Linearised Doubling in Geometry & Scheduling

• In geometric analysis, Linearised Doubling uses singular Jacobi solutions and catenoidal bridges to construct minimal surfaces with controlled area reduction.
• The method linearizes the mean curvature equation to manage nonlinear perturbations, yielding explicit topological invariants and precise catenoidal matching.
• In scheduling theory, the Coffman–Sethi LD algorithm is a rank-based heuristic that minimizes makespan in flowtime-optimal schedules with a tight worst-case ratio.

Searching arXiv for recent and relevant papers on "Linearized Doubling" to ground the article. Linearised Doubling (LD) denotes more than one technical construction in the arXiv literature. In geometric analysis, “Linearized Doubling” is a PDE-gluing methodology for constructing minimal surfaces, self-shrinkers, and related objects by starting from singular solutions of the Jacobi equation on a reference surface and replacing their logarithmic singularities with small catenoidal bridges (Kapouleas et al., 2020). In scheduling theory, “LD” also names the Coffman–Sethi heuristic for minimizing makespan among flowtime-optimal schedules on identical parallel machines, where its exact worst-case approximation ratio is $\frac{5m-2}{4m-1}$ (Ravi et al., 2015). These usages are unrelated in origin and content, and the shared abbreviation is terminological rather than structural.

1. Scope of the term

The literature represented here uses “LD” in several non-equivalent ways. Two uses are central to the phrase “Linearised Doubling” or “Linearized Doubling”: the geometric gluing method of Kapouleas and collaborators, and the Coffman–Sethi scheduling heuristic. Other papers use the abbreviation “LD” for entirely different notions, including Double Propositional Logic and left self-distributive algebras. A comparative review of doubling methods for Riccati-type matrix equations discusses repeated-squaring schemes closely related to linearisation, but explicitly states that it does not introduce or name any algorithm as “Linearised Doubling (LD)” (Poloni, 2020, Aristizábal, 2023, Dimonte, 2017).

Domain	Meaning of LD	Defining object
Minimal surfaces and self-shrinkers	Linearized Doubling	Singular Jacobi solutions with logarithmic singularities replaced by catenoidal bridges
Multi-processor scheduling	Coffman–Sethi LD algorithm	Rank-based heuristic for $P_m // C_{\max}; \sum C_j = F^*$
Riccati-type matrix equations	Not named LD in the paper	Doubling algorithms such as squared Smith, SDA, SDA-II, sign iteration
Propositional logic	Double Propositional Logic, LD	Logic with alternate affirmation and alternate negation
Large-cardinal algebra	LD-algebra	Left self-distributive algebra

This terminological multiplicity matters because “LD” does not designate a single transdisciplinary method. A plausible implication is that encyclopedia treatment must separate the geometric and scheduling traditions, while marking other uses as abbreviation overlap rather than substantive continuity.

2. Linearized Doubling in geometric analysis

In the geometric sense, Linearized Doubling is a gluing framework for constructing surfaces that resemble two nearby copies of a minimal background $\Sigma$ joined by many small catenoidal bridges. The ambient setting is a Riemannian $3$-manifold $(N,g)$ with $\Sigma \subset N$ a closed, embedded, two-sided minimal surface. The governing linear operator is the Jacobi operator

$$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$

and the central analytic objects are LD solutions: singular solutions $\varphi$ of $L_\Sigma \varphi = 0$ on $\Sigma \setminus L$, where $P_m // C_{\max}; \sum C_j = F^*$0 is a finite singular set and each singularity has logarithmic form $P_m // C_{\max}; \sum C_j = F^*$1 near $P_m // C_{\max}; \sum C_j = F^*$2 (Kapouleas et al., 2020).

The coefficient $P_m // C_{\max}; \sum C_j = F^*$3 has a dual role. It is both the strength of the logarithmic singularity and the parameter governing the waist radius of the catenoidal bridge inserted at $P_m // C_{\max}; \sum C_j = F^*$4. In local coordinates, the standard model catenoid is written as

$P_m // C_{\max}; \sum C_j = F^*$5

and its asymptotic graph matches the singular term $P_m // C_{\max}; \sum C_j = F^*$6. To organize local matching, the theory isolates the regular affine part of the expansion near each singular point and packages the discrepancy between the global LD solution and the local catenoidal model as a mismatch in the finite-dimensional space $P_m // C_{\max}; \sum C_j = F^*$7. This mismatch is linear in $P_m // C_{\max}; \sum C_j = F^*$8 and is the quantity compensated by finite-dimensional parameters in the gluing construction.

The method is “linearized” because the global geometry of a multi-neck surface is encoded first at the level of the linearized mean-curvature equation on $P_m // C_{\max}; \sum C_j = F^*$9, before the nonlinear surface PDE is solved on the glued manifold. The name “doubling” refers to the target geometry: two sheets $\Sigma$0 over the complement of small disks in $\Sigma$1, connected through neck regions.

3. General theorem, analytic mechanism, and quantitative outputs

The generalized LD theorem assumes that $\Sigma$2 is closed, embedded, minimal, two-sided, and that the Jacobi operator has trivial kernel: $\Sigma$3 Under this nondegeneracy hypothesis, and given a family of LD solutions with finite singular sets, small singularity strengths, and controllable unbalancing content, one constructs an initial surface made of two normal graphs over $\Sigma$4 together with truncated tilted catenoidal bridges near each $\Sigma$5. The graphical components are $\Sigma$6 and $\Sigma$7, with $\Sigma$8 chosen in a finite-dimensional obstruction space $\Sigma$9 so as to absorb the mismatch data (Kapouleas et al., 2020).

The perturbative step is formulated on the initial surface $3$0. Weighted Hölder norms are introduced to reflect the distinct scaling of the outer region and the necks, and a semi-local approximate inverse is iterated to produce a genuine bounded inverse

$3$1

If $3$2 denotes mean curvature and $3$3 the linearized operator on $3$4, then the nonlinear mean-curvature expansion has the form

$3$5

with a quadratic estimate for $3$6. Schauder’s fixed point theorem then yields an exact minimal perturbation.

The resulting minimal doubling has explicit topological and geometric invariants. If $3$7 is the doubled surface with bridge set indexed by $3$8, then

$3$9

Moreover, the area expansion is

$(N,g)$0

so in particular $(N,g)$1 for sufficiently small neck sizes. The negative leading correction comes entirely from the catenoidal contribution after cancellation of boundary terms between the neck and graph regions. This establishes that the doubled surface is strictly area-reducing relative to the formal double cover $(N,g)$2.

4. Rotationally invariant LD, balancing, and geometric applications

A concrete and highly developed instance is the doubling of the equatorial two-sphere $(N,g)$3. In that setting the Jacobi operator is

$(N,g)$4

and, after passing to the flat cylinder $(N,g)$5 with $(N,g)$6, the conformally related operator is

$(N,g)$7

Rotationally invariant LD solutions reduce to the ODE

$(N,g)$8

with explicit independent solutions

$(N,g)$9

The theory then allows piecewise $\Sigma \subset N$0-harmonic positive functions with jump latitudes, called RLD solutions, together with one-sided scale-invariant fluxes

$\Sigma \subset N$1

For smooth segments, $\Sigma \subset N$2 satisfies the Riccati equation

$\Sigma \subset N$3

which yields monotonicity and underlies the balancing of neck sizes and locations (Kapouleas et al., 2017).

From RLD data one constructs $\Sigma \subset N$4-symmetric LD solutions with singular set

$\Sigma \subset N$5

and residues related to the flux by the vertical balancing law

$\Sigma \subset N$6

The LD solution is decomposed as $\Sigma \subset N$7, where $\Sigma \subset N$8 captures the localized logarithmic singularities, $\Sigma \subset N$9 carries the global rotationally invariant component, and $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$0 is a controlled small remainder. Matched LD solutions impose local catenoidal asymptotics exactly, leading to explicit vertical and horizontal matching equations for the gluing parameters.

This machinery produces embedded minimal surfaces in $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$1 obtained by doubling the equatorial sphere along arbitrary numbers of parallel circles, with optional bridges at the poles or equator. For each $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$2 and large $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$3, Theorem 6.1 yields genus $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$4 examples. The generalized framework then extends to the Clifford torus, general $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$5-symmetric backgrounds, the spherical self-shrinker, the Angenent torus, the Euclidean catenoid, and the critical catenoid. In the generalized theorem, Part II constructs LD families for general $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$6-symmetric backgrounds, and Part III adapts the method to self-shrinkers, complete minimal surfaces of finite total curvature, and free boundary minimal surfaces (Kapouleas et al., 2020).

A recurring observation is “uniformization” of neck sizes in dense-neck regimes. In the equatorial-sphere constructions, the ratios $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$7 become controlled and tend to $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$8 in the regime $$L_\Sigma = \Delta_\Sigma + |A^\Sigma|^2 + \mathrm{Ric}(\nu_\Sigma,\nu_\Sigma),$$9 followed by $\varphi$0. This suggests that LD-based neck concentration does not naturally produce isolated singularities by allowing some necks to collapse while others remain macroscopically distinct.

5. The Coffman–Sethi LD algorithm in scheduling

In scheduling theory, LD is an unrelated acronym attached to the Coffman–Sethi heuristic for a bicriteria problem on $\varphi$1 identical parallel machines. Jobs are non-preemptive, all available at time $\varphi$2, and one studies flowtime-optimal schedules for which the total completion time

$\varphi$3

is minimized, then seeks the smallest makespan

$\varphi$4

among those schedules. When $\varphi$5, jobs are ordered in nonincreasing processing times $\varphi$6 and partitioned into ranks of size $\varphi$7. A schedule is flowtime-optimal if it has no idle time and satisfies the rank constraint that all jobs of rank $\varphi$8 start before any job of rank $\varphi$9, with the last rank starting at time $L_\Sigma \varphi = 0$0. The resulting bicriteria problem is

$L_\Sigma \varphi = 0$1

called the Flowtime–Makespan (FM) problem, and it is NP-hard (Ravi et al., 2015).

The LD algorithm is a natural extension of LPT list scheduling to this constrained setting. It initializes the machine profile at $L_\Sigma \varphi = 0$2, processes ranks in order $L_\Sigma \varphi = 0$3, and within each rank assigns the largest job to the machine with the smallest current completion time, the second largest to the second smallest, and so on, before re-sorting the profile. After all ranks are assigned, the schedule is reversed in time and left-justified, so that rank $L_\Sigma \varphi = 0$4 starts first and the result satisfies the flowtime-optimal rank ordering. The output makespan is $L_\Sigma \varphi = 0$5.

Coffman and Sethi conjectured in 1976 that the exact worst-case ratio of LD satisfies

$L_\Sigma \varphi = 0$6

where $L_\Sigma \varphi = 0$7 is the minimum makespan among all flowtime-optimal schedules. The 2015 paper proves this conjecture and shows the bound is tight for every integer $L_\Sigma \varphi = 0$8. The proof combines minimal-counterexample analysis, increasingly restrictive structural classes of counterexamples (Types I, IR, IR1, I2), and two reduction procedures, $L_\Sigma \varphi = 0$9 and $\Sigma \setminus L$0, which generate smaller instances while preserving or improving the approximation ratio. Earlier work had already settled the cases $\Sigma \setminus L$1 and $\Sigma \setminus L$2; the remaining case $\Sigma \setminus L$3 is excluded by incompatible inequalities derived from the structure of Type I2 instances.

In this scheduling usage, “Linearised Doubling” designates a rank-wise load-balancing heuristic with a precise approximation guarantee. Its relation to geometric Linearized Doubling is purely nominal.

6. Related doubling frameworks and recurrent misconceptions

One recurrent misconception is that every “doubling algorithm” obtained from a linearisation should automatically be called LD. The comparative review of Riccati-type matrix equations explicitly rejects that nomenclature for its own subject matter: it uses terms such as “doubling algorithm”, “structure-preserving doubling algorithm (SDA)”, “squared Smith method”, “sign iteration”, “cyclic reduction”, and “SDA-II”, but “never the label ‘linearised doubling’.” At the same time, it presents a unifying pattern in which a nonlinear matrix equation is first transformed into a linear invariant-subspace or fixed-point problem and then accelerated by repeated squaring so that $\Sigma \setminus L$4. If “Linearised Doubling” is used in that broader conceptual sense, then squared Smith, SDA for DARE, the Cayley-transform-based SDA for CARE, cyclic reduction, SDA-II, and sign iteration all fit the blueprint; however, that is a conceptual mapping rather than the paper’s own terminology (Poloni, 2020).

A second misconception is that the abbreviation “LD” itself signals Linearized Doubling. In the logic paper, LD means “Lógica Proposicional Doble” or Double Propositional Logic, a system with alternate affirmation and alternate negation, Kripke-style possible-worlds semantics, and Gamma-LD existential graphs (Aristizábal, 2023). In the large-cardinal paper, “LD-algebra” means a left self-distributive algebra satisfying

$\Sigma \setminus L$5

and the abbreviation refers to self-distributivity rather than doubling (Dimonte, 2017). These cases show that LD is an overloaded acronym whose local definition must be recovered from the disciplinary context.

Taken together, the arXiv record supports a sharply differentiated picture. “Linearized Doubling” is a well-defined methodology in geometric analysis and a historically separate heuristic in scheduling theory. Beyond those two settings, the phrase sometimes serves only as an interpretive lens for other doubling procedures, while the bare acronym “LD” often denotes something else entirely.