Bounding the first Dirichlet eigenvalue of a tube around a complex submanifold of $CP^n$ by the degrees of the polynomials defining it

TL;DR

This paper derives sharp upper bounds for the first Dirichlet eigenvalue of tubes around complex submanifolds in complex projective space, based on polynomial degrees and model eigenvalues, with applications to gap phenomena.

Contribution

It introduces bounds depending only on polynomial degrees and model eigenvalues, providing new comparison results and gap phenomena for specific models.

Findings

Bounds depend solely on polynomial degrees and model eigenvalues.

Bounds are sharp for the chosen model centers.

Application to gap phenomena and comparison results for specific models.

Abstract

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the models used are $CP^q$ or the complex hyperquadric, these bounds also give gap phenomena and comparison results.