مروری بر نظریهٔ سطح حداقلی کلاسیک (سری سخنرانی‌های دانشگاهی)

Meeks and PГ©rez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the focus of the account. The proof of the classification depends on the work of many currently active leading mathematicians, thus making contact with much of the most important results in the field. Through the telling of the story of the classification of minimal planar domains, the general mathematician may catch a glimpse of the intrinsic beauty of this theory and the authors' perspective of what is happening at this historical moment in a very classical subject. This book includes an updated tour through some of the recent advances in the theory, such as Colding-Minicozzi theory, minimal laminations, the ordering theorem for the space of ends, conformal structure of minimal surfaces, minimal annular ends with infinite total curvature, the embedded Calabi-Yau problem, local pictures on the scale of curvature and topology, the local removable singularity theorem, embedded minimal surfaces of finite genus, topological classification of minimal surfaces, uniqueness of Scherk singly periodic minimal surfaces, and outstanding problems and conjectures Meeks and Pérez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the focus of the account. The proof of the classification depends on the work of many currently active leading mathematicians, thus making contact with much of the most important results in the field. Through the telling of the story of the classification of minimal planar domains, the general mathematician may catch a glimpse of the intrinsic beauty of this theory and the authors'perspective of what is happening at this historical moment in a very classical subject. This book includes an updated tour through some of the recent advances in the theory, such as Colding–Minicozzi theory, minimal laminations, the ordering theorem for the space of ends, conformal structure of minimal surfaces, minimal annular ends with infinite total curvature, the embedded Calabi–Yau problem, local pictures on the scale of curvature and topology, the local removable singularity theorem, embedded minimal surfaces of finite genus, topological classification of minimal surfaces, uniqueness of Scherk singly periodic minimal surfaces, and outstanding problems and conjectures. 1. Introduction -- 2. Basic Results In Classical Minimal Surface Theory -- 3. Minimal Surfaces With Finite Topology And More Than One End -- 4. Limits Of Embedded Minimal Surfaces Without Local Area Or Curvature Bounds -- 5. The Structure Of Minimal Laminations With Isolated Singularities -- 6. The Ordering Theorem For The Space Of Ends -- 7. Conformal Structure Of Minimal Surfaces -- 8. Uniqueness Of The Helicoid I: Proper Case -- 9. Embedded Minimal Annular Ends With Infinite Total Curvature -- 10. The Embedded Calabi-yau Problem -- 11. Local Pictures, Local Removable Singularities And Dynamics -- 12. Embedded Minimal Surfaces Of Finite Genus -- 13. Topological Aspects Of Minimal Surfaces -- 14. Partial Results On The Liouville Conjecture -- 15. The Scherk Uniqueness Theorem -- 16. Calabi-yau Problems -- 17. Outstanding Problems And Conjectures. William H. Meeks Iii, Joaquín Pérez. Includes Bibliographical References. 1. Introduction 2. Basic results in classical minimal surface theory 3. Minimal surfaces with finite topology and more than one end 4. Limits of embedded minimal surfaces without local area or curvature bounds 5. The structure of minimal laminations of R3 6. The Ordering Theorem for the space of ends 7. Conformal structure of minimal surfaces 8. Uniqueness of the Helicoid I: Proper case 9. Embedded minimal annular ends with infinite total curvature 10. The embedded Calabi-Yau problem 11. Local pictures, local removable singularities and dynamics 12. Embedded minimal surfaces of finite genus 13. Topological aspects of minimal surfaces 14. Partial results on the Liouville Conjecture 15. The Scherk Uniqueness Theorem 16. Calabi-Yau problems 17. Outstanding problems and conjectures.