日本語 English
| 開講年度/ Academic YearAcademic Year |
20232023 |
| 科目設置学部/ CollegeCollege |
理学研究科/Graduate School of ScienceGraduate School of Science |
| 科目コード等/ Course CodeCourse Code |
LC156/LC156LC156 |
| テーマ・サブタイトル等/ Theme・SubtitleTheme・Subtitle |
|
| 授業形態/ Class FormatClass Format |
対面（全回対面）/Face to face (all classes are face-to-face)Face to face (all classes are face-to-face) |
| 校地/ CampusCampus |
池袋/IkebukuroIkebukuro |
| 学期/ SemesterSemester |
秋学期/Fall semesterFall semester |
| 曜日時限・教室/ DayPeriod・RoomDayPeriod・Room |
火5･4411/Tue.5･4411 Tue.5･4411 |
| 単位/ CreditCredit |
22 |
| 科目ナンバリング/ Course NumberCourse Number |
MAT6290 |
| 使用言語/ LanguageLanguage |
日本語/JapaneseJapanese |
| 備考/ NotesNotes |
CA198幾何学諸論 ２、RC156幾何学特論２と合同授業 |
| テキスト用コード/ Text CodeText Code |
LC156 |
In this course, we will discuss a theory of minimal surfaces, harmonic maps and surfaces of constant mean curvature in Euclid space of three dimension, after a brief review of fundamental invariants of a surfece.
Minimal surfaces and surfaces of a constant mean curvature (which will be denoted by CMC for brevity) are related to holomorphic function via the Weierstrass representation theorem. In this lecture, after a brief review of fundamental theorems of minimal surfaces and CMC, we will explain the Hopf 's theorem which says that an immersed CMC in ${\mathbb R}^3$ is a round sphere and a notion of conjugacy of CMC.
※Please refer to Japanese Page for details including evaluations, textbooks and others.
この講義では曲面の不変量を解説した後、３次元ユークリッド内の極小曲面と調和写像、平均曲率一定の曲面について解説する。
In this course, we will discuss a theory of minimal surfaces, harmonic maps and surfaces of constant mean curvature in Euclid space of three dimension, after a brief review of fundamental invariants of a surfece.
３次元ユークリッド空間における極小曲面や平均曲率一定の曲面はWeierstrassの表現定理を通して複素関数と密接な関係にある。本講義では、極小曲面や平均曲率一定に関する基本的な定理を解説したのち、３次元ユークリッド空間に埋め込まれた平均曲率一定の曲面は球面に限るというHopfによる定理と共役な平均曲率一定の曲面について解説する．
Minimal surfaces and surfaces of a constant mean curvature (which will be denoted by CMC for brevity) are related to holomorphic function via the Weierstrass representation theorem. In this lecture, after a brief review of fundamental theorems of minimal surfaces and CMC, we will explain the Hopf 's theorem which says that an immersed CMC in ${\mathbb R}^3$ is a round sphere and a notion of conjugacy of CMC.
| 1 | 曲面の不変量（第１基本形式と第２基本形式） |
| 2 | 等温座標の導入 |
| 3 | Weierstrassの表現定理 |
| 4 | 調和写像と平均曲率一定の曲面 |
| 5 | 変分原理（Euler-Lagrange方程式） |
| 6 | Noetherの定理（対称性） |
| 7 | 変数変換公式とその応用 |
| 8 | Hopf微分 |
| 9 | Hopfの定理 |
| 10 | Gauss-Codazzi方程式 |
| 11 | Gauss-Codazzi方程式の応用 |
| 12 | 極小曲面と平均曲率一定の曲面１ |
| 13 | 極小曲面と平均曲率一定の曲面２ |
| 14 | 幾つかの例 |
幾何学１と複素関数論の基礎的な知識を仮定する。
| 種類 (Kind) | 割合 (%) | 基準 (Criteria) |
|---|---|---|
| レポート試験 (Report Exam) | 60 | |
| 平常点 (In-class Points) | 40 |
最終レポート(Final Report)(40%) |
| 備考 (Notes) | ||
なし/None