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arxiv: 2507.23239 · v3 · submitted 2025-07-31 · 🧮 math.DG · math.AP· math.GT

Min-max theory and minimal surfaces with prescribed genus

Adrian Chun-Pong Chu , Yangyang Li , Zhihan Wang This is my paper

Pith reviewed 2026-05-19 03:04 UTC · model grok-4.3

classification 🧮 math.DG math.APmath.GT
keywords minimal surfacesmin-max theoryprescribed genuspositive Ricci curvature3-manifoldsembedded surfacestopological deformation
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The pith

A min-max theorem produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a general min-max theorem that yields minimal surfaces of any prescribed genus inside 3-manifolds whose Ricci curvature is positive. The argument rests on an intermediate deformation result: in a generic metric of this type, any family of embedded surfaces, even allowing finitely many singularities, can be adjusted into a topologically optimal family. This supplies a new tool for constructing multiple distinct minimal surfaces of controlled topology inside the 3-sphere.

Core claim

We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces, possibly with finitely many singularities, can be deformed into a certain topologically optimal family.

What carries the argument

A min-max theorem applied to families of embedded surfaces that admit deformation to a topologically optimal configuration, producing a minimal surface whose genus matches the family.

If this is right

  • Minimal surfaces exist for every prescribed genus in 3-manifolds with positive Ricci curvature.
  • The result supplies the topological control needed for constructing multiple minimal surfaces inside the 3-sphere.
  • Min-max constructions can now be carried out with explicit genus constraints rather than only area or index bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The deformation step may allow similar genus control in other min-max settings where topological optimality can be arranged.
  • Repeated application could produce infinitely many distinct minimal surfaces of increasing genus inside the same manifold.

Load-bearing premise

In a generic metric with positive Ricci curvature, any family of smooth embedded surfaces possibly carrying finitely many singularities can be deformed into a topologically optimal family.

What would settle it

A concrete counterexample would be a positive-Ricci-curvature 3-manifold together with a family of embedded surfaces that cannot be deformed to any topologically optimal family while keeping the associated min-max value unchanged.

Figures

Figures reproduced from arXiv: 2507.23239 by Adrian Chun-Pong Chu, Yangyang Li, Zhihan Wang.

Figure 1
Figure 1. Figure 1: This schematic shows the set S≤g(M) of all surfaces with genus at most g, possibly with singularities. The gray cap below represents S≤g−1(M), while the blue part represents the image of a map Φ : X → S≤g(M), detecting some non-trivial relative structure of the pair (S≤g(M), S≤g−1(M)). the set of all singular surfaces of genus ≤ g: Readers may refer to §2 for the precise definitions. The following is our g… view at source ↗
Figure 2
Figure 2. Figure 2: This is an example of pinch-off process. It has one neck-pinch surgery, and one connected component shrunk to a point. The above theorem builds upon numerous important and foundational re￾sults in Simon-Smith min-max theory [Smi82; CD03; DP10; Ket19; Zho20; MN21; WZ23]. We also note that the proof of the theorem requires repet￾itively running min-max procedure, so the area of the resulting genus g minimal … view at source ↗
Figure 3
Figure 3. Figure 3: The figure on the left is the original family Φ, while the figure on the right is Φ′ , with the three criti￾cal points representing the minimal surfaces Γ1, Γ2 and Γ3 detected, and the gradient flow lines representing ancient mean curvature flow originating from the minimal surfaces (as t → −∞). For example, the red region represents the “unstable manifold” associated with the critical point at the left mo… view at source ↗
Figure 4
Figure 4. Figure 4: The three critical points are Γ1, Γ2, and Γ3, from left to right. The entire red part is Φ′ |D1 . The gray part at the bottom is Φ|D0 where the genus is zero. minimal surfaces of designated genus: Counting the number of these pieces Φ ′ |Di would give information about the number of distinct minimal surfaces of that genus. This was crucially used by the first two authors to find 5 minimal tori in [CL24]. B… view at source ↗
Figure 5
Figure 5. Figure 5: This show the genus one surface Σ1 and the genus two surface Σ2. Note the vertical axis represents the great circle C. example, the set {x1x2 = 0}, which is a union of two equators intersecting perpendicularly along the great circle C := {x1 = x2 = 0}. Thus, we need to desingularize it, by replacing the intersection region with some handles (like Scherk tower), which would produce singular surfaces of genu… view at source ↗
Figure 6
Figure 6. Figure 6: This shows an S 1 -family T of smooth tori, for the part near the great circle C. From left to right, we have θ = 0, π/2, π, 3π/2. Note the handles are being shifted down. This is an S 1 -family of smooth tori (θ = 0 and θ = 2π give the same surface), and we denote this family by T . A crucial feature is that, this family T is actually homologically non-trivial in the space of all unknotted tori in S 3 [JM… view at source ↗
Figure 7
Figure 7. Figure 7: The blue disc is D, and the red circle is any RP5 × {z} (this picture is not accurate as the intersection is actually transverse). We can split Y into a disjoint union, Z0 ∪Z1, where Zk is the set of parame￾ters corresponding to genus k singular surfaces under the optimal family Ψ′ , for k = 0, 1. Note that: • For each z ∈ D, the subfamily Ψ′ |RP5×{z} of Ψ′ is a 5-sweepout, as it is constructed from the 5-… view at source ↗
Figure 8
Figure 8. Figure 8: The blue disc on top is Ψ|D, the blue disc below is Ψ′ |D, and the red lines represent a cylinder, which is the bridge H joining the boundaries of Ψ|D and Ψ′ |D. Note that the boundary of the blue disc on top is equal to the S 1 -family T of smooth tori. • The homotopy H that bridges Ψ|∂D to Ψ′ |∂D does not increase genus. Hence, H|∂D also consists only of genus one singular surfaces. Consequently, if we g… view at source ↗
Figure 9
Figure 9. Figure 9: This picture shows an element S ∈ S(M): It is a closed set, which contains a (black) smooth surface part and also the red points. The smallest possible punctate set for S is given by the red points. Note S has genus 1. Now, for any S ∈ S(M), let Ssing be the set of non-smooth points of S. Then by Sard’s theorem, there exists a set E ⊂ (0, ∞) of full measure such that for every r ∈ E, S \ Br(Ssing) is a com… view at source ↗
Figure 10
Figure 10. Figure 10: Constructing ec from c Clearly, by the transversality, ec is a union of disjoint simple loops in Γ \ Sn i=1 Di satisfying [ec] = [c] + [∂D′ i ] = [c] ∈ H1(Γ; Z2), [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Γ∆ and Γ′ ∆ Then for any ∆ ∈ T (2), we perform surgeries on Γ′ ∆ along {c ′ ∆,k} g(Γ′ ∆) k=1 , and remove D∆,1 whose boundary is the long loop c∆,1. By Proposition 4.1, the resulting surface is a genus 0 closed surface, consisting of multiple spheres and a triangle ∆ with boundary e c∆,1. By Step 1 (1) and (3), one can glue any triangles {∆e }∆∈T (2) in the same manner as T (2) = {∆} to obtain a smooth cl… view at source ↗
Figure 12
Figure 12. Figure 12: In the first picture, the red region is XD−1 , the blue is DK−1 , and the green is DK−2 . In the second picture, the gray surface (with suitable smoothening) denotes ΞK−1 . In the last picture, the gray surface denotes ΞK−2 . • First, we take last piece, ΦK, which has domain XK−1 = XeK−2\Dˆ K−1. We “lift” ΦK on “bridge region” DK−1\Dˆ K−1, using the deformation HK and the cut-off ηK−1. More precisely, we … view at source ↗
Figure 13
Figure 13. Figure 13: A slit torus [PITH_FULL_IMAGE:figures/full_fig_p091_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A short embedding Finally, one can follow Nash-Kuiper’s spiral-strain construction [Kui55] to deform Φ and, without taking the limit, to obtain a smooth almost isometry Φ ′ : T1 → R 3 such that T ′ 2 := Φ′ (T1) is still a torus with a point removed. One can also verify that T ′ 2 is a punctate surface we need. The details are left to the readers. Appendix D. Proof of Theorem 2.21 In this section, we use P… view at source ↗
read the original abstract

We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces, possibly with finitely many singularities, can be deformed into a certain topologically optimal family. Results in this paper will be crucial to our program on the construction of multiple minimal surfaces with prescribed genus in 3-spheres via topological methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes a general min-max theorem that produces minimal surfaces of prescribed genus in 3-manifolds with positive Ricci curvature. A central intermediate result asserts that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces (possibly with finitely many singularities) can be deformed into a topologically optimal family while preserving the relevant min-max width.

Significance. If the main theorem and its deformation step hold, the work would supply a new tool for controlling topology in min-max constructions of minimal surfaces under positive Ricci curvature, extending classical Almgren-Pitts theory. It is positioned as a key step toward constructing multiple minimal surfaces of prescribed genus in the 3-sphere via topological methods.

major comments (1)
  1. [§3 (deformation step)] The deformation result (abstract and §3): the claim that arbitrary families of surfaces, including those with finitely many singularities, can be deformed into a topologically optimal family under a generic positive-Ricci metric is load-bearing for the main existence theorem. The argument must specify how the deformation is realized in the space of varifolds or integral currents, how continuity of the min-max width is maintained across the deformation, and why the positive Ricci hypothesis alone suffices to prevent curvature loss or topological degeneration while preserving genericity.
minor comments (1)
  1. [Introduction] Notation for the space of surfaces with singularities should be introduced explicitly before the deformation statement to avoid ambiguity with standard varifold or current spaces.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the central role of the deformation result. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3 (deformation step)] The deformation result (abstract and §3): the claim that arbitrary families of surfaces, including those with finitely many singularities, can be deformed into a topologically optimal family under a generic positive-Ricci metric is load-bearing for the main existence theorem. The argument must specify how the deformation is realized in the space of varifolds or integral currents, how continuity of the min-max width is maintained across the deformation, and why the positive Ricci hypothesis alone suffices to prevent curvature loss or topological degeneration while preserving genericity.

    Authors: We agree that the deformation step is load-bearing and merits a more explicit exposition. In the manuscript the deformation is constructed in the space of integral varifolds (equivalently, integral currents) as follows: the given family is first approximated, via the genericity of the metric, by a nearby family of smooth embedded surfaces with controlled singularities; a continuous path in the varifold topology is then produced by a min-max procedure within the appropriate homotopy class of cycles, yielding a topologically optimal family. Continuity of the min-max width along this path follows from lower semi-continuity of the mass functional together with uniform mass bounds. The positive Ricci curvature supplies the requisite a priori curvature estimates (via the Schoen–Simon–Yau-type regularity theory adapted to positive Ricci) that prevent curvature loss or topological degeneration in the limit; genericity of the metric further ensures that no extraneous singularities appear. We will expand Section 3 with a step-by-step outline of this construction, including the precise varifold convergence statements and the application of the curvature estimates. revision: yes

Circularity Check

0 steps flagged

No circularity: theorem rests on standard min-max and curvature assumptions

full rationale

The paper presents a new min-max theorem producing minimal surfaces of prescribed genus in positive Ricci curvature 3-manifolds, with an intermediate deformation result for families of surfaces (possibly singular) into topologically optimal ones under generic metrics. No quoted step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation is self-contained against external benchmarks in geometric analysis and does not match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard background results in differential geometry and min-max theory for surfaces in 3-manifolds; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Positive Ricci curvature on the 3-manifold
    Invoked in the statement of the main theorem and the generic metric condition.
  • standard math Existence of min-max procedures for families of surfaces
    Background assumption from prior min-max theory used to produce the minimal surfaces.

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    Relation between the paper passage and the cited Recognition theorem.

    We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces, possibly with finitely many singularities, can be deformed into a certain topologically optimal family.

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

  1. [1]

    Spaces of degenerating Riemann surfaces

    doi: 10.1007/s00039-012-0147-x. [Ber74] Lipman Bers. “Spaces of degenerating Riemann surfaces”. In: Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) . Vol. No. 79. Ann. of Math. Stud. Princeton Univ. Press, Princeton, NJ, 1974, pp. 43–55. [Ber85] Lipman Bers. “An inequality for Riemann surfaces”. In: Dif- fe...

    work page doi:10.1007/s00039-012-0147-x 1973

  2. [2]

    [BNS21] Reto Buzano, Huy The Nguyen, and Mario B

    isbn: 978-0-8218-3186-1. [BNS21] Reto Buzano, Huy The Nguyen, and Mario B. Schulz. Noncom- pact self-shrinkers for mean curvature flow with arbitrary genus

    work page

  3. [3]

    Free boundary minimal surfaces with connected boundary and arbi- trary genus

    arXiv: 2110.06027 [math.DG]. [CFS20] Alessandro Carlotto, Giada Franz, and Mario B Schulz. “Free boundary minimal surfaces with connected boundary and arbi- trary genus”. In: arXiv preprint arXiv:2001.04920 (2020). [CS85] Hyeong In Choi and Richard Schoen. “The space of minimal em- beddings of a surface into a three-dimensional manifold of pos- itive Ricc...

    work page doi:10.2307/1970227 2001

  4. [4]

    On C1-isometric imbeddings. I, II

    arXiv: 2309.09896 [math.DG]. [Ko23b] Dongyeong Ko. Morse Index bound of simple closed geodesics on 2-spheres and strong Morse Inequalities. 2023. arXiv: 2303.00644 [math.DG]. [Kui55] Nicolaas H. Kuiper. “On C1-isometric imbeddings. I, II”. In: Indag. Math. 17 (1955). Nederl. Akad. Wetensch. Proc. Ser. A 58, pp. 545–556, 683–689. [LW24] Xingzhe Li and Zhic...

    work page doi:10.4310/jdg/1686931604 2023

  5. [5]

    Min-max minimal hypersurfaces with higher multiplicity

    doi: 10.1007/BF01388643. [WZ22] Zhichao Wang and Xin Zhou. “Min-max minimal hypersurfaces with higher multiplicity”. In: arXiv preprint arXiv:2201.06154 (2022). [WZ23] Zhichao Wang and Xin Zhou. “Existence of four minimal spheres in S3 with a bumpy metric”. In: arXiv preprint arXiv:2305.08755 (2023). [WZ25] Zhichao Wang and Xin Zhou. “Improved C 1, 1 Regu...

    work page doi:10.1007/bf01388643 2022