Spectral scalar curvature and uniform Urysohn width

1 Introduction

Positive scalar curvature is expected to force a manifold to lose two dimensions at a controlled geometric scale. A spectral lower bound mixes scalar curvature with the energy of a test function and permits negative scalar curvature on parts of the manifold. We show that the uniform width conclusion survives this weakening. The geometric input is the stabilized width theorem of [14], stated precisely in Theorem 2.1; the work here constructs its hypotheses from the spectral inequality.

Let (M^n,g) be a connected complete smooth Riemannian manifold without boundary. We use \Delta_g=\diver_g\nabla_g and integrate against the Riemannian volume density. The normalized spectral scalar-curvature bound is

Thus -4\Delta_g+\Scal_g\ge1 in the quadratic-form sense. This is the definition on noncompact manifolds as well; it does not require a lowest eigenvalue to be attained.

For a metric space X and an integer d\ge0, its Urysohn d-width is

where f is continuous and K is a simplicial complex of dimension at most d with its usual polyhedral topology. The diameter refers to the entire fiber, including all its components; empty fibers may be omitted.

The constant depends only on the dimension. The theorem applies to compact and noncompact manifolds, without orientability, spin, or bounded-geometry assumptions. If the right side of (1.1) is multiplied by \lambda>0, scaling the metric to \lambda g gives the fiber bound C_n/\sqrt\lambda.

1.1 History and relation to earlier work

Gromov’s macroscopic dimension program relates positive scalar curvature to the dimension of a space seen at large scales. The quantitative formulation asks whether a fixed positive scalar-curvature normalization forces a dimension-dependent bound for the fibers of a map to an (n-2)-dimensional polyhedron. This formulation goes back to [4, § 2.A(c)]; see also [5, § 2\frac12] and [7]. Uniform width is stronger than merely asking for a finite fiber bound for each individual metric. The stabilized partition theorem used here is the geometric input from [14] that gives the uniform conclusion.

Several earlier results illustrate the distinction between the geometric and topological versions of the problem. Early three-dimensional distance-level diameter estimates of Gromov–Lawson [6] use vanishing rational first homology, as clarified in Gromov’s later account [7]. That account also gives a uniform Urysohn 1-width bound for arbitrary complete three-manifolds with scalar curvature bounded below by a positive constant. Under such a uniform lower bound, Liokumovich–Maximo [11] and Liokumovich–Wang [12] construct Morse functions with simultaneous area and diameter bounds on connected level components, in the closed and complete noncompact cases, respectively. In dimensions four and five, Chodosh–Li–Liokumovich proved finite Urysohn 1-width for universal covers of closed positive-scalar-curvature manifolds under the respective assumptions \pi_2=0 and \pi_2=\pi_3=0 [2, Theorem 4]. That conclusion is qualitative and uses the topology of a compact quotient. The setting of Theorem 1.1 instead allows arbitrary complete manifolds and asks for one constant depending only on the dimension.

The coefficient 4 in (1.1) is natural in weighted scalar curvature. On closed manifolds, Perelman’s \mathcal F-functional and its lowest eigenvalue formulation involve -4\Delta+\Scal, together with the expression \Scal+2\Delta t-|\dd t|^2 [15, §§1–2]. Gromov’s torus-stabilized scalar curvature makes the same connection through diagonal warped products [8, §§1.A–1.B]. Spectral curvature bounds also enter the band and cube inequalities of Hirsch–Kazaras–Khuri–Zhang [9], whose hypotheses depend on the dimension, topology, and coefficient of the scalar-curvature term. Those inequalities control separations of boundary sets, rather than the fibers of a map on an arbitrary complete manifold.

Our starting point is the partition theorem in [14], including its locally finite stabilized input. The spectral-to-positive-solution passage and the warped-product formula are established tools. The additional work is to create a quadratic reserve from an unrestricted logarithmic drift and to adapt the local stretching argument to a signed drift. These constructions retain the original metric as the metric in which the final fibers are measured. In particular, no control of a conformal factor or of the weight at infinity is needed.

1.2 The new constructions and the proof

The spectral hypothesis supplies a positive smooth function v with -4\Delta_g v+\Scal_gv\ge v. For a metric h and a smooth real function t, write

Then w=-2\log v satisfies \Dcal(g,w)\ge1. This identity produces a weighted curvature bound, but gives no control on the value or oscillation of w. The stretching construction needs an additional reserve that grows quadratically with its drift.

The first new ingredient supplies that reserve without estimating w. For an arbitrary complete metric h and smooth w, we construct a smooth function u, a global graph metric h^+=h+\dd u^2, and a new signed drift t^+ such that

Here C(n) is independent of the geometry and of w. The graph equation has pressure proportional to the discrepancy u-w; subtracting the graph height in the drift update turns that discrepancy into the required square. Local barriers and an angle estimate give a global solution even when w and the geometry are unbounded at infinity. The precise statement and proof are in Proposition 3.2.

The second ingredient stretches selected pairs of sets by increasing the metric on a compact neighborhood. Its pressure depends on a possibly negative drift. For a dimension-dependent T, the reserve is V(t)=t^2+T^2. When the drift increases from t to t'\ge t, the local estimate retains the positive part of V(t')-V(t), because increasing a signed drift need not increase V. Its loss is paid by a decrease of \tr_h h_0, where h_0 is fixed during one round of stretching, together with an arbitrarily small summable error. The inverse construction of the pressure is given in Proposition 4.1, and the resulting geometric estimate is Lemma 5.1.

To assemble these tools, choose a radius D depending only on n and maximal D-separated centers x_i in the original metric. Their closed radius-D balls cover M, and their closed radius-2D balls are locally finite. The local lemma is applied in rounds until each inner ball is as far as prescribed from the exterior of its outer ball. The trace decreases telescope within each round, independently of the number of overlapping balls. Each index i requires only finitely many rounds, and every compact set meets only finitely many outer balls, so the final metric and drift are smooth and eventually unchanged on each compact neighborhood.

The final bound \Dcal(h,t)\ge1/2 is converted into the stabilized scalar-curvature inequality by expressing -t as a locally finite sum of compactly supported logarithmic warps and subdividing each summand until the sum of squared gradients is small. Theorem 2.1 then supplies a map whose entire fibers are bounded in g. Section 6 carries out the parameter choice and this last passage. Thus the graph reserve and the signed stretching estimate are the analytic contributions; the topological dimension drop is the conclusion of the stated geometric input.

2 A partition theorem for stretched data

The passage from scalar curvature to a map of controlled fiber diameter uses the partition construction of [14]. We state the precise consequence needed here. Its distances are measured in two metrics: a metric g fixes the scale of the eventual fibers, while a larger metric h separates prescribed pairs of subsets. The work of the subsequent sections is to construct such an h from the spectral hypothesis. Here h\ge g means quadratic-form domination: h_x(v,v)\ge g_x(v,v) for every x\in M and v\in T_xM.

For a smooth metric h and a finite family of smooth real functions p_\alpha, the scalar curvature of the diagonal circle metric

is given by the standard multiple-warp formula [8, Section 1.A]:

We call p_\alpha a log warp. The same expression makes sense for a countable family whose compact supports are locally finite: every point has a neighborhood meeting only finitely many supports. On that neighborhood, (2.1) is the scalar curvature of a metric on its product with finitely many circles. Adjoining a circle of constant length one changes neither the formula nor the weight e^P\,\dd\vol_h.

We use this theorem as an imported result. It follows from Lemma 6.1 and Proposition 6.6 of [14], followed by Proposition 7.1 and Equation (7.14) there. The threshold D_{\mathrm{nat}}(n) retains the scale chosen in that paper, where the initial scale may be enlarged.

The pointwise scalar-curvature theorem of [14] first constructs h and the log warps from \Scal_g\ge1. Its partition construction begins with these data already in hand. Lemma 6.1 there uses the separations in (2.2) and the initial scalar requirement

which follows from 2/5. Its scalar hypothesis concerns R(h,(p_\alpha)); a lower bound for \Scal_g is not required at this stage.

The countable version in [14] expressly permits local finiteness of the initial log warps. The measure e^P\,\dd\vol_h is smooth and positive, and every compact region has a neighborhood on which only finitely many factors occur. The cited construction first separates M into pieces confined to compact radial regions; label 0 is reserved for this step. Product cylinders are then attached along boundary hypersurfaces, with the metric and warp data independent of the cylinder coordinate. The localization and induction in Propositions 6.4 and 6.6 ensure that each analytic piece at every later finite stage has only finitely many relevant initial warps on its entire extent, including its product ends. Thus the initial data need no uniform bounds on the geometry or the weight at infinity. The finiteness and geometric bounds needed on the later pieces are conclusions of the imported construction.

The reconstruction produces a continuous map q:M\to M with d_g(x,q(x))<1 and continuous coordinates a_i:M\to[0,1] with locally finite supports. They satisfy

and at each point some coordinate equals 1 while at most n-2 coordinates lie strictly between 0 and 1. These vectors lie in a cubical complex of dimension at most n-2. Its compatible staircase triangulations give K. Local finiteness of the coordinate supports places the map locally in a finite subcomplex, which gives continuity for the polyhedral topology. Equality of images preserves the full coordinate vector. Thus two points in one fiber have a common coordinate equal to 1; their q-images lie in the same original outer ball, and their g-distance is less than 4D+2.

To apply Theorem 2.1 under the spectral hypothesis, we must construct the data in (2.2) at a uniform initial stretching scale even though the positive supersolution associated with the spectral inequality can have unbounded logarithm and gradient.

3 From spectral positivity to a quadratic reserve

Recall the weighted scalar expression \Dcal, and define its associated weighted divergence by

The spectral hypothesis first supplies a function w with \Dcal(g,w)\ge1. The difficulty is that it supplies no uniform control of w. Our aim in this section is to enlarge the metric and replace w by a new function for which the scalar inequality contains a positive quadratic term. This term will pay for the later distance enlargements.

The passage from a quadratic-form inequality to a positive solution is part of Allegretto–Piepenbrink theory [1, 13]. We include the exhaustion argument in the form needed here.

The reserve construction applies to arbitrary smooth functions, not only to the function supplied by the spectral hypothesis.

We shall use the metric induced by a graph of height u in (M,h)\times\R, prescribing its weighted mean curvature by

The pressure measures the discrepancy between the graph height and the original function, even when both are unbounded. For a candidate height u, put s=|\dd u|_h and l=(1+s^2)^{-1/2}. The induced metric is h_u=h+\dd u\otimes\dd u, and its volume density is l^{-1}\dd\vol_h. Thus t_u=t-\log l is exactly the change preserving the weighted density:

For t=w and pressure p=10(u-w), our final drift will satisfy

This is the quantity whose square the graph equation must supply. The scalar calculation below produces a positive p^2 term, together with slope and derivative terms that we will also estimate. We first construct a smooth solution of the graph equation; the scalar comparison and the square completion will then be carried out consecutively.

3.1 The angle equation and elliptic estimates

To construct the prescribed graph, we need an a priori gradient estimate on a possibly noncompact manifold. The useful gradient variable is the vertical component of the graph normal: a positive lower bound for this component is an upper bound for the slope. We record the equation it satisfies, allowing angle-dependent pressure for use in the later local construction. The use of normal variation and comparison to estimate graph gradients has a classical predecessor in Korevaar’s treatment of prescribed mean curvature [10]; we give the weighted calculation and all estimates used here.

We will use the following local elliptic facts in both graph constructions. In coordinates, the equations have the form

where the volume density may be included in \mathfrak A and \mathfrak B. Suppose these functions are smooth on a neighborhood of a compact range of (x,u,Du), with bounded derivatives there, and that \mathfrak A^i_{p_m} is uniformly elliptic on that range. For a family of equations, the stated coefficient and ellipticity bounds are assumed uniform over the parameters under consideration. Uniform height and gradient bounds for smooth solutions then give interior bounds of every order. For completeness, differentiating in x_j and writing b=\partial_j u gives

Subscripts x_j on the coefficient functions mean explicit derivatives, with the other arguments held fixed. Thus this is a uniformly elliptic divergence equation with bounded coefficients, bounded lower-order terms, bounded scalar data, and the divergence of bounded vector data. The interior De Giorgi–Nash estimate [3, Theorem 8.24] gives a uniform Hölder bound for b, and hence a C^{1,\alpha} bound for u, for some \alpha>0. The nondivergence form of (3.4) now has Hölder coefficients and right-hand side. Interior Schauder estimates [3, Theorem 6.2 and Corollary 6.3] give C^{2,\alpha} bounds, and differentiation yields bounds of all higher orders; compare also [3, Theorem 13.1] for this differentiation argument. On a closed manifold a finite coordinate cover makes these estimates global.

In particular, a continuation on a fixed closed manifold is closed once uniform height and gradient bounds keep the solutions in a compact subset of the open equation domain. It is open when the linearization has strictly negative zeroth-order coefficient. Indeed, for the resulting smooth uniformly elliptic operator, interpolation with \Delta-1, the maximum principle, and Schauder estimates give invertibility C^{2,\alpha}\to C^{0,\alpha}. The starting operator \Delta-1 is invertible by the energy method and elliptic regularity. The implicit function theorem then gives openness, and elliptic regularity gives smooth solutions. All constants in this discussion may depend on the individual compact manifold or the fixed coordinate charts.

3.2 A global graph with linear discrepancy pressure

We now solve the graph equation without imposing any global size condition on w. The positive derivative of the pressure with respect to height will provide both comparison and the angle bound.

3.3 Scalar comparison for a graph enlargement

The global graph is now available, and we turn to its scalar curvature. For the later localized construction we allow a more general metric which stretches only the line normal to a level set of the height function. Writing this length multiplier as 1/l, we again use t_u=t-\log l to preserve the weighted density. The following calculation extends the graph comparison underlying the local deformation in [14, Section 3.5, Lemma 3.4].

Identity (3.9) explains the role of M_1: it is the weighted divergence of k_1e, divided by k_1. For the ordinary graph choice \theta=s^2/(1+s^2), the vector field is k_1e=\nabla_hu/\sqrt{1+|\dd u|_h^2}. If this divergence is prescribed to be p, then M_1=p/k_1 and the term \theta M_1^2 in (3.10) is precisely p^2. The terms involving K and e(M_1) still have to be controlled. For the graph just constructed, the linear discrepancy pressure will control them through the following square completion.

3.4 Completion of the reserve estimate

We can now turn the discrepancy pressure into the square in Proposition 3.2. The argument is pointwise; consequently the compact-data constants needed to construct u will not enter the reserve constant.

The new function t^+ may have either sign and need not be bounded below. Proposition 3.2 controls its square with a constant depending only on the dimension. The local enlargement must therefore use a pressure law that works for every real value of the drift parameter; this is the purpose of the next section.

4 A pressure law for a signed drift

The local metric enlargement will be obtained from a graph equation whose right-hand side depends on a height discrepancy and on the drift. We need this dependence to be increasing in the discrepancy, with poles that confine the graph height. We also need quantitative control of its drift derivative for every real value of the drift. The following construction adapts the pressure method of [14, Theorem 2.1] to this signed parameter.

In the local scalar-curvature comparison, the alternative permits a negative term proportional to F_x to be controlled either by a small multiple of the reserve V(t) or by a positive contribution from -FF_t. Where the graph is nearly horizontal, the comparison also contains a positive multiple of F^2. The last inequality lets us absorb the \epsilon F^2 part of the derivative loss into this positive term and reduce the remaining reserve loss by making the metric modification small there.

5 Local stretching with a signed drift

We now enlarge the distance between two closed sets while retaining the quadratic reserve constructed in Proposition 3.2. The enlargement must have compact support: this will allow infinitely many applications on a noncompact manifold. We adapt the local graph construction of [14, Theorem 3.1], using Proposition 4.1 to allow the drift to take either sign.

The loss in the following statement has two parts. One is charged to a decrease of a metric trace; these trace decreases telescope in the iteration. The other can be chosen arbitrarily small. We write a_+=\max\{a,0\} for the positive part of a real number.

The proof constructs a graph that follows a large height function near A and becomes small before leaving its W-neighborhood. At appreciable slope, the graph metric increases lengths. At small slope, a modified graph metric makes the curvature error arbitrarily small. After solving the graph equation on a compact extension, we estimate its scalar curvature and then cut off the change.

5.1 Slope modification and spatial cutoffs

Fix the constants

Apply Proposition 4.1 with these parameters, obtaining S_0,T,F. Throughout this section, N(t)=\sqrt{t^2+T^2} and V(t)=N(t)^2. We may take W to be a numerical constant at least 10^5; this size allows the cutoffs below and the final length comparison.

Let 0<a\le1/2. This parameter will be fixed small enough in terms of the prescribed loss d. Choose a smooth nondecreasing function \sigma:[0,\infty)\to[0,1] that is zero on [0,s_0], one on [5s_0,\infty), and satisfies 0\le\sigma'\le60. For s\ge0 set

Thus the graph modification of Lemma 3.5 is the ordinary graph metric when s\ge5s_0, whereas its size is reduced by a^2 near zero slope. For the graph equation, we use the multiplier

It is smooth as a function of the gradient covector, and j(0)=as_0>0. This keeps the derivative of the pressure with respect to height positive at critical points of the graph. The same multiplier will also make the small-slope curvature error proportional to a^2.

We next arrange the spatial support. Write r_A(x)=\dist_{h_0}(A,x). Choose smooth compactly supported functions \psi\ge0 and \eta_1,\zeta\in[0,1] such that

These cutoffs require only completeness of h_0. Indeed, finite-radius closed neighborhoods of the compact set A are compact. Start with Lipschitz functions of r_A, leaving positive buffers around each specified constant region. In particular, smooth (W/3-r_A)_+ with value error less than W/100 and gradient bound 1.05, then divide by 1.05 to obtain \psi. The profile for \eta_1 can be smoothed with gradient at most 12/W\le s_0/8. To carry out the smoothing, use convolutions in sufficiently small charts on the compact neighborhood, where metric distortion is as small as desired, and a partition of unity. Choose the local approximation errors small enough to control the partition derivatives. Nonnegative kernels preserve the range, and the buffers preserve the stated constant regions. The same procedure supplies \zeta.

For 0<\rho\le1/2, put

Thus \eta=1 near \supp\psi, while \eta=\rho outside \{r_A<.8W\}. Moreover, since h\ge h_0,

The square in (5.6) gives the second estimate uniformly as \rho\downarrow0. For each \rho we will construct a smooth graph height u_\rho close to \psi. The arrangement in Figure 1 leaves a region where the graph tends to zero before the final cutoff \zeta begins to vary.

5.2 Solving the graph equation and obtaining locality

Work for the moment on a closed auxiliary manifold extending a neighborhood of \{r_A\le1.1W\}. If M is compact, use M itself. Otherwise, enclose this compact neighborhood in a relatively compact smooth domain, modify the data to product form near its boundary, and double. Extend h,t smoothly and the compactly supported cutoffs by zero. Define \eta from the extended \eta_1 as above. This auxiliary manifold is fixed as \rho tends to zero; all geometric constants in this subsection may depend on it.

For 0\le\lambda\le1, consider the equation

We solve it within the open domain |u-\lambda\psi|/\eta<S_0. Both \mathcal A and j are smooth functions of the covector \dd u, including at zero. The flux derivative has tangential and radial eigenvalues k_1/s and (k_1)_s, respectively; both are positive for s>0 and tend to a at zero.

We first obtain estimates for every smooth solution in this domain. There is S_1<S_0, independent of \rho,\lambda, such that

To see this, compare with \lambda\psi\pm S\eta for S<S_0. Their first two derivatives are uniformly bounded as S\uparrow S_0, whereas j\ge as_0 and the poles of F are uniform on the compact range of t. For S sufficiently close to S_0, these functions are strict upper and lower barriers. At a positive maximum of a violation, the gradients agree, ellipticity orders the second derivatives, and F_x>0 orders the right sides, a contradiction. The same fixed S therefore works for all \rho,\lambda.

There is also a gradient bound independent of \rho,\lambda. At a point of maximum slope, there is nothing to prove if s<1. Otherwise \mu=1 in a neighborhood, and (5.8) is the ordinary graph equation of Lemma 3.3, with

By (5.9), the values of F,F_t,F_x range in a compact set uniformly in \rho,\lambda. Holding \omega fixed and writing \xi=(u-\lambda\psi)/\eta, we have on the graph

Thus the zeroth order coefficient of the angle operator is bounded below by -C/\eta, after discarding the nonnegative term |\mathrm{II}|^2. Here C is independent of \rho,\lambda. On the other hand,

at the point under consideration. At this minimum of \omega, the angle equation gives

It follows that \omega is uniformly bounded below, as required.

For each fixed \rho>0, these estimates confine the solutions to compact jet ranges strictly inside the domain of the equation. The elliptic estimates described in Section 3.1 give all higher bounds. The linearization of the left side minus the right side has zeroth order coefficient -jF_x/\eta<0, so it is invertible on the closed manifold by the maximum principle and linear elliptic theory. The method of continuity therefore applies from the solution u=0 at \lambda=0 to \lambda=1. Write u_\rho for the resulting smooth solution. The higher estimates used for this continuation may depend on \rho; the height and gradient estimates do not.

The graph equation has now produced a height close to \psi. Its next property is what permits a compactly supported change of metric:

In such a region \eta=\rho. Expanding (5.8) in a coordinate chart gives

The previously established gradient bound makes (a^{ij}) uniformly elliptic and bounded and makes b^i bounded. The first order terms have this form because the flux vanishes at zero covector. By (5.9) and Proposition 4.1, c_\rho is bounded above and below by positive constants, independently of \rho.

On a coordinate box with closure in this region, let \ell_f be the inward affine distance from its face f. For sufficiently small fixed \gamma>0, the function

is a supersolution of the operator in (5.11), for all small \rho. Indeed, its second derivative terms are bounded by a constant times \gamma^2/\rho, its first derivative terms by a constant times \gamma/\sqrt\rho, and its negative zeroth order term dominates both. Choose C to dominate the uniform height bound on the faces and compare with u_\rho and -u_\rho. This proves exponential decay of the height on every interior subbox.

To turn this into derivative decay, rescale around an interior point by x=x_0+\rho y and u_\rho(x)=\rho v(y). The equation takes the divergence form

Here |v|\le S_1 by (5.9), while Dv=Du_\rho is uniformly bounded. All coefficient derivatives on these ranges are uniformly bounded, and ellipticity is uniform. The interior estimates from Section 3.1 give bounded derivatives of v of every order on smaller fixed balls. Returning to the original coordinates yields bounds C_j\rho^{1-j} for derivatives of order j. For fixed nested subboxes K\subset\operatorname{int}K', interior interpolation gives

The C^4 norm is at most C\rho^{-3}, while the C^0 norm decays exponentially in 1/\sqrt\rho. The right side tends to zero, proving (5.10).

5.3 Curvature balance

For the slope functions (5.4),

Indeed, -s\beta_s=s\theta_s/[2(1-\theta)]; on the switching interval use s\le.05 and \sigma'\le60, and outside it the formula is immediate. Thus Lemma 3.5 applies to u=u_\rho. Denote its output by

Here \dd u^2 denotes \dd u\otimes\dd u. These expressions are smooth across s=0: the coefficient of \dd u^2 extends as \mu/[1+(1-\mu)s^2]. All comparisons with h_0 below take place in the original neighborhood of A; no extension of h_0 is needed.

On \{s>0\}, put e=\nabla_hu/s and b=\sqrt{h_0(e,e)}\le1. The rank-one inverse metric formula gives

Use J,H,K,M_1 as in Lemma 3.5: J is the second fundamental form of the levels of u, H=\tr J, K=-e\log k_1, and M_1=H-et-K. In particular,

Increasing the drift by -\beta has a controlled effect on the reserve:

Indeed, if b_*=-\beta\ge0, the numerator is at most 2N(t)b_*+b_*^2. When \theta\le1/2 one has b_*\le\theta. When \theta>1/2, one has s>1 and b_*\le\tfrac12\log(1+s^2)\le\sqrt s; these inequalities give (5.14).

Together with (5.13), this identifies the estimate needed at slopes s\ge s_0: it suffices to prove E\ge4(1+s)N(t)-\delta b^2V(t_u). Below s_0, we will instead use the adjustable small error in Lemma 5.1.

To express the graph equation in the normal variables, put, for s>0,

The factor \chi normalizes the pressure after division by k_1, and m_1 records its derivative. Directly from these formulas,

For s\le s_0, substitute \theta=a^2s^2/(1+s^2); for s\ge s_0, use \theta\ge a^2s^2/(1+s^2) to bound \chi.

Equation (5.8) at \lambda=1 says M_1=\chi F, with F evaluated at (\xi,t) and \xi=(u-\psi)/\eta. Since e\log\chi=-m_1K and et=H-M_1-K, differentiation gives

Completing the H- and K-squares now yields

For clarity, the discarded quantity is exactly

which is nonnegative. The choice of j now has two useful effects. At small slope, m_1\le-1/2 leaves the coefficient of F^2 at least 3/4, while \theta\chi^2=O(a^2) makes its eventual error small. At larger slope, the two alternatives in the pressure law supply the balance: a large value of F_x is accompanied by a favorable term -FF_t.

Slopes at least s_0. The pressure bounds imply F_t^2\le\epsilon|FF_t|. Since n\epsilon\le1/8 and 0\le1+m_1\le1, they give

Indeed, if FF_t\ge0, its coefficient has absolute value at most 4; if FF_t<0, that coefficient contributes at least 2|FF_t|, whereas the last term costs at most n\epsilon|FF_t|.

If \eta\ne1, then \psi=0 in a neighborhood, so

using |\xi|\le S_1<1, (5.7), and s\ge s_0. Equations (5.17)–(5.18) therefore give E\ge(3s/2-20\epsilon)F_x\ge sF_x. If \eta=1, then \dd\eta=0 and e\xi\ge s-b by |\dd\psi|_{h_0}\le1. For s>8b the same estimates give E\ge(7s/4-20\epsilon)F_x\ge sF_x.

It remains to consider s_0\le s\le8b with \eta=1. Here 2\chi(s-b)-s\ge-4, so E\ge sF_x-10F_x. On the pressure alternative -FF_t\ge P_0F_x, the positive term \chi^2P_0F_x more than pays this loss and gives E\ge sF_x. Otherwise F_x\le b_0N(t)^2, and the deficit is at most 10b_0N(t)^2. Since s\le8, we have 0\le-\beta<3, and T\ge10 implies

This comparison is valid for signed t: for 0\le c<3, 2V(t+c)-V(t)=(t+2c)^2+T^2-2c^2>0. Moreover, b\ge s_0/8, and hence

We have proved throughout \{s\ge s_0\} that

where the last inequality uses F_x\ge1000N(t) and s\ge.01. Together with (5.14) and (5.13), this proves the required curvature inequality without the small loss d on this slope range.

Slopes below s_0. Here 1+m_1\le1/2. The pressure estimate |F_t|\le\epsilon|F| therefore gives

When \eta=1, we have e\xi/\chi\ge-1. When \eta\ne1 and e\xi<0, we have s\le|\dd\eta|_h and, by (5.16) and (5.7),

Using the last bound in Proposition 4.1, we conclude that

The small metric modification now pays for this error. Namely, \theta\chi^2\le2a^2s_0^2 and b_*=-\beta\le\theta\le a^2s_0^2, so

Fix a\in(0,1/2] small enough that

Then the scalar error and the increase in reserve together cost at most (d/2)V(t_u). This choice is made before taking \rho small in the locality argument.

Combining the two slope ranges gives

The inequality extends to boundary points of \{s>0\} by smoothness. On the interior of \{s=0\} the metric and drift are unchanged, so it holds there as well. We have thus retained the full positive increase of the reserve, apart from the trace loss and half the prescribed small error. The remaining half will absorb the final cutoff.

5.4 Compact support and distance gain

Define the actual output on the original manifold by

extending the increments by zero. The resulting data are smooth and satisfy h'\ge h and t'\ge t; their changes from h,t have compact support in \{r_A<W\}. On the compact annulus \{.81W\le r_A\le.99W\}, which contains every point where 0<\zeta<1, (5.10) gives C^3 convergence u_\rho\to0. Consequently both the uncut and cut metric and drift increments tend to zero in C^2 there.

To verify the inequality after cutoff without assuming any monotonicity of V, consider its residual

By (5.21), \mathcal R(h_u,t_u)\ge(d/2)V(t_u)\ge dT^2/2. The residual is continuous in the indicated C^2 data, including the positive-part term. Thus \mathcal R(h',t')-\mathcal R(h_u,t_u)\to0 uniformly on the annulus. For sufficiently small \rho, (5.1) follows there. Outside this annulus, \zeta is locally constant, with value one or zero. The data then agree locally with h_u,t_u, where (5.21) applies, or with the original h,t. Completeness follows from h'\ge h_0.

It remains to quantify the distance gained. Use the effective height \zeta u_\rho, extended by zero. By (5.9) and (5.5), it exceeds W/5-1 on A and vanishes on B. Decrease \rho further so that |\dd(\zeta u_\rho)|_h\le.01 throughout the cutoff transition. Suppose a piecewise smooth curve from A to B had h'-length at most L_*W. Its h-length is also at most L_*W, and is at least W. The absolute height variation in the cutoff transition, together with that on \{\zeta=1,s<.05\}, is at most .06L_*W; repeated visits are included by integrating the absolute derivative along the curve. Where \zeta=0 there is no height variation. Thus the remaining portion, where \zeta=1 and s\ge.05, carries variation greater than

On that portion \mu=1 and h'=h+\dd u_\rho^2. Everywhere else h'\ge h. Applying the integral triangle inequality to the pair consisting of the h-speed and the absolute height derivative restricted to this portion gives

a contradiction. This proves (5.2) and completes the proof of Lemma 5.1.

6 Stretching rounds and the width bound

We now construct the metric and log warps required by Theorem 2.1. The local stretching lemma increases one separation at a time. Its loss is measured by a decrease in the trace of a fixed background metric, so a whole family of separations can be treated without a bound on the overlap of their neighborhoods. Changing the working scale after each round restores the quadratic reserve.

6.1 Summing the local losses

The following form of the round argument accommodates a drift of either sign. It extends the iteration in [14, Section 4] using the positive-part term in Lemma 5.1.

6.2 Proof of Theorem 1.1

For a quadratic-form lower bound \lambda>0, the metric \lambda g has normalized lower bound 1, since its scalar curvature and Laplacian are \lambda^{-1} times those of g. Its distances are \sqrt\lambda times those of g, giving the bound C_n/\sqrt\lambda stated in the introduction.