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Minimal Surfaces of Finite Genus: Classification, Dynamics and Laminations

Joaquín Pérez

TL;DR

This work surveys a long-term program to classify complete embedded minimal surfaces in $\\mathbb{R}^3$ with finite genus, spotlighting the role of Riemann’s minimal examples and laminations. It develops a four-step uniqueness framework for genus-zero, infinite-topology surfaces, leveraging flux, moduli spaces, and a Shiffman–KdV analytic engine to identify the Riemann minimal examples as the unique models. The paper also extends moduli classifications for singly and doubly periodic surfaces, analyzes minimal laminations via curvature and topology rescalings (the Dynamics Theorem), and connects these ideas to the Calabi–Yau problem and the Hoffman–Meeks conjecture, while outlining open problems. Overall, it highlights deep interactions among topology, flux, curvature bounds, and moduli-space structure in the global theory of minimal surfaces.

Abstract

This article explains a program to study complete and properly embedded minimal surfaces in $\mathbb{R}^3$ developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with the same title and supplies details and references to the original papers. After recalling the role of the classical Riemann minimal examples in minimal surface theory, we explain our four-step classification of properly embedded minimal surfaces of genus zero and infinite topology in $\mathbb{R}^3$: the periodic case, the quasi-periodicity of the two-limit-ended case, the non-existence of one-limit-ended examples, and the final classification. We then review the lamination techniques (limit-leaf stability, local removable singularity, and singular structure theorems), the dynamics theorem, bounds on topology and index for complete embedded minimal surfaces of finite total curvature, and the resolution of the embedded Calabi-Yau problem for finite genus and countably many ends. Throughout we emphasize the interaction between topology, flux, curvature estimates, and the structure of related moduli spaces. We end this article with a list of some open problems.

Minimal Surfaces of Finite Genus: Classification, Dynamics and Laminations

TL;DR

This work surveys a long-term program to classify complete embedded minimal surfaces in with finite genus, spotlighting the role of Riemann’s minimal examples and laminations. It develops a four-step uniqueness framework for genus-zero, infinite-topology surfaces, leveraging flux, moduli spaces, and a Shiffman–KdV analytic engine to identify the Riemann minimal examples as the unique models. The paper also extends moduli classifications for singly and doubly periodic surfaces, analyzes minimal laminations via curvature and topology rescalings (the Dynamics Theorem), and connects these ideas to the Calabi–Yau problem and the Hoffman–Meeks conjecture, while outlining open problems. Overall, it highlights deep interactions among topology, flux, curvature bounds, and moduli-space structure in the global theory of minimal surfaces.

Abstract

This article explains a program to study complete and properly embedded minimal surfaces in developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with the same title and supplies details and references to the original papers. After recalling the role of the classical Riemann minimal examples in minimal surface theory, we explain our four-step classification of properly embedded minimal surfaces of genus zero and infinite topology in : the periodic case, the quasi-periodicity of the two-limit-ended case, the non-existence of one-limit-ended examples, and the final classification. We then review the lamination techniques (limit-leaf stability, local removable singularity, and singular structure theorems), the dynamics theorem, bounds on topology and index for complete embedded minimal surfaces of finite total curvature, and the resolution of the embedded Calabi-Yau problem for finite genus and countably many ends. Throughout we emphasize the interaction between topology, flux, curvature estimates, and the structure of related moduli spaces. We end this article with a list of some open problems.
Paper Structure (16 sections, 21 theorems, 9 equations)

This paper contains 16 sections, 21 theorems, 9 equations.

Table of Contents

  1. Introduction and background
  2. Riemann's examples and their uniqueness program
  3. The uniqueness program of the $R_t$ in four steps
  4. Step 1: The singly periodic case
  5. Step 2: Quasiperiodicity in the two-limit-end case
  6. Step 3: Nonexistence of one-limit-end PEMS
  7. Step 4: The classification of properly embedded minimal planar domains
  8. Moduli spaces of periodic minimal surfaces: additional classifications
  9. Singly periodic, genus zero, Scherk-type ends
  10. Doubly periodic, genus zero, Scherk-type ends
  11. Doubly periodic, genus one, Scherk-type parallel ends
  12. Minimal laminations and local pictures
  13. Rescalings by curvature and topology, dynamics in the set of limits
  14. The Hoffman-Meeks conjecture
  15. Properness versus completeness
  16. ...and 1 more sections

Key Result

Theorem 1.1

Every PEMS $M\subset \mathbb{R}^3$ of finite topology with at least two ends has finite total curvature.

Theorems & Definitions (30)

  • Theorem 1.1: Collin col1
  • Theorem 1.2: López, Ros lor1
  • Theorem 1.3: Meeks, Rosenberg mr8
  • Theorem 1.4: Frohman, Meeks fme2
  • Theorem 1.5: Collin, Kusner, Meeks, Rosenberg ckmr1
  • Corollary 2.1
  • Theorem 2.2: Quadratic Curvature Decay Theorem, Meeks, Pérez, Ros mpr10
  • Theorem 3.1: Meeks, Rosenberg mr3mr2mr10mr13
  • Theorem 3.2: Pérez, Traizet PeTra1
  • Theorem 3.3: Lazard-Holly, Meeks lm2
  • ...and 20 more