Calculus of Variations and Geometric Measure Theory

A. Hassannezhad

Eigenvalues of the Laplacian and extrinsic geometry

created by hassannezhad1 on 04 Dec 2020



Inserted: 4 dec 2020

Year: 2012

ArXiv: 1210.7714 PDF


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.