Background and scope

Stable minimal surfaces provide a classical link between scalar curvature and the geometry of three-manifolds. Schoen and Yau used the stability inequality and the Gauss equation to obtain curvature and topological restrictions on such surfaces [13]. Fischer-Colbrie and Schoen developed the associated positive-solution method for Schrödinger operators [3]. The stable-surface radius estimate of Schoen and Yau [14] and the distance-level estimates of Gromov and Lawson [7] show how local surface stability can control the large-scale geometry of the ambient three-manifold. The latter distance-level argument uses a topological condition on the first homology; it is not a diameter bound for an arbitrary distance sphere in an arbitrary three-manifold.

Gromov’s torus-stabilized scalar-curvature framework gives the closest precedent for the spectral formulation here. Torus stabilization means considering metrics on \(M\times\mathbb T^N\) of the form \(g+\sum_{i=1}^N\varphi_i^2\,d\theta_i^2\), with smooth positive functions \(\varphi_i\) on \(M\), while measuring width in the base metric \(g\). In his Four lectures on scalar curvature, Gromov gives a three-dimensional graph-width theorem in this framework [5]. That statement is formulated for orientable complete three-manifolds, with a mean-convex boundary allowed in the torus extension. The relation between torus stabilization and the coefficient \(4\) in \(-4\Delta+\mathop{\mathrm{Scal}}\) is made explicit in [6]. Thus the spectral viewpoint and the scalar-curvature-to-width principle are established parts of this theory. Our purpose is to give a direct weighted proof of Theorem 1, including its nonorientable and noncompact cases, without a global bound on the auxiliary weight or its derivatives.

The prescribed-mean-curvature method used below also belongs to this development. Gromov’s \(\mu\)-bubbles minimize area with an added volume term, whose density prescribes the mean curvature. Chodosh and Li developed weighted versions and a slice-and-dice construction that first simplifies the topology and then produces pieces of bounded diameter [1]. The two-dimensional differential inequality underlying our closed-surface diameter estimate is equivalent to theirs [1]. Under a pointwise positive scalar-curvature lower bound, Liokumovich and Maximo constructed singular foliations of compact three-manifolds with simultaneous area, diameter, and genus control [10]. Liokumovich and Wang extended area and diameter control to complete noncompact three-manifolds [11]. These are stronger geometric conclusions in their pointwise setting, and they also imply graph-width bounds. Their level-set statements concern connected components of fibers of proper Morse functions; passing to their Reeb graphs distinguishes such components. The graph target in Theorem 1 likewise permits a uniform bound on whole fibers.

There are also sharp spectral band inequalities. For example, Hirsch, Kazaras, Khuri, and Zhang obtain quantitative bounds for bands from positivity of \(-\Delta+c\mathop{\mathrm{Scal}}\), including \(c=1/4\), under the relevant orientability and topological hypotheses [8]. Such an end-to-end distance estimate is a different conclusion from a graph-width bound. We use the weighted surface argument directly, keeping the spectral condition, the topology of separating surfaces, and the final fiber estimate within one proof.

The proof and its useful components

After scaling, assume \(\lambda=1\). A positive supersolution of \((-4\Delta_g+\mathop{\mathrm{Scal}}_g)v\ge v\) gives a smooth function \(t=-2\log v\) such that \[ D(g,t):=\mathop{\mathrm{Scal}}_g+2\Delta_g t-|dt|_g^2\ge1. \tag{2}\] The function \(t\) supplies the positive area density \(e^{-t}\), without changing the metric in which the final width is measured. All minimization and smoothing arguments involving this density take place in compact regions.

The first step is a weighted form of the stable-surface estimate. If a closed connected two-sided surface is stable for weighted area minus the weighted volume integral of a smooth pressure \(\mu\), its equation is \(H-\nu t=\mu\), where \(H\) is its mean curvature and \(\nu\) is the chosen unit normal. A positive function for its stability operator converts the ambient inequality into a two-dimensional inequality of the same form as (2). A conformal geodesic index argument then gives an intrinsic diameter bound in the original surface metric whenever \(D(g,t)+\mu^2-2|d\mu|_g\) has a positive lower bound. The local version, proved on relatively compact surface domains in Section 2, also prevents stable minimizing sheets from escaping a sufficiently large fixed neighborhood.

We next cut along a locally finite family of closed stable weighted minimal surfaces. The family is chosen so that no finite subfamily disconnects \(M\), and is maximal with this property. Every component \(P\) of the cut manifold then has the following separation property: each compact connected two-sided surface meeting \(\partial P\) transversely, with its boundary exactly on \(\partial P\), separates \(P\). Section 3 proves this by a localized minimizing argument. On the orientation cover, antisymmetric integral currents retain coorientations downstairs. A local area-nonincreasing retraction at a cut boundary permits full-neighborhood comparisons, so ordinary interior regularity applies even at boundary contacts.

The separation property allows small surfaces to control large distance slabs. To construct those surfaces without a boundary regularity problem, Section 4 doubles \(P\) and smooths the metric and density compatibly with reflection. The useful smoothing variable is the logarithm of the weighted area density of the normal slices. Weighted minimality makes its first normal derivative vanish at the doubling locus, and the curvature inequality survives smoothing. A reflection-invariant weighted \(\mu\)-bubble then restricts to a finite collection of compact separating surfaces in \(P\), each with uniformly bounded diameter. The graph with complementary regions as vertices and interface components as edges is a tree, because each interface separates. The bubble places the basepoint and a fixed distant slab component in different regions, so one interface meets every path between them. Section 5 uses this common interface to bound the slab component’s diameter.

Finally, overlapping slabs give a cover of multiplicity at most two, and a subordinate partition of unity maps \(P\) to a graph. Section 6 joins these maps across the original cuts, following the bounded-interface gluing principle of [5]. Each cut surface is a sphere or a projective plane, so its map to a finite graph is nullhomotopic. The homotopies are kept inside finite subgraphs whose vertices correspond to cover sets meeting the relevant collar. This extra condition ensures that all collar pieces mapping to a given old graph point lie near one common cover set. New edges join only the two collars belonging to a single cut. These observations control entire fibers after gluing, rather than only their connected components.

The weighted surface estimate, the compatible smoothing of doubles, and the collar construction each isolate a useful part of the argument. Their proofs require local elliptic and geometric measure theory, but no uniform bounds on the geometry at infinity.

From the quadratic form to a positive density

We use the standard ground-state construction from a nonnegative quadratic form; see Fischer-Colbrie–Schoen [3]. We recall the exhaustion argument in the form needed here.

Fix this function \(t\) on \(M\). Its density \(e^{-t}\) determines the weighted area used to cut \(M\). We first carry out the surface calculation for arbitrary ambient data \((h,T)\), since a modified metric and weight will also occur later.

The stability inequality

Let \(\Sigma\) be a smooth two-sided surface, possibly with boundary, in a Riemannian three-manifold \((X,h)\), and choose its unit normal \(\nu\). All surface variations and test functions will be compactly supported in \(\operatorname{int}\Sigma\). Write \(k=h|_\Sigma\) and use the conventions \[B(V,W)=h(\nabla_V\nu,W),\qquad H=\operatorname{tr}_k B.\] For a smooth ambient function \(T\), weighted area means \(\int_\Sigma e^{-T}\,d\mathop{\mathrm{Area}}_k\). If \(\Sigma\) is the boundary of a set \(E\), with outward normal \(\nu\), the functional \[ \int_{\partial E}e^{-T}\,d\mathop{\mathrm{Area}}_h -\int_E\mu e^{-T}\,d\mathop{\mathrm{vol}}_h \tag{5}\] has Euler–Lagrange equation \[ H-\nu T=\mu, \tag{6}\] where \(\mu\) is a smooth function on \(X\). For a surface not globally bounding a set, the same equation and second variation are defined locally using its chosen normal. Stability always means unconstrained stability: all normal variations compactly supported in the surface interior are allowed, with no volume constraint.

The weighted surface Laplacian and the potential in the second variation are \[ \begin{split} \Delta_{\Sigma,T}u &=\Delta_ku-\langle d(T|_\Sigma),du\rangle_k,\\ Q&=\mathop{\mathrm{Ric}}_h(\nu,\nu)+\mathop{\mathrm{Hess}}_hT(\nu,\nu)+|B|_k^2+\nu\mu. \end{split} \tag{7}\] In particular, stability is the inequality \[ \int_\Sigma\bigl(|du|_k^2-Qu^2\bigr)e^{-T}\,d\mathop{\mathrm{Area}}_k\ge0 \qquad(u\in C_c^\infty(\operatorname{int}\Sigma)). \tag{8}\] To check the signs, a normal variation of speed \(u\) satisfies \[H'=-\Delta_ku-(\mathop{\mathrm{Ric}}_h(\nu,\nu)+|B|_k^2)u, \qquad \nu'=-\nabla_ku.\] Differentiating \(H-\nu T-\mu\) therefore gives \(-\Delta_{\Sigma,T}u-Qu\). The first variation of (5) has factor \(H-\nu T-\mu\), so its second variation at a stationary surface is exactly (8).

In particular, the lower bound \[ D(h,T)+\mu^2-2|d\mu|_h\ge c>0 \tag{12}\] along a stable surface gives \(D(k,U)\ge c\) on the closed surface or the chosen relatively compact domain in Lemma 3. We next convert this curvature inequality into a length estimate in the original induced metric \(k\), not merely in a conformal metric.

The conformal length estimate

The conformal geodesic argument is part of the stable-surface radius method of Schoen–Yau [14]; the differential inequality used here also appears in Chodosh–Li [1]. We give the index calculation explicitly to track length in the original metric.

For the original data \((M,g,t)\), weighted minimality means \(\mu=0\). Since \(D(g,t)\ge1\), Proposition 5 bounds every closed connected two-sided stable weighted minimal surface by \(2\pi\) and identifies its topology. The same constant in Lemma 6 prevents an open stable sheet from extending arbitrarily far inside a region where it has compact intersections with ambient balls.

A maximal locally finite family

Order by inclusion the families of pairwise disjoint closed connected embedded two-sided stable weighted minimal surfaces for which no finite subfamily disconnects \(M\). The empty family is admissible, and the union of a chain is admissible because each finite subfamily lies in one member of the chain. Zorn’s lemma gives a maximal family \(\mathcal S\).

We first verify local finiteness. By Proposition 5, every member has intrinsic, and hence ambient, diameter at most \(2\pi\). Consequently, all members meeting a fixed compact set \(A\subset M\) lie in the compact closed \(2\pi\)-neighborhood of \(A\). Here completeness is used through the properness of the Riemannian distance on \(M\).

For any finite subfamily \(S_1,\ldots,S_k\), the classes \([S_i]\in H_2(M;\mathbb Z/2)\) are linearly independent. Indeed, the connected complement of the subfamily contains a path joining the two sides of \(S_i\); closing it by a short transverse arc gives a loop meeting \(S_i\) once and missing the other surfaces. Modulo two, intersection with this loop pairs to one with \([S_i]\) and to zero with the other classes. This uses only the intersection pairing of compact cycles, so neither orientability nor compactness of \(M\) is required. A compact smooth submanifold neighborhood of the closed \(2\pi\)-neighborhood of \(A\) has finite-dimensional \(H_2(-;\mathbb Z/2)\). Its image in \(H_2(M;\mathbb Z/2)\) therefore bounds the size of every finite subfamily meeting \(A\). Thus only finitely many members meet \(A\).

Local finiteness makes cutting well-defined. In a tubular chart at a cut surface, \(q\) glues the two closed half-charts along their boundary; away from the cuts it is a diffeomorphism. These charts show that \(q\) is a quotient map and is proper locally over relatively compact neighborhoods. A finite such cover of a compact subset of \(M\) proves global properness. The pulled-back metric induces the manifold topology, also for its intrinsic length metric on each component \(P\). A closed bounded subset of \(P\) maps into a bounded subset of \(M\); it is closed in the compact inverse image of a sufficiently large closed ball. It is therefore compact. This proves properness of \(P\).

The boundary assertions in Proposition 7 now follow from Proposition 5. It remains to prove separation. Fix a connected component \(P\) of \(N\). We first treat closed hypersurfaces in the interior; this is the only step requiring minimization.

Producing an additional stable surface

From closed to neat hypersurfaces

We finish the proof of Proposition 7. A compact connected two-sided neat hypersurface has a bicollar, and its complement has at most two components. If it does not separate, the bicollar and a path joining its two sides produce a loop in the interior crossing it once. Suppose \(F\) is such a nonseparating neat hypersurface. Its bicollar defines a smooth map \[u:P\longrightarrow S^1\] which is constant outside a compact set, goes once around the circle along the transverse collar coordinate, and has \(F\) as a regular level. The map has nonzero degree on a loop crossing \(F\) once. One may use a transverse profile that is constant near both ends of the bicollar, so extension by the same constant value outside the collar is smooth.

Every boundary component of \(P\) has finite fundamental group. The restriction of \(u\) to a collar of any such component therefore has a smooth real lift under \(\mathbb R\to S^1\). Only finitely many boundary components meet the compact region where \(u\) is not constant, by local finiteness. On their collars, multiply the difference between this lift and a chosen lift of the outside constant by a smooth cutoff that is zero near \(\partial P\) and one near the interior end. Exponentiating gives a compactly supported homotopy of \(u\) to a smooth map \(u_1\) that equals the same constant on a neighborhood of all of \(\partial P\). Its degree on the crossing loop is unchanged.

Choose a regular value of \(u_1\) distinct from the constant value. Its inverse image is a compact cooriented hypersurface in \(\operatorname{int}P\), with finitely many connected components. Perturb the crossing loop to be transverse to it. The sum of its signed crossings with these components is the nonzero degree of \(u_1\) on the loop. If every component separated \(P\), every such signed crossing number would be zero. Thus a component would contradict Lemma 8. This proves the final assertion of Proposition 7.

Smoothing the double

Let \(Z\) be the double of \(P\) along \(\partial P\), and let \(\sigma\) exchange its two copies. If \(\partial P\) is empty, take two disjoint copies.

Write \(d\) for the intrinsic length metric of \(h|_P\). It is proper by (18) and properness of the original length metric on \(P\). Fix \(o\in\operatorname{int}P\) and put \[ r(x)=d(o,x)\qquad(x\in P). \tag{22}\] Extend \(r\) evenly to \(Z\). The folding map \(Z\to P\) preserves the length of a path on either copy and does not increase length across the seam. This follows directly from \(h=ds^2+k(s)\), with \(k\) even, by replacing \(s\) with \(|s|\) along an absolutely continuous path. In particular, the even extension of \(r\) is proper and \(1\)-Lipschitz on each component of \(Z\). We will also use the consequence \[ d(x,y)\le\operatorname{length}_h(\gamma) \quad(x,y\in P) \tag{23}\] for every path \(\gamma\) in \(Z\) from \(x\) to \(y\).

Separating interfaces in a distance annulus

We use a weighted perimeter-minus-volume minimization, as in [1], to obtain the following output for the slab argument. An open set here is understood relative to \(P\) when it meets \(\partial P\).

A locally finite partition and its graph

Fix a component \(P\) of \(N\), with the cover \(\mathcal U_P\) from (38). Choose a locally finite smooth partition of unity \(\{\psi_\alpha\}\) whose supports are compact and with each support contained in some assigned \(U(\alpha)\in\mathcal U_P\). Such a partition follows by taking a locally finite relatively compact refinement of the cover and then a subordinate partition of unity. Group functions assigned to the same set: \[ \phi_U=\sum_{\alpha:U(\alpha)=U}\psi_\alpha. \tag{40}\] There are only finitely many summands in each of these sums. Indeed, their nonempty supports all meet the compact set \(\overline U\), whereas the original family of supports is locally finite. Consequently \(\mathop{\mathrm{supp}}\phi_U\) is a compact subset of \(U\). Moreover, the family \(\{\mathop{\mathrm{supp}}\phi_U\}\) remains locally finite: a neighborhood meeting only finitely many of the original supports meets only finitely many grouped supports. Discard identically zero functions.

Give each remaining function a vertex \(v\), and write \(U_v\) and \(\phi_v\) for the corresponding set and function. Include the edge \([v,w]\) precisely when \(\phi_v\) and \(\phi_w\) are simultaneously positive somewhere. This defines a simplicial graph \(K_P\). It is locally finite, because the compact support of \(\phi_v\) meets only finitely many supports of other functions. Since the original cover has multiplicity at most two, at most two of the functions are positive at any point. The formula \[ F_P(x)=\sum_v\phi_v(x)v \tag{41}\] therefore defines a map to \(K_P\), interpreted in barycentric coordinates on its vertices and edges. Near each point only finitely many coordinates occur, so this map is continuous for the usual polyhedral topology, not merely for a topology on an infinite coordinate product.

Use disjoint vertex sets for different components \(P\), and put \(K_0=\coprod_P K_P\). The maps \(F_P\) give a continuous map \(F_0:N\to K_0\). If \(y\) belongs to the relative interior of a simplex of \(K_0\) and \(v\) is any vertex of that simplex, then \[ F_0(x)=y\quad\Longrightarrow\quad \phi_v(x)>0\quad\Longrightarrow\quad x\in U_v. \tag{42}\] In particular, the whole fiber is controlled by a single set in (39).

Collar homotopies with controlled anchors

Choose disjoint closed collars \(\mathcal C_B\) of all boundary components \(B\) of \(N\), with the family of collars locally finite. Take them from sufficiently small tubular neighborhoods of the cut surfaces in \(M\). Each collar is compact, and its normal \(g\)-length may be taken at most one. If \(B^+,B^-\) are the two copies of a cut surface \(S\), the intrinsic diameter estimate \(\mathop{\mathrm{diam}}_g S\le2\pi\) from Proposition 7 gives \[ \mathop{\mathrm{diam}}_g q(\mathcal C_{B^+}\cup\mathcal C_{B^-}) \le L,\qquad L:=2\pi+2. \tag{43}\] Indeed, project any two collar points normally to \(S\), join the projections in \(S\), and add the two normal segments.

Write each collar as \(B\times[0,1]\), with \(B\times\{0\}\) the actual boundary and \(B\times\{1\}\) its inner end. Let \[h_B:B\longrightarrow K_0,\qquad h_B(x)=F_0(x,1).\] Its image lies in a finite connected subgraph \(L_B\) of the appropriate \(K_P\): take the smallest subcomplex containing \(h_B(B)\). Compactness of \(B\) and local finiteness of \(K_P\) imply finiteness of this subcomplex, and connectedness of the image implies connectedness of the subcomplex. More importantly, its choice ensures that \[ v\in L_B\quad\Longrightarrow\quad (B\times\{1\})\cap\{\phi_v>0\}\ne\varnothing. \tag{44}\] For a vertex in the image this is immediate. Any other vertex of \(L_B\) is an endpoint of an edge whose interior meets the image, and both endpoint coordinates are positive at such a point. Thus (44) concerns the same cover sets \(U_v\) as the original partition, even if the ensuing homotopy visits parts of \(L_B\) outside \(h_B(B)\).

Each \(B\) is a sphere or a real projective plane by Proposition 7, so \(\pi_1(B)\) is finite. Since the fundamental group of a connected graph is free and torsion free, \((h_B)_*\) is trivial. The map \(h_B\) therefore lifts to the universal covering tree of \(L_B\). Contracting this lift to a lift of a chosen vertex \(a_B\in L_B\) and projecting back gives a homotopy \[ H_B:B\times[0,1]\longrightarrow L_B, \qquad H_B(x,0)=h_B(x),\quad H_B(x,1)=a_B. \tag{45}\] In particular, the homotopy stays inside a subgraph all of whose vertices satisfy (44).

Adding edge paths and descending to the manifold

For each cut surface \(S\) add an edge path \(A_S\) from \(a_{B^+}\) to \(a_{B^-}\), with interior disjoint from \(K_0\) and from every other added path. More explicitly, introduce two distinct new vertices \(c_S,d_S\) and the three edges \[[a_{B^+},c_S],\qquad[c_S,d_S],\qquad[d_S,a_{B^-}].\] All new vertices are distinct for different \(S\). This produces an ordinary simplicial graph even if the two endpoints agree or other paths have the same endpoints. Parametrize this path by \(\alpha_S:[0,1]\to A_S\), taking the first chosen endpoint at zero and the second at one; when these endpoints agree, the parametrization is injective on \([0,1)\) and only its endpoints are identified. Let \(m_S=\alpha_S(1/2)\).

Call the enlarged graph \(K\). It is locally finite. The only point requiring verification is the number of added paths at an old vertex \(v\). An endpoint attached there is some \(a_B=v\), and (44) implies that the corresponding collar meets \(\mathop{\mathrm{supp}}\phi_v\). A compact set meets only finitely many members of a locally finite family of collars, proving the assertion.

Define \(\widetilde F:N\to K\) to agree with \(F_0\) outside the collars. On the homotopy half of a collar, adjacent to its inner end and given by \(1/2\le s\le1\), put \[\widetilde F(x,s)=H_B(x,2-2s).\] For the remaining halves of the paired collars of \(S\), put \[ \begin{aligned} \widetilde F(x,s)&=\alpha_S(\tfrac12-s) &&\text{on }\mathcal C_{B^+},\quad 0\le s\le\tfrac12,\\ \widetilde F(x,s)&=\alpha_S(\tfrac12+s) &&\text{on }\mathcal C_{B^-},\quad 0\le s\le\tfrac12. \end{aligned} \tag{46}\] The formulas agree at \(s=1/2\) and with \(F_0\) at \(s=1\). Local finiteness of the collars proves continuity on all of \(N\). Both boundary copies of \(S\) are mapped constantly to \(m_S\). Since \(q:N\to M\) is the quotient that identifies these copies, \(\widetilde F\) descends to a continuous map \[ f:M\longrightarrow K,\qquad f\circ q=\widetilde F. \tag{47}\]

The diameter of an entire fiber

Fix \(y\in K_0\), and let \(v\) be a vertex of the unique simplex whose relative interior contains \(y\). Every point of \(f^{-1}(y)\) lies at distance at most \(L\) from the one set \(q(U_v)\). To prove this, lift the point to \(N\). If its value is unchanged from \(F_0\), use (42). Otherwise it lies in some collar \(\mathcal C_B\). On the homotopy half, its image lies in \(L_B\); because \(L_B\) is a subcomplex and contains \(y\), it contains \(v\). On the remaining half the image can belong to \(K_0\) only at the endpoint \(a_B\), so again \(v\in L_B\). By (44), choose \(z\in B\times\{1\}\) with \(\phi_v(z)>0\), hence \(z\in U_v\). Equation (43) gives the asserted distance to \(q(z)\).

For any two points of this fiber, their anchor points therefore belong to the same set \(q(U_v)\). It follows from (39) that \[ \mathop{\mathrm{diam}}_g f^{-1}(y)\le440+2L =440+2(2\pi+2)<500. \tag{48}\] If instead \(y\in K\setminus K_0\), it belongs to the open part \(A_S\setminus K_0\) of exactly one added path. Its entire preimage is contained in the images of the two collars of \(S\), so (43) gives the stronger bound \(L\). This also covers the new vertices and the common value \(m_S\). Empty fibers require no estimate. Thus (48) controls whole fibers, without any connectedness assumption.