant term may and will be zero. Derivatives through order four of the exponential of this fixed quadratic polynomial at the point cost at most Ce^{C\epsilon E}, by the ordinary product rule. These are fixed-order multilinear estimates.

First consider \tau\ge e^{-E} in tilted logarithmic coordinates. The metric inverse has norm at most Ce^{C\epsilon E}/\tau. By (5.21), (5.20), (5.23), and q_x\ge c on this range, the retained curvature on a rank-two matrix H is at least

in these fixed coordinates, for a fixed C_1. The cubic table (4.13) shows explicitly that the only possibly nonzero connection blocks are

Consequently their operator norm is at most Ce^{C_1E}q, since r is bounded. Allowing a larger fixed power of q in the following estimates would give the same result.

The inequality q\ge1 and the left normalization imply F\le x. Thus

After the tilt, all shear jets needed for the metric and curvature, of orders two, three, and four, are therefore at most

where e^{-J/2}e^{-\psi/2}\le1 was used. For every fixed integer r_1, the scalar estimate (5.8), also valid as an upper bound for small \tau, gives

Indeed q^{r_1}\le C_{r_1}(e^{r_1b_0E} +\log(1+\tau)^{r_1}); on [e^{-E},1] use \tau^{-1/2}\le e^{E/2}, and on [1,\infty) the quotient \log(1+\tau)^{r_1}/\sqrt\tau is bounded.

For clarity, this bound controls the quadratic perturbation as well as the linear terms. Write \delta=e^{-CE} and absorb all other fixed E powers into e^{cE}. The relative metric error is bounded by \delta e^{cE}/\sqrt\tau. Each linear curvature error from (4.23), divided by (5.25), is bounded by \delta e^{cE}q^{r_1}/\sqrt\tau for some fixed r_1. The quadratic contraction error before division by that margin is at most \delta^2e^{cE}q^{2r_1}; after division its bound is

Equation (5.27) makes all these quantities arbitrarily small by a single sufficiently large C. It follows that the metric is at least half its old value and that the curvature errors are smaller than a fixed fraction of the retained margin, uniformly on this whole fiber range.

For 0<\tau\le e^{-E}, exponentiate the tilted logarithmic coordinate and use the affine fiber coordinate with scale e^{-(\psi(s_0)+E)/2} at the fixed spatial point s_0. Denote its value by z. The zero constant term of the tilt and (5.10) give

For example, |F-x|\le Ce^E\tau follows by integration of (q^{-1}-1)d\tau/\tau on this range, and proves the comparison. The metric blocks in affine coordinates are h and P/|z|^2. Both have inverse norm at most e^{C_3E}. The curvature tensor transforms tensorially: its fiber diagonal block divides by |z|^4, its W block by |z|^2, and its spatial block is unchanged. Since

the retained rank-two curvature margin in affine coordinates is at least e^{-C_3E}|H|^2, after increasing C_3.

The connection in the new coordinates is also bounded by e^{C_3E}, without an apparent singularity at z=0. Indeed the logarithmic block \Gamma^0_{00}=P_\tau becomes (P_\tau-1)/z, and P_\tau-1=O(e^E\tau); thus it is bounded by C\sqrt{e^E\tau}. The mixed spatial block becomes Ph^{-1}G/z, bounded by e^{C_3E} using (5.28). The spatial-output, two-fiber-input block is zero because P_i=0, and the spatial connection is already bounded. These checks cover all cubic entries. The same bounds extend to zero by the smoothness proved in Lemma 4.1.

In these affine coordinates the shear jets of orders two through four are bounded by

The estimate includes derivatives of the tilt and uses that |z| is bounded on this range. Apply Lemma 4.4, with the inverse metric, connection, and reciprocal margin bounded by fixed powers of e^{C_3E}. Increasing the same C makes all relative metric errors and all curvature errors smaller than one half their respective retained bounds, including at the axis. The estimates overlap at \tau=e^{-E}, so they cover the entire fiber.

Finally, all exponents just used arise from derivatives of order at most four, a fixed finite number of products in (4.22), and operator norms of linear maps. Rank-two contractions introduce only the absolute constants in (4.14). None of these operations counts spatial indices. This proves the asserted independence from n.□

Proof. The first three identities follow by changing the independent variable from \nu to u. Since d\Psi/d\nu=t a_\psi and H=Y/(t a_\psi), the fourth follows as well. Differentiating \log H=\log Y-\nu-\log a_\psi gives the fifth. The lower bound on \kappa implies

In u time, dY/du=\kappa(1+Y)\le1.1(1+Y), so Y remains finite and positive on each finite interval. The remaining variables are obtained by integrating the displayed equations there. In a cap, dY/d\nu=C_\kappa Y, which gives the asserted global \nu continuation. Dividing the last equation of (6.3), with p=1, by the second proves (6.4). Finally, \kappa\ge1 gives Y(u)\ge Y(u_0)e^{u-u_0}, and integration of d\nu/du=1/Y gives the last assertion.□

Proof. At s=(\sqrt t,0,\ldots,0) the metric has entries

and its radial and transverse eigenvalues are a(1+Y) and a. The unitary symmetry in the transverse variables and the Kähler symmetries leave only a transverse block, a radial–transverse bisectional block, and a pure radial entry. After removing the common factor Y/(ta)=H/\lambda, their respective values in V-unit axes are

For completeness, the transverse coefficient is a'/a^2=Y/(ta). The radial–transverse coefficient is

For the pure radial entry, differentiate the logarithm of the radial metric along the radial complex line. Since

its second \nu derivative divided by ta(1+Y) is exactly the last expression in (6.6) times Y/(ta). These calculations also follow directly by subtracting the cubic contractions in the Kähler curvature formula.

Write a Hermitian test matrix in radial/transverse blocks as

The contraction after removing Y/(ta) is

The assumption on \kappa_Y gives D\ge\kappa. Absorb the cross term with 0.9Dr^2. Its cost is at most

Since \operatorname{rank}S\le2, |\operatorname{tr}S|\le\sqrt2|S|. The transverse reserve is therefore at least 5|S|^2/9, in addition to 0.1Dr^2 and 2\kappa|b|^2. As \kappa\le1.1, these bound an absolute positive multiple of \kappa(r^2+2|b|^2+|S|^2). This proves (6.5).□

Proof. The eigenvalues of G_0 are a_\psi transversely and a_\psi(1+pY) radially, giving (6.8). In these axes dt and d^2t have operator norms at most Ct/Y, and higher derivatives of t vanish. The first bound includes the fact that the only nonzero first derivative is radial; thus its axis length includes the extra factor (1+pY)^{-1/2}.

Set a dot to mean d/d\nu. We have

The last two estimates use, respectively, the first and second indicated derivatives of \kappa. Also \dot p=O(\delta Y) and \ddot p=O(\delta Y^2). Direct differentiation gives

Consequently |a^{(j)}|\le Ca(Y/t)^j for 0\le j\le3. The same formulas for a_\psi use pY in place of Y and its derivatives. They imply the corresponding estimate for a_\psi, and comparison at the point, where a=\lambda a_\psi, improves their difference by the factor \delta.

The chain rule for a radial potential in at most four spatial slots is a finite sum of products of its t derivatives and copies of dt,d^2t. For a term containing its lth t derivative, the product of these latter factors is bounded by C(t/Y)^l. Since B^{(l)}=a^{(l-1)}, every such product is at most Cat/Y=C\lambda/H, and the same argument gives C/H for \Psi. The improved difference estimate proves the metric-jet assertion. For u, its t derivatives through order four are O((Y/t)^j), by the displayed bounds on Y and its derivatives; the same chain rule proves the third assertion. All sums here have a number of terms bounded in terms of the derivative order, not the dimension.

Finally, the largest axis length in (6.8) is \sqrt{t/Y}\le1. Applying two, three, or four such axes to the extra potential therefore gives at most CD_e t/Y=CD_e/(a_\psi H).□

Proof. Since q has no spatial dependence, \mathcal N=G and h=V+\tau G. Lemma 6.3 shows that, through two metric jets, h differs from the constant multiple (\lambda+\tau)V/\lambda by at most C\delta(\lambda+\tau)/H. In particular

Lemma 6.2 gives a covariant curvature lower bound c\lambda|T|^2/H for V in these scaled axes. Multiplication of a metric by a positive constant multiplies its covariant curvature by that constant. The curvature difference is at most C\delta(\lambda+\tau)|T|^2/H: indeed its fourth-derivative term has this bound, and its cubic contractions have the same bound by the inverse estimate |h^{-1}|\le CH/(\lambda+\tau) and the first-jet bounds. Lemma 4.4 makes this calculation explicit. Reducing \delta_0 therefore gives

Let \theta=\lambda/(\lambda+\tau). The identity \nabla^h(V+\tau G)=0 implies

The curvature table gives A=Pq^2D_q, vanishing spin and A_i terms, and K=P\nabla^hG. In G-unit normalization it gives

Here V_G denotes V expressed in G-unit axes; its eigenvalues are (1+O(\delta))\lambda. In particular W>0. Spending a fixed small portion of W on the K coupling costs at most

which is an arbitrarily small portion of (6.12) after reducing \delta_0.

It remains to pay for 2t_f(W:T). Spending 0.98A t_f^2 costs (W:T)^2/(0.98A). First suppose the second alternative in (6.10) holds. Split the two terms in (6.13) using (a+b)^2\le1.1a^2+11b^2. The r term costs at most

The horizontal curvature contains the positive term

As |\operatorname{tr}_G T|\le\sqrt2|T|_G, their difference retains more than P|T|_G^2/2. The remaining trace cost is at most

For \tau\le1, (6.9) bounds P/D_q by C_E. For \tau\ge1, it bounds (6.14) by

For the last inequality, put z=\tau/\lambda and use boundedness of z(1+z)^{-3} and z(1+z)^{-3}(\log z)^2 on (0,\infty). Taking M_E sufficiently large absorbs this cost into a small portion of (6.12).

Now suppose \lambda=1. The profile estimates give, for sufficiently large E,

To see the uniformity of this assertion, \|\mathcal Y\|\le(1+C\delta)/(1+\tau). On 2e^{-E}\le\tau\le1, the first summand of q_0 gives a logarithmic slope bounded below by a large fixed multiple of one. On 1\le\tau\le e^{3b_0E}, it gives r\ge0.9b_0(1+\tau)^{-1/16} after increasing E. Beyond this interval the logarithmic summand gives r\ge c/q, which exceeds 40/(1+\tau) by its linear-in-\log\tau upper bound and the starting size \tau\ge e^{3b_0E}. These statements are unchanged by taking \delta_0 smaller.

By (6.15), the trace expenditure outside the exceptional interval is at most

For 0\le s\le\sqrt2, the ratio (s+\sqrt2/18)^2/[0.98(1+s^2)] is strictly less than one, by an absolute amount. Thus a fixed portion of P|T|_G^2 remains. On \tau\le2e^{-E}, the explicit left-end profile gives q\le C, r\le C, and D_q\ge c\tau e^E, where constants may depend on the fixed profile but not E. Hence the trace cost is at most Ce^{-E}|T|^2/H^2. Since H\ge\sigma, this is a small portion of (1+\tau)|T|^2/H for E sufficiently large in terms of \sigma.

All cases retain fixed portions of A, W, (6.12), and the symmetrized PG^2 term, proving (6.11). Lemma 4.3 gives nonpositive real sectional curvature off the fiber axis. The metric is positive there and h\ge cV. Smoothness of the original affine fiber potential and its positive limiting metric extend both conclusions across the axis.□

Proof. A prime on \beta now means differentiation in \Psi. Direct differentiation of \psi-F^\beta gives

Here D_\beta is the derivative at fixed x; thus all dependence of the primitive on the spatial cutoff is included. The last term and its first two spatial jets are O(C_f/H^2), because all positive derivatives of \Psi in scaled axes are O(1/H). The derivatives of the scalar D\beta' beyond its value are O(C_f/H). With d_f=1+D\beta'\ge1, these facts show

The constants in \asymp can be absolute by making H large enough; the value d_f itself is bounded by C_f.

First consider x\le X. Since \tau\le e^X, the first two metric jets of h-V are O(C_f/H). Hence

The second estimate follows from Lemma 6.2, the scaled jet estimates, and the curvature perturbation identity, taking \lambda\gg C_f.

The mixed form in Lemma 4.2 is

The last term equals

The spin form, divided by Pq, equals

In fact q_\beta/q^2=-\mathfrak s_\beta; the remaining terms contain two first derivatives of \Psi. Lemma 6.3 and (7.13) give |\nabla_h^{2,0}\Psi|\le C/H with absolute C. Also |\tau\mathcal N h^{-1}\mathcal N|\le C_f/(\lambda H). Choose d_0 first so that the domination in (7.11) is sufficiently strong in comparison with this absolute C, and then choose \lambda,H large compared with C_f. It follows that

This argument also applies to the reverse mixture, using absolute values of \mathfrak s_\beta.

By (7.8) and convexity of the mixture,

Left-end decay and boundedness on the rest of this interval give

It follows from the curvature table that

where the relative error tends uniformly to zero as \lambda,H increase. To verify uniformity at \tau=0, the subtracted term in A is at most C_fP^2\min(1,\tau)^2/(\lambda H), while the leading term is Pq^2D_q. The ratio tends to zero uniformly under the stated choices. Similarly (7.14) and (7.15) make the A_i coupling arbitrarily small in units (AW)^{1/2}.

Finally |K/P|\le C_f/H. Spending fixed portions of A,W according to Lemma 4.3 costs at most

In these displayed bounds C_f is enlarged to include e^X; also P/D_q is bounded on x\le X, including the left endpoint. Both costs are small compared with (7.13). Thus fixed fractions of A, W, and \lambda|T|^2/H remain.

Now let x>X. Since the two profiles coincide exactly here, F^\beta(x)-F^0(x) is independent of x. Consequently \mathcal N=\mathcal N_\infty is independent of \tau, and

There is no infinite tail in this expression. Its first two spatial jets differ from those of V by O(C_f/H). Treating d_f=1+D\beta' at the point as constant, the jets of \mathcal N_\infty differ from those of d_fG_0 by at most C\delta d_f/H, after taking H\gg C_f/\delta; similarly those of V_f differ from \lambda G_0 by a sufficiently small relative amount after taking \lambda\gg C_f/\delta.

The aligned argument can now be applied with the two spatial scales \lambda and d_f. More explicitly, let \theta_f=\lambda/(\lambda+\tau d_f). The comparisons and the identity \nabla^h(V_f+\tau\mathcal N_\infty)=0 give

The K cost is bounded by C\delta^2(\lambda+\tau d_f)|T|^2/H. The rI part of the trace cost uses the same rank-two symmetrized P\mathcal N_\infty^2 reserve as before. The remaining cost is at most

For the last step, use (6.9) and the scalar inequality from the proof of Proposition 6.4 with \lambda/d_f in place of \lambda. We have 1\le d_f\le C_f and may require \lambda\ge C_f, so the logarithm is bounded by \log\lambda. Taking H\ge M_f(1+\log\lambda)^2 with sufficiently large M_f absorbs this last cost. Spin and A_i vanish in this region because q=q_0 is spatially constant.

We have proved positivity and nonpositive curvature off the axis, with fixed fractional reserves. The profiles are smooth in \tau at zero, equal to one there, and have the prescribed left normalization. Thus the potential and metric are smooth in the original affine fiber coordinate, whose metric at the axis has entries V,e^\psi. Curvature extends by continuity. This proves the lemma.□

Completion of Proposition 7.1. Choose d_0 to dominate the absolute spin constants in the interpolation estimate, and choose K_0 so that K_0/4-d_0/2>K_0/9 and K_0/9>10. The latter explicit exponent is larger than the exponent 8 in (7.24). For fixed old data and E, choose L sufficiently large for (7.20) and for the moving-center metric and curvature errors to be less than the retained reserves. Next compute the finite constants C_f for both switches and choose \lambda and then H according to (7.2), with the larger thresholds needed in Lemma 7.2. This is the required noncircular order of choices.

Perform the forward switch with constant old center, the center interpolation with frozen profile q_1 and offset, and finally the reverse switch with constant new center. Lemma 7.2 controls the two switches and the preceding argument controls the actual center motion. The cutoffs are flat-ended, so all joins are smooth. At the end the profile is q_0, the center is the prescribed new constant, and the added offset is w^+(1)+w^-(1)\ge0. All profiles are at least one, have smooth left endpoints with value one, and have normalized primitives tending to infinity on the right; thus they define smooth fiber potentials for all affine fiber values throughout. The spatial projection bound follows on each part, with a fixed fractional loss during the center motion. No bound on the final constant offset is required or asserted.□

Proof. Homogeneity and the spherical upper bound give |(\mathcal H,p_N)(s/R)|\leq Ne^l. Cauchy estimates at scale c_0\sqrt t/N bound its derivatives of order at most four by this quantity times a fixed power of N, since t\geq1. In a differentiated logarithm there are only a fixed number of factors and denominator powers; the denominator in (8.4) is at least e^{-2c}. This proves (8.5) with an absolute exponent; fixed-dimensional constants and d affect only its prefactor.

The spherical lower bound also gives |(\mathcal H,p_N)(s/R)|\geq e^lN^{-C_n}. Once l exceeds a sufficiently large multiple of \log N, subtraction of (0,1) leaves norm at least e^lN^{-C_n}/2. The factors e^l in derivatives of the logarithm can then be canceled. The remaining costs are fixed powers of N, with exponents depending on the already fixed n. Increase L_n to pay them and the degree threshold to pay the prefactor.□

Proof. On the spatial region following the previous center-change interval, the center is \zeta_j and \ell(s,\zeta_j)=B(s), including on shear supports. The same identity holds on the initial cap at the first stage. Thus the bound there follows from the fixed cap continuation up to \nu_B+2/Y_B. The earlier spatial region is a fixed compact set K with already chosen profiles and centers; future stages do not change q(x,s) on K. The reservation gives |\psi-\psi_{\rm old}|\leq1 there, including the current and all later wells. Both centers are in the unit disk, so |\zeta_j-\zeta(s)|\leq2. Choose A>\log4+\sup_K\psi_{\rm old}+1. For each s\in K, F(x,s)\to\infty as x\to\infty. Continuity, a finite cover, and monotonicity in x give a common finite X with F(X,s)>A on K. Inverting the fiber relation therefore gives \tau\leq e^X. Since q\geq1, the radial potential is bounded by \sup_K B+e^X; the fixed shear adds at most 2\sup_K|c_{\rm sh}|. At center coincidences this argument is understood by the smooth limit \tau=0. These bounds are independent of c and of the future wells. Add two and increase to at least j.□

Proof. For raw estimates use Euclidean axes. Here t\geq1, Y\asymp Y_B, and \log Y_B\leq C u_s(\nu_B); the last estimate follows by integrating d\log Y/du_s=\kappa(1+Y^{-1})\leq2.2 for Y\geq1. Differentiation of a'=aY/t bounds the required radial metric jets by e^{C_{\rm rad}u_s}. Their inverses cost no more. The radial curvature estimate gives

since V\geq aI and \kappa H=C_\kappa Y/((1+Y)ta)\geq c/(ta). The initial severity pays these bounds, the old data and the future allowance. The trigger cost e^{4E/k_b} fits the quiet raw exponent. The current well is small before saturation. Also G\geq V, since all nonconstant extra twist terms in this interval are plurisubharmonic.

For the covariant condition, write

and use primes for \nu derivatives within this proof. On the severity-varying interval,

where \varepsilon_b can be made as small as required. Away from the pole, h_b and its relevant derivatives are bounded, whereas \mathcal B'=Y/H and \mathcal B''=Y(1+Y)/H. The choice \sigma M_bk_b^2\ll1 makes these background terms dominate. Near the pole let D_b=Y_B/(1-\xi). Then

and |E^{(r)}|\leq Ck_b^rD_b^r for 1\leq r\leq3, including saturation. These prove (8.27) and make (8.28) small by first enlarging M_b and then decreasing \sigma. The fixed intervening interval of \xi is covered by the first estimates. We also have \mathcal U''\geq1.

At fixed \tau, the radial part of the Schur metric is the Hessian of

where the bar denotes averaging from zero to \tau in the \tau variable. Bounded severity derivatives of \bar F and (8.28) show that the first two derivatives of \mathcal U-\bar F are positively comparable with those of \mathcal U and its third derivative is bounded by C(\mathcal U'')^{3/2}. Use affine axes normalized by the Hessian of \mathcal U, with radial radius \sqrt{t/\mathcal U''} and transverse radii \sqrt{t/\mathcal U'}. At a radial point, the only nonzero holomorphic connection types for a radial Hessian metric are radial–radial to radial and radial–transverse to transverse. Differentiation gives the bounds

They are bounded by (8.27). No estimate of a radial/transverse condition number is used.

Write the remaining Hessian as \tau G_e, with G_e\geq0. Its size and first jets in these axes are bounded: old Euclidean bounds are dominated by the large initial a, and the current and future wells have the reserved absolute bounds. The normalizing radii are at most a^{-1/2} because this background is at least V\geq aI. If h_r denotes the radial Schur metric, the exact connection difference is, with the usual index placement,

Since h_r\geq c\tau I, \tau h^{-1} is uniformly bounded for all \tau>0. Thus the full connection is uniformly bounded as well.

In these axes the first derivative of E has norm |E'|/\sqrt{\mathcal U''}; the complex Hessian entries are bounded by |E'|/\mathcal U' and |E''|/\mathcal U'', and the only pure second derivative has size |E''-E'|/\mathcal U''. Consequently subtracting the connection times the first derivative gives a small \nabla_h^{2,0}E. Since the actual G is at least the normalizing background, the same smallness holds in G units. This proves all quiet smallness conditions uniformly in \tau.

After saturation, severity derivatives vanish. Through l=4c the well’s raw costs are bounded by e^{C(c+\log N)}; the post term below has cost at most a fixed multiple of N^4e^{2m_zc}. Both fit \epsilon E_* by (8.9), while the inverse and radial curvature estimates persist.□

Proof. Each derivative of a function of l costs a fixed power of N. Thus Z_{j,\rm out} has positive-order bounds through order four of size C N^4e^{m_z\max(0,l-2c)}. These and (8.5) fit the raw quiet exponent by (8.9) and (8.36). Other radial and old-data bounds remain fixed on the cap continuation. Through l=4c, severity is constant and the preceding lemma applies.

Where severity varies, take \mathcal U=\Psi+Z_{j,\rm out}. Before division by t, its transverse and radial Hessian entries have sizes

For l\geq4c, m_z(l-2c) exceeds C_{\rm pol}(l+c+\log N) by a fixed positive multiple of c+l. Thus this Hessian dominates the current well and its first Hessian jets in normalized axes. The affine Z_{j,\rm in} tail has coefficient at most a fixed factor times e^{E_*/k_b}\ll N; old terms and future tails are covered as before. Moreover \mathcal U''/\mathcal U',|\mathcal U'''|/\mathcal U''\leq Cm_zN, while \sqrt{\mathcal U''}\gtrsim m_zN e^{m_z(l-2c)/2}. Severity derivatives of orders r\leq3 are at most CB_*N^r. Thus the connection ratios hold and all severity ratios are small. Equations (8.29) and (8.30) again prove the pure Hessian condition uniformly for all \tau. Every extra Hessian is positive semidefinite in this interval.

Past T_*+4 the new well satisfies (8.6), and the Z terms are affine in \nu. Their positive-order Euclidean jets are bounded on the entire outer range by their endpoint bounds up to fixed constants. Their logarithmic costs are at most C(1+m_z)T_*+4\log N+C_{\rm old} with an absolute C. The absolute constant C_* in (8.9) was chosen to pay this inequality too. Equation (8.36) then implies (8.37), after increasing c for the finite sum of old bounds, the number of generated terms, and dimensional prefactors. The dimension-dependent L_n multiplies T_* on both sides and changes no leading comparison.□

Proof. Initially grow H with \kappa=\kappa_* and p=\lambda=1. Use the \lambda=1, H\geq\sigma alternative of Proposition 6.4. Ultimately we will lower p slightly below one for a bump, thereby increasing \lambda, and then restore p=1 and return smoothly to \kappa=1.

We first justify the order in which the targets are chosen. During the growth and the bump, p\leq1 and \kappa=\kappa_*, so d\log H/du_s\geq.08. Thus

This bounds the total increase of \Psi before the final transition independently of the size of either target. The transition from \kappa_* to one takes a fixed \log Y interval and leaves H nondecreasing, adding another fixed bound to \Psi. Moreover \kappa\geq1 gives d\log Y/du_s\geq1, hence

through the growth and holding intervals. The old nonradial twists therefore have bounded values on one fixed compact spatial interval, independently of the targets. Their scaled derivatives are bounded by the fixed Euclidean bounds, since the scaled coordinate radii are at most one. Derivatives of \Psi in these axes are O(1/H)\leq O(1/\sigma). The forward switch and center move each occupy a fixed amount of the variable \Psi; these amounts can be included in the same bounds. Include the reserved future value and jet allowance as well. These are exactly old-twist bounds independent of \lambda,H, as required by Proposition 7.1.

Choose its shift L_0 from these bounds and E_{\rm post}. This fixes its profile constants C_f and a sufficient target \lambda, and that target determines a sufficient target H. This is the order of parameter selection in Proposition 7.1. The radial path reaches the targets in the other order: grow H first until it exceeds every required multiple of

Choose a smooth bump of 1-p\geq0 with small amplitude and small derivatives in \log Y, and with integral in u_s equal to the desired \log\lambda. Such a bump exists by making its middle plateau arbitrarily long; the endpoint ramps can have fixed small amplitude and sufficiently long fixed length, while the plateau length supplies the remaining integral. The conversion d\log Y/du_s=\kappa(1+Y^{-1}) and its bounded derivatives show that bounds through order two in \log Y follow from the same bounds in u_s. Throughout this bump H increases, and the already chosen lower bound for H pays the entire range 0\leq\log\lambda(s)\leq \log\lambda. Apply the large-H alternative of Proposition 6.4.

Restore p=1 and lower \kappa smoothly to one. The downward transition can be made slowly enough in \log Y that Y\kappa_Y\geq-\kappa, with absolute derivative bounds. Since \kappa\geq1 here, H stays large. Now \kappa=p=1, so both H and \lambda are constant. Perform the forward profile switch, the center move, and the reverse switch of Proposition 7.1. Interpolate the two center values by a flat-ended convex combination, keeping the center in the unit disk. The old bounds and targets have been chosen in the order just proved. The changes take a bounded interval of \Psi, hence a finite interval of u_s because d\Psi/du_s=1/H and H is fixed, even though this interval can be very long. Restore q_0 and retain the new constant offset. The center is now \zeta_{j+1}. The offset may be arbitrarily large, but it is now constant and changes no derivative of the twist. Its value becomes part of the old data when the next testing height is chosen, before the next accuracy parameter c.

Raise \kappa to \kappa_* again and use a bump with 1<p<\kappa_* to bring \log\lambda down to zero. A small positive plateau and fixed ramps give any prescribed finite integral of p-1; stop precisely when that integral equals the existing \log\lambda. Thus \lambda never falls below one. Also

so the large-H aligned condition remains valid. Restore p=1.

Finally reduce H to \sigma. Transition \kappa to 1/2, hold there for an adjustable interval, then return to one from below. For p=1 the exact relation is

Its value is -1 on the middle plateau. The endpoint transitions have fixed finite effects on \log H; the initial H can be made as large as needed. Choose the plateau length so that the final value is exactly \sigma. Throughout this descent H\geq\sigma, \lambda=1, and 1/2\leq\kappa\leq\kappa_*, so the \lambda=1 aligned alternative applies. Choose the downward transitions slowly enough to preserve the same derivative inequality as above.

All these intervals have finite u_s or finite \log Y length. Equations (8.1) show that Y remains finite on each finite u_s interval and that the radius advances by a finite positive amount. A putative pole of an indefinitely continued accelerated equation is therefore never reached during an episode. This proves the claim.□

Proof. The spatial scale u_s may now be enormous, and E_{\rm post} need not dominate it. Proposition 5.2 permits any fixed holomorphic affine frame at the evaluation point. We first use V-unit affine axes, where the curvature scale is the fixed \sigma and the relevant jets are bounded independently of u_s. This will justify the initial severity increase before arranging the size inequality needed for the next trigger.

First hold \kappa=p=\lambda=1 and H=\sigma. Let E increase smoothly as a function of u_s, starting at E_{\rm post}, with a fixed flat-ended transition to slope 4C_\uparrow. Its derivatives through order four in u_s are bounded by constants depending on the fixed parameters, not on \sigma or the possibly enormous u_s at entry. Continue long enough that

and any desired fixed threshold. The greater slope ensures that this occurs after a finite interval.

We verify the quiet criterion even before (8.41). Use V-unit affine axes. The scaled-jet radial estimates give metric jets of orders r=1,2 bounded by CH^{r/2} in these axes, and scaled derivatives of u_s through order four are bounded by an absolute constant. Therefore

The extras in (8.39) satisfy the same bounds after this normalization, and G\geq V. For \sigma small, the severity terms in \mathcal N are small relative to G, so h\asymp(1+\tau)V, with first derivatives bounded in these axes by C(1+\tau)\sqrt H. Its connection is consequently O(\sqrt H) uniformly in \tau. The first, complex second, and pure covariant second severity derivatives are small for every \tau.

The raw curvature margin on this hold is C^V(T,T)\geq c\sigma|T|^2 in V-unit axes, because \kappa=1 and H=\sigma. All other raw data are bounded there by fixed constants. The previously chosen E_{\rm post} exceeds the fixed profile and \sigma thresholds, so it pays this margin even when it is much smaller than the current u_s. Thus there is no gap before severity catches up.

After (8.41), keep the same slope while turning \kappa into a cap. Here is an explicit admissible transition. If Y_0 is its starting clock value, set

where \eta_0 is smooth, equals one near 1, decreases nonnegatively to zero, and is zero for arguments at least two. For Y\leq Y_0 use A=1+Y. Then 1\leq A\leq1+Y, so (1+Y)^{-1}\leq\kappa\leq1. Moreover

The first two derivatives of \kappa in \log Y have absolute bounds, because Y/Y_0 ranges over a fixed interval while \eta_0 changes. For example, writing b=Y/(1+Y) and D=Y\partial_Y, one has D\kappa=b(\eta_0-\kappa) and D^2\kappa=b(1-2b)(\eta_0-\kappa)+bD\eta_0. At the end A is the new constant C_\kappa. Since \kappa\leq1 and p=1, H is nonincreasing and stays at most \sigma.

The same normalized jet and connection estimates continue through the transition and the cap. To check the raw curvature margin there, note that Y\geq t: it holds initially, and d\log Y/d\nu=\kappa(1+Y)\geq1 preserves it. Also u_s\geq t after choosing the initial constant large enough, since du_s/dt=Y/t\geq1. Hence

and (8.41) with large C_\uparrow pays the quiet lower bound e^{-\epsilon E}. The positive-order extras stay bounded relative to V on an arbitrarily long cap. More explicitly, a fixed Euclidean jet bound D_{\rm out} costs at most C(D_{\rm out}/a)H^{r/2} in V-unit axes for Hessian jets of order r\leq2, since aH=Y/t\geq1. The ratio only decreases as a grows. Future wells are included in the fixed allowance until their own activations.

Coast as far in \nu as desired. Shortly before the next chosen trigger, turn off the severity slope smoothly over a fixed-length u_s interval. The margin in (8.41) is sufficient to retain E_0\geq C_\uparrow u_s(\nu_B) after this bounded final increment, by making the cap endpoint large enough. The existing raw jet bounds can also be absorbed by making that endpoint large. This produces (8.17) and the next cap data.□

Proof of Proposition 8.1. The recursion gives a smooth strictly plurisubharmonic potential on \mathbb C^{n+1}, with nonpositive real sectional curvature by the local lemmas and completeness by the preceding argument. Every required pair (\zeta,h) is tested infinitely often. The stages satisfy R\geq1, N\to\infty, c\to\infty, \log N\leq.005c<c/100, D_j\to\infty, and D_j/c_j\to0. Lemma 8.3 supplies the height bound. The node disks, sphere derivative bound, and shear amplitude have the fixed exponents (8.12). Their phases were chosen as permitted by Proposition 3.5, and their geometric realization allows all those choices. That proposition proves the asserted Liouville property.□

Completion of the proof of Theorem 1.1. Apply Lemma 2.1 to the base just constructed. Its tube is connected, complete, and diffeomorphic to Euclidean space, hence simply connected. Its real sectional curvatures are at most -1 by that lemma, and its bounded holomorphic functions are constant by Proposition 8.1. The complex dimension is the fixed finite m=n+2.

There is no finite lower sectional-curvature bound. At s=0,y=\zeta_1, the center and profile are constant nearby and there is no shear. On the center axis the spatial curvature tensor of the base equals that of V: the added radial-fiber potential vanishes to second order in y-\zeta_1, and its cubic derivatives with two spatial indices and one fiber index vanish there. The initial radial metric has a strictly negative holomorphic-plane curvature in a spatial direction. Corollary 2.2 therefore shows that the tube’s sectional curvatures are unbounded below. This proves every assertion of Theorem 1.1.□