Topics in Differential Geometry, Spring 2022

Course Information

• NTU COOL
• Lectures: Tuesday and Thursday, 10:20 ~ 11:40 at Astro-Math 304
• Office hour: Monday, 11:00 ~ 12:00 at Astro-Math 458
• Grading scheme:
  • Homework 50%
  • Final Report 50%
• Course prerequisite:
  • mathematical analysis
  • linear algebra and algebra
  • geometry: besides the surface theory, we will assume some Riemannian geometry, in particular, geometry of submanifold.
• Textbook:
  • [M] John Douglas Moore, Introduction to global analysis. Minimal surfaces in Riemannian manifolds. MR
• Other references:
  • Richard Melvin Schoen and Shing-Tung Yau, Lectures on harmonic maps. MR

Lecture summaries

• (Week 1) infinite-dimensional calculus, Banach manifold, Sobolev embedding and multiplication. Reference: [M] §1.2, §1.3, §1.4.
• (Week 2) tangent bundle and the other tensors, tangent bundle of mapping space, Riemannian/Finsler metric on Hilbert/Banach manifold. Reference: [M] §1.7, §1.8, §1.9.
• (Week 3) vector field and the flow, Palais-Smale condition, critical point and ambient isotopy. Reference: [M] §1.10, §1.11, §1.12.
• (Week 4) minimax principle, properties of the energy functional on L₁²(S¹,M): property C and smoothness of critical state. Reference: [M] §1.12, §2.1, §2.2.
• (Week 5) some homotopy theory, topology of the space of maps from circles, Fet-Lusternik theorem, second variational formula. Reference: [M] §1.5, §2.3.
• (Week 6) Fredholm operator, Sard-Smale theorem, parametric transversality theorem. Reference: [M] §2.5, §2.6.
• (Week 7) bumpy metric for geodesic, energy and harmonic map. Reference: [M] §2.7, §4.1.
• (Week 8) conformal invariance in 2D, energy and area functionals, Hopf differential, S² case. Reference: [M] §4.1, §4.2.
• (Week 9) the variation of the energy function in conformal structures: T² and higher genus case. introduction to the α-energy of Sacks&Uhlenbeck. Reference: [M] §4.3, §4.4.
• (Week 10) α-energy and Condition C, regularity of critical point, Bochner formula for harmonic maps. Reference: [M] §4.4, §4.6.
• (Week 11) ε-regularity theorem, limit as α→1. Reference: [M] §4.6.
• (Week 12) removable of singularities, bubbling and harmonic S². Simons identity. Reference: [M] §4.6, §4.7. Simons 1968 paper.
• (Week 13) index and nullity of minimal submanifolds in the spheres, the gap theorem of Simons. Reference: Simons 1968 paper.
• (Week 14) minimal cones and their stability. Reference: Simons 1968 paper.
  analysis of harmonic maps in higher dimensions: monotonicity, and sup-norm estimate. Reference: Schoen 1984 paper.
• (Week 15) singular set of harmonic map: closedness and Hausdorff measure; second variational formula of energy: global property into negatively curved space; some local regularity estimates. Reference: Schoen 1984 paper.

Homework

• Homework 01: due February 24.
• Homework 02: due March 3.
• Homework 03: due March 10. (This homework contains mistakes.)
• Homework 04: due March 17.
• Homework 05: due March 24.
• Homework 06: due March 31.
• Homework 07: due April 14.
• Homework 08: due April 21.
• Homework 09: due April 28.
• Homework 10: due May 5.
• Homework 11: due May 12.
• Homework 12: due May 19.