Skip to main content
MathOverflow

Return to Question

added 277 characters in body
Source Link
Analysis Now
Analysis Now
  • 1.5k
  • 13
  • 25

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus $g_n$ and varying hyperbolic metrics $d_n$ such that its ($2g_n - 3$)rd eigenvalue $\lambda_n$ is of the order $o(g_n ^3)?$

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between $\lambda_n$ and $l_n,$ of the form $c(g) l_n \leq \lambda_n \leq d(g) l_n,$ which I do not want to use, since I do not understand fully the proof of $\lambda_n \geq c(g)l_n\ldots$.I might ask a question about it later. The quantity l_n is defined below : consider all the family F of sets of disjoint closed geodesics on the compact hyperbolic surface M such that they divide M into ( n+1 ) components. Then l_n = infimum of sum of all the closed such geodesics in all such sets in the family F.

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus $g_n$ and varying hyperbolic metrics $d_n$ such that its ($2g_n - 3$)rd eigenvalue $\lambda_n$ is of the order $o(g_n ^3)?$

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between $\lambda_n$ and $l_n,$ of the form $c(g) l_n \leq \lambda_n \leq d(g) l_n,$ which I do not want to use, since I do not understand fully the proof of $\lambda_n \geq c(g)l_n\ldots$.I might ask a question about it later.

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus $g_n$ and varying hyperbolic metrics $d_n$ such that its ($2g_n - 3$)rd eigenvalue $\lambda_n$ is of the order $o(g_n ^3)?$

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between $\lambda_n$ and $l_n,$ of the form $c(g) l_n \leq \lambda_n \leq d(g) l_n,$ which I do not want to use, since I do not understand fully the proof of $\lambda_n \geq c(g)l_n\ldots$.I might ask a question about it later. The quantity l_n is defined below : consider all the family F of sets of disjoint closed geodesics on the compact hyperbolic surface M such that they divide M into ( n+1 ) components. Then l_n = infimum of sum of all the closed such geodesics in all such sets in the family F.

latex $s, tag
Source Link
Victor Protsak
Victor Protsak
  • 14.5k
  • 4
  • 68
  • 94

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus g_n$g_n$ and varying hyperbolic metrics d_n$d_n$ such that its ( 2g_n - 3 $2g_n - 3$) rdrd eigenvalue lambda_n$\lambda_n$ is of the order o ( g_n ^ 3 ) ?$o(g_n ^3)?$

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between lambda_n$\lambda_n$ and l_n ,$l_n,$ of the form c(g) l_n <= lambda_n <= d(g) l_n ,$c(g) l_n \leq \lambda_n \leq d(g) l_n,$ which I do not want to use, since I do not understand fully the proof of lambda_n >= c(g)l_n...$\lambda_n \geq c(g)l_n\ldots$.I might ask a question about it later.

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus g_n and varying hyperbolic metrics d_n such that its ( 2g_n - 3 ) rd eigenvalue lambda_n is of the order o ( g_n ^ 3 ) ?

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between lambda_n and l_n , of the form c(g) l_n <= lambda_n <= d(g) l_n , which I do not want to use, since I do not understand fully the proof of lambda_n >= c(g)l_n....I might ask a question about it later.

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus $g_n$ and varying hyperbolic metrics $d_n$ such that its ($2g_n - 3$)rd eigenvalue $\lambda_n$ is of the order $o(g_n ^3)?$

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between $\lambda_n$ and $l_n,$ of the form $c(g) l_n \leq \lambda_n \leq d(g) l_n,$ which I do not want to use, since I do not understand fully the proof of $\lambda_n \geq c(g)l_n\ldots$.I might ask a question about it later.

Source Link
Analysis Now
Analysis Now
  • 1.5k
  • 13
  • 25

Examples of compact hyperbolic surfaces/manifolds with very small or very large eigenvalues

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus g_n and varying hyperbolic metrics d_n such that its ( 2g_n - 3 ) rd eigenvalue lambda_n is of the order o ( g_n ^ 3 ) ?

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between lambda_n and l_n , of the form c(g) l_n <= lambda_n <= d(g) l_n , which I do not want to use, since I do not understand fully the proof of lambda_n >= c(g)l_n....I might ask a question about it later.