Extremal Metric for the First Eigenvalue on a Klein Bottle
نویسندگان
چکیده
منابع مشابه
A Unique Extremal Metric for the Least Eigenvalue of the Laplacian on the Klein Bottle
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [15]: on the Klein bottle K, the metric of revolution g0 = 9 + (1 + 8 cos v) 1 + 8 cos v „ du 2 + dv 1 + 8 cos v « , 0 ≤ u < π 2 , 0 ≤ v < π, is the unique extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof l...
متن کاملGreatest Least Eigenvalue of the Laplacian on the Klein Bottle
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [12]: For any Riemannian metric g on the Klein bottle K one has λ1(K, g)A(K, g) ≤ 12πE(2 √ 2/3), where λ1(K, g) and A(K, g) stand for the least positive eigenvalue of the Laplacian and the area of (K, g), respectively, and E is the complete elliptic integral of the second kind. Moreover, the equality...
متن کاملThe Klein Bottle: Variations on a Theme
It is a well-known fact that K is a genus 2 nonorientable closed surface with Euler characteristic χ = 0, which is topologically equivalent to a couple of Möbius bands glued together along the border, and that it is nonembeddable in R. It is possible, however, to immerse it in R, that is, to map it into R obtaining an image with no singular points. To give an immersion of K in R, it suffices to...
متن کاملDrawing Disconnected Graphs on the Klein Bottle
Résumé We prove that two disjoint graphs must always be drawn separately on the Klein bottle in order to minimize the crossing number of the whole drawing.
متن کاملFive-coloring graphs on the Klein bottle
We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list. 31 August 2007, revised 18 January 2012. Partially supported by NSF under Grant No. DMS-0200595. Partially supported by NSF under Grants No. DMS-0200595, DMS-0354742, and DMS-0701077. Partially supported by NSF Graduate Fello...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Canadian Journal of Mathematics
سال: 2006
ISSN: 0008-414X,1496-4279
DOI: 10.4153/cjm-2006-016-0