Kapouleas–McGrath Minimal Surface Doublings

• Kapouleas–McGrath Construction is a singular perturbation method that produces minimal surface doublings by joining parallel copies of a minimal surface with scaled catenoids.
• It reformulates the gluing problem into a finite-dimensional matching condition using variational techniques and Coulomb-type energy associated with a Schrödinger operator.
• In S³, the method yields explicit doublings of the Clifford torus with catenoidal bridges arranged along torus knots, resulting in high-genus surfaces with controlled area.

The Kapouleas–McGrath construction is a singular perturbation construction for producing minimal surface doublings: sequences of high-genus embedded minimal surfaces converging to a given minimal surface with multiplicity two. In its original geometric form, one starts from two nearby parallel copies of a fixed two-sided embedded minimal surface and connects them by many small catenoidal bridges; in the later Kapouleas–McGrath framework, the existence theory is reduced to finding suitable families of ansatz data on the base surface. A variational reformulation relates this reduction to a Coulomb-type interaction energy for a Schrödinger operator on the base surface, and in a generic closed $3$-manifold it implies that every two-sided embedded minimal surface of index one admits infinitely many doublings. In the round $S^3$, the same framework yields explicit families of doublings of the Clifford torus with bridges arranged along parallel copies of torus knots (Chu et al., 23 Sep 2025, Kapouleas et al., 2024).

1. Singular perturbation and the geometric doubling ansatz

Twenty years ago, N. Kapouleas introduced the singular perturbation construction known as “doubling,” which produces sequences of high-genus minimal surfaces converging to a given minimal surface with multiplicity two. The basic geometric input is a fixed two-sided minimal surface $\Sigma^2\subset N^3$, together with a finite set of neck-centers $p_i\in\Sigma$ and small waist parameters $\sigma_i$.

The approximate surface is obtained by taking two parallel normal graphs over $\Sigma$ at heights $\pm \phi$, deleting small disks near the chosen points $p_i$, and inserting scaled catenoids. Concretely,

$M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$

where $\Cat_{\sigma}(p)$ is the standard catenoid of waist $S^3$0 centered at $S^3$1, and $S^3$2. The gluing function $S^3$3 is chosen so that

$S^3$4

with $S^3$5 small weights comparable to the $S^3$6.

Near each $S^3$7, on the overlap region $S^3$8, both the normal-graph height $S^3$9 and the catenoid profile $\Sigma^2\subset N^3$0 provide valid approximations. Their discrepancy is encoded by the mismatch data

$\Sigma^2\subset N^3$1

and the balancing condition is $\Sigma^2\subset N^3$2.

Expanding the mean-curvature operator about $\Sigma^2\subset N^3$3 produces a linear problem of the form

$\Sigma^2\subset N^3$4

and solvability forces the balancing equations $\Sigma^2\subset N^3$5. In this form, the construction is a gluing problem with singular sources on $\Sigma^2\subset N^3$6, logarithmic asymptotics near the neck-centers, and finite-dimensional matching conditions.

2. Ansatz-data reduction in the Kapouleas–McGrath framework

The deep work of Kapouleas–McGrath reduces the existence theory to the problem of finding suitable families of ansatz data on the initial minimal surface. Rather than working directly with immersed surfaces, one works with

$\Sigma^2\subset N^3$7

To each $\Sigma^2\subset N^3$8, one associates a linearized doubling solution and a mismatch map

$\Sigma^2\subset N^3$9

which encodes exactly the Taylor-expansion data of $p_i\in\Sigma$0 near each $p_i\in\Sigma$1.

The reduction theorem requires a family of data for which the following hold: the $p_i\in\Sigma$2 are comparable and small in $p_i\in\Sigma$3; the $p_i\in\Sigma$4 are well-separated; the linear-mismatch map $p_i\in\Sigma$5 is a diffeomorphism onto a small ball in the target; and $p_i\in\Sigma$6 has the correct sign away from the necks. Under these hypotheses, the nonlinear problem can be solved by a fixed-point argument, producing a bona-fide minimal surface.

In the $p_i\in\Sigma$7 implementation, this reduction is presented as a three-step “Linearized Doubling” method. There one works with continuous families

$p_i\in\Sigma$8

parameterized by a compact convex set $p_i\in\Sigma$9, where $\sigma_i$0 is a $\sigma_i$1-invariant finite set of neck-centers, $\sigma_i$2 assigns neck-sizes, $\sigma_i$3 is an LD solution with logarithmic singularities, $\sigma_i$4 is a cut-off scale, $\sigma_i$5 is a finite-dimensional obstruction space, and $\sigma_i$6 is a linear isomorphism reading vertical unbalancing as a parameter in $\sigma_i$7 (Kapouleas et al., 2024).

3. Coulomb-type variational reformulation

The variational reformulation is built from a quadratic interaction energy on $\sigma_i$8,

$\sigma_i$9

where $\Sigma$0 is the Green’s function and $\Sigma$1 the Robin function for the base-surface Schrödinger operator

$\Sigma$2

Its differential recovers the gluing mismatch: $\Sigma$3

$\Sigma$4

Accordingly, critical points of $\Sigma$5 are equivalent to zero-mismatch data and hence to gluing data.

An equivalent projectivized formulation uses the quotient

$\Sigma$6

with $\Sigma$7 and

$\Sigma$8

This places the construction in a Coulomb-gas setting on $\Sigma$9, with the Green’s function $\pm \phi$0 governing the interaction of the neck-centers and the potential $\pm \phi$1 entering through the Schrödinger operator.

The variational theorem states that any sufficiently nondegenerate critical point of $\pm \phi$2—with Hessian bounded away from $\pm \phi$3 in weighted norm, separation, $\pm \phi$4-comparability, and sign conditions—yields via the Kapouleas–McGrath theorem an embedded minimal doubling (Chu et al., 23 Sep 2025).

4. Minimization, equilibrium measures, and generic metrics

For each $\pm \phi$5, one considers

$\pm \phi$6

equivalently $\pm \phi$7 up to a computable change of scale. The asymptotic regime is governed by the equilibrium energy

$\pm \phi$8

with $\pm \phi$9.

For $p_i$0, the minimization theory gives

$p_i$1

and the infimum is achieved by some $p_i$2, with

$p_i$3

The counting measures

$p_i$4

converge to an equilibrium measure $p_i$5, and a Hessian-lower-bound argument establishes nondegeneracy in the weight-directions.

A further conformal perturbation in the ambient metric achieves generic nondegeneracy in the $p_i$6-directions as well. One can then apply the precise gluing theorem to each minimizer, producing for each large $p_i$7 an embedded minimal doubling

$p_i$8

with $p_i$9 necks.

At the global level, for a comeager set of ambient metrics $M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$0, every two-sided index-one minimal $M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$1 admits infinitely many doublings. The proof combines White’s bumpy-metric theorem with transversality of the property $M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$2 in the Banach manifold of minimal-metric pairs. In this form, the variational viewpoint yields the first generic-metric existence theorem for doublings of index-one surfaces in closed $M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$3-manifolds (Chu et al., 23 Sep 2025).

5. The Clifford torus in the round three-sphere

In the round three-sphere, the construction is specialized to

$M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$4

the Clifford torus, with catenoidal bridges arranged along parallel copies of an $M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$5-torus knot. The neck set is taken to be

$M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$6

defined from the torus knot

$M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$7

together with $M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$8 parallel copies.

One chooses a unique $M_0 = \Graph_{\Sigma\setminus \cup_i D_\delta(p_i)}(\pm \phi) \;\cup_{i=1}^n\; \Cat_{\sigma_i}(p_i),$9-invariant LD solution $\Cat_{\sigma}(p)$0 with singular set $\Cat_{\sigma}(p)$1 and all $\Cat_{\sigma}(p)$2. Its averaged part is

$\Cat_{\sigma}(p)$3

where $\Cat_{\sigma}(p)$4 is the geodesic distance to the nearest copy of $\Cat_{\sigma}(p)$5. One then sets

$\Cat_{\sigma}(p)$6

cuts off near each $\Cat_{\sigma}(p)$7 at scale $\Cat_{\sigma}(p)$8, and defines the approximate surface $\Cat_{\sigma}(p)$9 from the two graphs and the catenoidal bridges.

The linearized operator on the Clifford torus is

$S^3$00

and since $S^3$01 and $S^3$02, one has

$S^3$03

Under the symmetry group $S^3$04, the invariant kernel vanishes: $S^3$05 The approximate Jacobi fields are the cutoff-affine functions

$S^3$06

and the obstruction space is chosen so that $S^3$07 is an isomorphism.

The linear equation is solved with a parametrix and matching coefficients $S^3$08, while the remaining gluing parameters are the neck-waists $S^3$09 and the unbalancing mode $S^3$10. The map

$S^3$11

is shown to be a contraction for $S^3$12, and a Banach-fixed-point argument yields unique parameters for which all matching equations vanish. The resulting existence theorem states that for each relatively prime pair $S^3$13, each integer $S^3$14 apart from a few small exceptional $S^3$15, and each $S^3$16, there exists a smooth, embedded, $S^3$17-invariant minimal surface

$S^3$18

of genus

$S^3$19

consisting of two parallel copies of the Clifford torus joined by $S^3$20 small catenoidal necks arranged along $S^3$21 parallel $S^3$22-torus knots. Moreover,

$S^3$23

and

$S^3$24

(Kapouleas et al., 2024).

6. Area asymptotics, genus growth, and relation to adjacent constructions

The area of a doubling admits a precise asymptotic expansion. In the variational formulation,

$S^3$25

and

$S^3$26

For a bumpy metric $S^3$27, min-max produces an index-one $S^3$28 with area $S^3$29. Each doubling $S^3$30 then satisfies

$S^3$31

hence

$S^3$32

Its genus is

$S^3$33

Thus $S^3$34 admits infinitely many embedded minimal surfaces of genus $S^3$35 and area uniformly $S^3$36 (Chu et al., 23 Sep 2025).

In the round three-sphere, the Clifford-torus doubling family is used to prove a new quadratic lower bound for the number of embedded minimal surfaces in $S^3$37 with prescribed genus. This improves upon bounds recently given by Ketover and Karpukhin-Kusner-McGrath-Stern, and contributes to a question of Yau about the structure of the space of minimal surfaces in $S^3$38 with fixed genus. The same work verifies Yau’s conjecture for the first eigenvalue of minimal surfaces in $S^3$39 for all minimal surface doublings of the equatorial two-sphere constructible by the earlier general theorem and for all the Clifford Torus doublings constructed there. It also notes that Ketover constructed similar minimal surfaces by min-max methods as suggested by Pitts-Rubinstein, but those methods apply only to surfaces which are lifts of genus two surfaces in lens spaces, while the doubling construction described there is not constrained this way (Kapouleas et al., 2024).

These results place the Kapouleas–McGrath construction at the intersection of singular perturbation, Schrödinger-operator analysis, Coulomb-type variational principles, and global questions about the abundance and geometry of embedded minimal surfaces.