Journal for History of Mathematics (한국수학사학회지)

The Korean Society for History of Mathematics (한국수학사학회)
권호 표지

Modern History of Parabolic Equations on a Riemannian manifold

리이만 다양체에서 포물형 편미분 방정식에 관한 근현대사 고찰

  • Received : 2011.01.05
  • Accepted : 2011.02.19
  • Published : 2011.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

Partial differential equations on a Riemannain manifold is one of the most important areas in differential geometry. In this article, we survey the role of parabolic equations on some of the main results of differential geometry and topology, especially in the modern mathematical history. Also, we introduce some recent research in this area.

라이만 다양체 위에서의 편미분 방정식의 연구는 미분기하학에서 중요한 연구 분야로 인식되어 왔다. 본 논문에서는 특히 최근에 미분기하학과 위상수학 분야에서 중요한 역할을 하고 있는 리이만 다양체 위에서의 포물형 방정식에 관한 역사적으로 주목받고 있는 중요한 연구 결과를 정리해 보고, 아울러 이 분야의 최근 연구 결과를 고찰한다.

Keywords

References

  1. 도널 오셔 저, 전대호 역, 푸앵카레의 추측, 출판사 까치, 2007.
  2. Jeongwook Chang and Jinho Lee, Harnack type inequalities for the porous medium equation, preprint
  3. Bennett Chow et al., The Ricci flow: techniques and applications, Part I. Geometric aspects, Mathematical Surveys and Monographs 135, AMS., Provindence, RI, 2007.
  4. Bennett Chow et al., The Ricci flow: techniques and applications, Part II. Geometric aspects, Mathematical Surveys and Monographs 144, AMS., Provindence, RI, 2008.
  5. Bennett Chow et al., The Ricci flow on the 2-sphere, J. Differential Geom., 33 (1991), no. 2, 325-334. https://doi.org/10.4310/jdg/1214446319
  6. E. Di Benedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rational Mech. Anal., 100 (1988), no. 2, 129-147. https://doi.org/10.1007/BF00282201
  7. James Jr. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. https://doi.org/10.2307/2373037
  8. Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), no. 2, 255-306.
  9. Richard S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom., 37 (1993), 225-243. https://doi.org/10.4310/jdg/1214453430
  10. Richard S, Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol, II (Cambridge, MA, 1993), 7-136, Internat, Press, Cambridge, MA, 1995.
  11. Richard S, Hamilton, Non-Singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom., 7 (1999), no, 4, 695-729.
  12. Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59 (2001), no. 3, 353-437. https://doi.org/10.4310/jdg/1090349447
  13. Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:Math.DC/0211159.
  14. Grisha Perelman, Ricci flow with surgery on three-manifolds, arXiv:Math. DC/ 0303109.
  15. Grisha Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:Math. DC/0307245.