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 of four parts:

PART 1. FUNDAMENTALS

  • Session 1: The variational problem and model examples (IMAG, 2 h)

    • 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?

    Taught by Prof. Francisco Martín Serrano (Universidad de Granada)

  • Session 2: Second variation, stability and Jacobi operator (IMAG, 2 h)

    • 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.

    Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

  • Session 3: Monotonicity, rescaling and compactness (IMAG, 2 h)

    • 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.

    Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

    PART 2.LOCAL STRUCTURAL TOOLS

    • Session 4: Embedded minimal disks and one-sided estimate (IMAG, 2 h)

      • 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.

      Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

    • Session 5: Multi-valued graphs (IMAG, 2 h)

      • 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.

      Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

    • Session 6: Local structure near large curvature (IMAG, 2 h)

      • 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.

      Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

      PART 3. LAMINATIONS AND GLOBAL STRUCTURE

      • Session 7: Minimal laminations and singular set (IMAG, 2 h)

        • 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.

        Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

      • Session 8: Lamination theorem for disks (IMAG, 2 h)

        • 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.

        Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

      • Session 9: Fixed genus, necks and decomposition (IMAG, 2 h)

        • 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.

        Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

        PART 4. SYNTHESIS AND APPLICATIONS

        • Session 10: Applications, scope and open problems (IMAG, 2 h)

          • 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.

          Taught by Prof. Francisco Martín Serrano (Universidad de Granada).

          RECIPIENTS, REQUIREMENTS AND EVALUATION

            This course is aimed at PhD students in Mathematics and master’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’s level.

            Training activities and evaluation:

            • Short problem sheet on first and second variation, stability and scaling (30%).
            • Written reconstruction of the argument of the one-sided curvature estimate (30%).
            • Guided reading of an article from the series on fixed genus surfaces.
            • Final presentation of 10-15 minutes on a practical or theoretical application (40%).

              • ORGANIZERS

              Francisco Martín Serrano (IMAG)