{"id":14533,"date":"2026-09-16T08:44:03","date_gmt":"2026-09-16T06:44:03","guid":{"rendered":"https:\/\/wpd.ugr.es\/~imag\/?page_id=14533"},"modified":"2026-09-16T11:02:19","modified_gmt":"2026-09-16T09:02:19","slug":"minimal-surfaces-doc-course-introduction-to-the-structural-theory-of-colding-and-minicozzi","status":"publish","type":"page","link":"https:\/\/wpd.ugr.es\/~imag\/minimal-surfaces-doc-course-introduction-to-the-structural-theory-of-colding-and-minicozzi\/","title":{"rendered":"MINIMAL SURFACES DOC-COURSE: INTRODUCTION TO THE STRUCTURAL THEORY OF COLDING AND MINICOZZI"},"content":{"rendered":"<p><a href=\"https:\/\/wpd.ugr.es\/~imag\/cms\/wp-content\/uploads\/Blanco-Negro-Rectangulo-Simple-Formato-para-Imprimir-Etiqueta-1.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wpd.ugr.es\/~imag\/cms\/wp-content\/uploads\/Blanco-Negro-Rectangulo-Simple-Formato-para-Imprimir-Etiqueta-1.png\" alt=\"\" width=\"900\" height=\"600\" class=\"aligncenter size-full wp-image-14565\" srcset=\"https:\/\/wpd.ugr.es\/~imag\/cms\/wp-content\/uploads\/Blanco-Negro-Rectangulo-Simple-Formato-para-Imprimir-Etiqueta-1.png 900w, https:\/\/wpd.ugr.es\/~imag\/cms\/wp-content\/uploads\/Blanco-Negro-Rectangulo-Simple-Formato-para-Imprimir-Etiqueta-1-300x200.png 300w, https:\/\/wpd.ugr.es\/~imag\/cms\/wp-content\/uploads\/Blanco-Negro-Rectangulo-Simple-Formato-para-Imprimir-Etiqueta-1-768x512.png 768w, https:\/\/wpd.ugr.es\/~imag\/cms\/wp-content\/uploads\/Blanco-Negro-Rectangulo-Simple-Formato-para-Imprimir-Etiqueta-1-465x310.png 465w, https:\/\/wpd.ugr.es\/~imag\/cms\/wp-content\/uploads\/Blanco-Negro-Rectangulo-Simple-Formato-para-Imprimir-Etiqueta-1-695x463.png 695w\" sizes=\"(max-width: 900px) 100vw, 900px\" \/><\/a><\/p>\n<p>As an activity developed within the framework of the PhD Program in Mathematics at the University of Granada, we offer an advanced doctoral course of 20 in-person hours on the structural theory of minimal surfaces.<\/p>\n<h1>MINIMAL SURFACES: INTRODUCTION TO THE STRUCTURAL THEORY OF COLDING AND MINICOZZI<\/h1>\n<p>The program will consist of four parts:<\/p>\n<h4>PART 1. FUNDAMENTALS<\/h4>\n<ul>\n<li><strong>Session 1: The variational problem and model examples (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Area, first variation and minimal surface equation. Plane, catenoid, helicoid and Scherk examples. Embedding, immersion, completeness and properness. Guiding questions: what information does embedding add?, why is the helicoid the model of concentrated curvature?<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada)<\/p>\n<li><strong>Session 2: Second variation, stability and Jacobi operator (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Second variation of area and index. Jacobi functions and fields coming from isometries. Estimates for stable surfaces and maximum principle. Application: curvature control far from the boundary.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<li><strong>Session 3: Monotonicity, rescaling and compactness (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Monotonicity formula and density. Blow-up at large curvature points. Smooth convergence with multiplicity; role of area and curvature bounds. Introduction to the language of minimal laminations.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<h4>PART 2.LOCAL STRUCTURAL TOOLS <\/h4>\n<ul>\n<li><strong>Session 4: Embedded minimal disks and one-sided estimate (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>One-sided curvature estimate theorem. Analogy with the half-space theorem and chord-arc consequences. Proof outline: stability between sheets, separation and contradiction by rescaling. Exercises on scaling and normalization.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<li><strong>Session 5: Multi-valued graphs (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Polar coordinates in the universal cover of the punctured plane. Separation between sheets and gradient estimates. The helicoid as a double spiral staircase. Formation and extension of multi-valued graphs on embedded disks.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<li><strong>Session 6: Local structure near large curvature (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Blow-up pairs and scale selection. Approximate axis and graphical regions outside it. Estimates off the axis. Guided reading of a central theorem of the Colding-Minicozzi series.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<h4>PART 3. LAMINATIONS AND GLOBAL STRUCTURE<\/h4>\n<ul>\n<li><strong>Session 7: Minimal laminations and singular set (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Definition and examples of minimal lamination. Convergence away from the curvature concentration set. Limit leaves, stability and removable extension. Transversal structure of the singular set.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<li><strong>Session 8: Lamination theorem for disks (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Local and global formulation. Convergence to a foliation by planes and transversal singular curve. Interpretation as a limit of rescaled helicoids. Difference between global phenomena and pathology in bounded domains.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<\/li>\n<li><strong>Session 9: Fixed genus, necks and decomposition (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>From disks to fixed genus surfaces. Difference between ULSC regions and neck formation. Pants decomposition and graphical pieces. The Riemann examples as a bridge between the two regimes.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<h4>PART 4. SYNTHESIS AND APPLICATIONS<\/h4>\n<ul>\n<li><strong>Session 10: Applications, scope and open problems (IMAG, 2 h)<\/li>\n<p><\/strong><\/p>\n<p>Uniqueness of the helicoid: role of structural theory. Properness property and Calabi-Yau conjectures in the embedded case. Connections with classification, moduli spaces and geometric flows. Short presentations by students and discussion of open problems.<\/p>\n<p>Taught by Prof. Francisco Mart\u00edn Serrano (Universidad de Granada).<\/p>\n<h4>RECIPIENTS, REQUIREMENTS AND EVALUATION<\/h4>\n<ul>\n<p>This course is aimed at PhD students in Mathematics and master&#8217;s students with a solid background in differential geometry and partial differential equations. It is recommended to have taken differential geometry of curves and surfaces, real analysis, and partial differential equations at master&#8217;s level.<\/p>\n<h4>Training activities and evaluation:<\/h4>\n<ul>\n<li>Short problem sheet on first and second variation, stability and scaling (30%).<\/li>\n<li>Written reconstruction of the argument of the one-sided curvature estimate (30%).<\/li>\n<li>Guided reading of an article from the series on fixed genus surfaces.<\/li>\n<li>Final presentation of 10-15 minutes on a practical or theoretical application (40%).<\/li>\n<h4>ORGANIZERS<\/h4>\n<p>Francisco Mart\u00edn Serrano (IMAG)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>As an activity developed within the framework of the PhD Program in Mathematics at the University of Granada, we offer an advanced doctoral course of 20 in-person hours on the structural theory of minimal surfaces. MINIMAL SURFACES: INTRODUCTION TO THE STRUCTURAL THEORY OF COLDING AND MINICOZZI The program will consist<span class=\"more-link\"><a href=\"https:\/\/wpd.ugr.es\/~imag\/minimal-surfaces-doc-course-introduction-to-the-structural-theory-of-colding-and-minicozzi\/\">Continue Reading<\/a><\/span><\/p>\n","protected":false},"author":4,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"translation":{"provider":"WPGlobus","version":"3.0.0","language":"en","enabled_languages":["en"],"languages":{"en":{"title":true,"content":true,"excerpt":false}}},"_links":{"self":[{"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/pages\/14533"}],"collection":[{"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/comments?post=14533"}],"version-history":[{"count":19,"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/pages\/14533\/revisions"}],"predecessor-version":[{"id":14658,"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/pages\/14533\/revisions\/14658"}],"wp:attachment":[{"href":"https:\/\/wpd.ugr.es\/~imag\/wp-json\/wp\/v2\/media?parent=14533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}