On the metric of a surface which maximizes the eigenvalues
- 발행기관 서강대학교 일반대학원
- 지도교수 Ho, Pak Tung
- 발행년도 2015
- 학위수여년월 2015. 2
- 학위명 석사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/sogang/000000055241
- 본문언어 영어
- 저작권 서강대학교 논문은 저작권보호를 받습니다.
초록/요약
We review the definition of the Laplacian and the spectrum on Riemannian manifolds. We survey that metrics which maximize the k-th eigenvalue on surfaces without boundary can be mapped onto minimal surfaces in the unit sphere Sn. The simple example of this fact is the case of the 2-dimensional unit sphere S2: the standard metric maximizes the first eigenvalue of S2 by the theorem proved by Hersch [7]. We also give an analogous result for the case of surfaces with boundary using the Steklov eigenvalue and free boundary minimal surfaces. We introduce the simple example for the case of the flat disk D2 to see that the result holds.
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