Estimates of eigenvalues of Laplacian by a reduced number of subsets

TL;DR

This paper improves bounds on Laplacian eigenvalues in Alexandrov spaces by reducing the required input subsets, and discusses a related conjecture on universal eigenvalue inequalities.

Contribution

It demonstrates that the input subsets in Chung-Grigor'yan-Yau's inequality can be minimized in Alexandrov spaces with CD(0,∞), advancing understanding of eigenvalue bounds.

Findings

Reduced the number of subsets needed for eigenvalue bounds

Extended inequality applicability to Alexandrov spaces with CD(0,∞)

Discussed a conjecture on universal eigenvalue inequalities

Abstract

Chung-Grigor'yan-Yau's inequality describes upper bounds of eigenvalues of Laplacian in terms of subsets ("input") and their volumes. In this paper we will show that we can reduce "input" in Chung-Grigor'yan-Yau's inequality in the setting of Alexandrov spaces satisfying CD$(0,\infty)$. We will also discuss a related conjecture for some universal inequality among eigenvalues of Laplacian.