Eigenvalues of the Laplacian and extrinsic geometry

TL;DR

This paper generalizes bounds relating Laplacian eigenvalues to extrinsic invariants for submanifolds, extending results to complex projective spaces and more stable invariants, with sharp bounds for the first non-zero eigenvalue.

Contribution

It introduces new stable extrinsic invariants for bounding Laplacian eigenvalues and extends the analysis to complex submanifolds in projective space.

Findings

Derived eigenvalue upper bounds depending only on submanifold dimension.

Extended previous results to complex projective space.

Established sharp bounds for the first non-zero eigenvalue.

Abstract

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space $\mathbb{C} P^N$ instead of submanifolds of $\mathbb{R}^N$ and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue.