Universal eigenvalue ratio conjecture for weighted compact Alexandrov spaces

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Let (X,μ)(X,\mu) be a weighted compact finite-dimensional Alexandrov space of CD⁡(0,∞)\operatorname{CD}(0,\infty), and let kk be a natural number. Eigenvalue ratio conjecture. There exists a universal constant c>0c>0 such that

λk+1(X,μ)≤cλk(X,μ).\lambda_{k+1}(X,\mu)\leq c\lambda_k(X,\mu).

This conjecture proposes a uniform upper bound on the ratio of consecutive Laplacian eigenvalues in this class of metric-measure spaces. The supplied source does not indicate whether the conjecture has been resolved.

References

Primary source

Kei Funano, “Estimates of eigenvalues of Laplacian by a reduced number of subsets”, arXiv:1601.07581 (2016).

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