学部・大学院区分 Undergraduate / Graduate   多・博前   時間割コード Registration Code   3211112   科目区分 Course Category   Ａ類Ⅲ（集中講義） Category A3   科目名 【日本語】 Course Title   幾何学特別講義Ⅱ   科目名 【英語】 Course Title   Special Course on Geometry Ⅱ   コースナンバリングコード Course Numbering Code     担当教員 【日本語】 Instructor   ＭＩＣＡＬＬＥＦ Ｍ ○   担当教員 【英語】 Instructor   MICALLEF Mario ○   単位数 Credits   1   開講期・開講時間帯 Term / Day / Period   秋集中 その他 その他 Intensive(Fall) Other Other   授業形態 Course style  
  学科・専攻 Department / Program     必修・選択 Required / Selected    
授業の目的 【日本語】 Goals of the Course(JPN)   At the end of this course, students should know
• the link between minimal surfaces in Euclidean space and meromorphic 1forms,
• the relation between energy and area of a map of a surface,
• the proof of the existence of a minimal disk spanning a regular Jordan curve in Euclidean space,
• about the existence of minimal surfaces of higher topological type,
• about bubbling when trying to prove the existence of minimal surfaces in a general Riemannian manifold,
• the proof of the Bernstein theorem under a Gauss map assumption and under a stability assumption,
• about curvature estimates as a local version of Bernstein's theorem,
• about Morse theoretic aspects of minimal surface theory. 
  授業の目的 【英語】 Goals of the Course   At the end of this course, students should know
• the link between minimal surfaces in Euclidean space and meromorphic 1forms,
• the relation between energy and area of a map of a surface,
• the proof of the existence of a minimal disk spanning a regular Jordan curve in Euclidean space,
• about the existence of minimal surfaces of higher topological type,
• about bubbling when trying to prove the existence of minimal surfaces in a general Riemannian manifold,
• the proof of the Bernstein theorem under a Gauss map assumption and under a stability assumption,
• about curvature estimates as a local version of Bernstein's theorem,
• about Morse theoretic aspects of minimal surface theory. 
  到達目標 【日本語】 Objectives of the Course(JPN))   This course will provide an introduction to the theory of minimal surfaces. Basic existence theorems and differential geometric properties of minimal surfaces will be presented. Applications of minimal surface theory to the study of the interaction between the curvature and the topology of a manifold will be discussed. 
  到達目標 【英語】 Objectives of the Course   This course will provide an introduction to the theory of minimal surfaces. Basic existence theorems and differential geometric properties of minimal surfaces will be presented. Applications of minimal surface theory to the study of the interaction between the curvature and the topology of a manifold will be discussed. 
  授業の内容や構成 Course Content / Plan   1. Differential Geometry of Minimal Surfaces
1.1 First variation of area (for a varying metric on a surface); second fundamental form as infinitesimal variation of metric. Minimal surfaces as surfaces of zero mean curvature.
1.2 Isothermal coordinates and minimal surfaces as conformal harmonic maps.
1.3 Minimal surfaces in R^n and flat tori via integration of meromorphic differentials satisfying a quadratic condition and period conditions; EnneperRiemannWeierstrass formulas. Brief mention of examples.
1.4 Osserman's generalisation of Bernstein's Theorem.
1.5 Holomorphic curves as minimal surfaces.
2. Existence Theorems
2.1 Formulation of Plateautype problems. EnneperRiemannWeierstrass formulas not particularly helpful for solving Plateau problem.
2.2 Relation between energy and area.
2.3 Disktype solution to the Plateau problem for a curve in R^n; Riemann mapping theorem as special case. Mention of DouglasRado controversy.
2.4 Area minimising property of Douglas solution.
2.5 Branch points. Osserman's theorem and MicallefWhite (no detailed proof). Rado's graphical theorem.
2.6 Energy as a function on the space of maps and conformal structures.
2.7 Existence theorems of SacksUhlenbeck and SchoenYau; bubbling (level of detail will depend on PDE background of students).
2.8 Mention of Teichmüller harmonic map flow.
3. Morse index of minimal surfaces
3.1 Second variation formula
3.2 Bernstein Theorem for oriented stable minimal surfaces in R^3 (if time permits, generalisation to stable CMC surfaces in simply connected space forms; Yau's isoperimetric inequality)
3.3 Curvature estimates for stable minimal surfaces in R^3; mention generalisation to stable minimal surfaces in 3manifolds. Compactness results.
3.4 Wirtinger's inequality and area minimizing property of holomorphic curves.
3.5 Complex version of secondvariation formula.
3.6 Relation between stability and holomorphicity of minimal surfaces in Euclidean space, flat tori and K3 surfaces.
3.7 Relation between Morse index and total scalar curvature; EjiriMicallef, Choe, ChodoshMaximo
4. Relations between curvature and topology via minimal surfaces
4.1 Topology of 3manifolds with positive scalar curvature (SchoenYau). Rigidity of area minimizing surfaces in 3manifolds with lower bound on scalar curvature. (MicallefMoraru, BrayBrendleNeves, Nunes.)
4.2 Upper bound on index in SacksUhlenbeck existence theory.
4.3 Topology of manifolds with positive isotropic curvature (MicallefMoore, FraserSchoen). 
  履修条件 Course Prerequisites   Background knowledge (some of which will be in a handout as stated in the textbook section below):
GaussCodazzi equations for a surface.
Isothermal coordinates on a Riemannian surface.
Riemann curvature tensor.
Exponential map from the tangent space of a Riemannian manifold.
Poisson integral formula for a harmonic function on the unit disc.
AscoliArzela Theorem.
Basic algebraic topology, at least the fundamental group.
Desirable, but not required
Cartan's method of moving frames.
Gradient estimate for harmonic functions.
de Rham cohomology, higher homotopy groups, homology groups.
Basics of Morse Theory in finite dimensions.
Sobolev spaces, Sobolev inequality and Rellich compact embedding theorem.
この講義は英語で行います。
This course will be taught in English. 
  関連する科目 Related Courses     成績評価の方法と基準 Course Evaluation Method and Criteria   Assignments will be set during the course and the students will be required to hand in their solutions a week after the course. Also, on each of the first three days, easy exercises to check the students' understanding will be set and the students will be advised to hand in their solutions the following day and then the assignments with comments will be returned the following day. 
  教科書・テキスト Textbook   The course will not follow any particular book, as minimal surface theory is a huge subject and no book covers all its aspects. A handout describing Cartan’s method of moving frames and its application to the equations of submanifold geometry and definition of the curvature operator will be prepared. Some other material, e.g. gradient estimate of harmonic functions, will also be distributed. 
  参考書 Reference Book   In the list below, Lawson's book and Osserman's book are a good starting point for a differential geometer. The book by Colding and Minicozzi is more anlytical. The lecture notes by Brian White and Rick Schoen and the surveys in Volume 90 of the Encyclopaedia of Mathematical Sciences provide broader perspectives. The books by Hildebrandt et al. are very detailed and technical.
Blaine Lawson Jr., H., Lectures on minimal submanifolds. Vol. I, 2nd ed., Mathematics Lecture Series, vol. 9, Publish or Perish Inc., Wilmington, Del., 1980.
Osserman, R., A survey of minimal surfaces, 2nd ed., Dover Publications Inc., New York, 1986
Colding, T. H. and Minicozzi II, W. P., A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011.
Encyclopaedia of Mathematical Sciences, Vol 90, Geometry V, Miimal Surfaces, edited by Osserman, R.
Dierkes, U., Hildebrandt, S. and Sauvigny, F., Minimal surfaces, 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 339, Springer, Heidelberg, 2010. With assistance and contributions by A. Küster and R. Jakob.
Dierkes, U., Hildebrandt, S. and Tromba, A. J., Global Analysis of Minimal Surfaces, 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 341, Springer, Heidelberg, 2010.
White, B., Lectures on Minimal Surface Theory. https://arxiv.org/pdf/1308.3325.pdf
Schoen, R., Minimal Submanifolds in Higher Codimension.
https://geometrysummer.math.uconn.edu/wpcontent/uploads/sites/2312/2018/06/minsurf_5.pdf 
  課外学習等（授業時間外学習の指示） Study Load(Selfdirected Learning Outside Course Hours)     注意事項 Notice for Students     他学科聴講の可否 Propriety of Other department student's attendance     他学科聴講の条件 Conditions of Other department student's attendance     レベル Level     キーワード Keyword   mean curvature, minimal surface, Plateau problem, bubbling, Bernstein theorem, curvature estimate, Morse index, RiemannRoch, sphere theorem 
  履修の際のアドバイス Advice   It is desirable that the students are sufficiently familiar with mathematics of undergraduate level. See Prerequisite for details. 
  授業開講形態等 Lecture format, etc.   Blackboard and chalk, pdf (Beamer) slides 
  遠隔授業（オンデマンド型）で行う場合の追加措置 Additional measures for remote class (ondemand class)    
