Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold

For any bounded regular domain $\Omega$ of a real analytic Riemannian manifold $M$, we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $\lambda_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.Comment: To appear in Illinois J. Mat

CRITICAL METRICS OF THE TRACE OF THE HEAT KERNEL ON A COMPACT MANIFOLD

This paper is devoted to the study of critical metrics of the trace of the heat kernel on a compact manifold. We obtain various characterizations of such metrics and investigate their geometric properties. We also give a complete classification of critical metrics on surfaces of genus zero and one

Eigenvalues upper bounds for the magnetic Schrödinger operator

We study the eigenvalues λk(HA,q) of the magnetic Schro ̈dinger operator HA,q associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neu- mann boundary conditions if ∂M ̸= ∅. We obtain various bounds on λ1(HA,q),λ2(HA,q) and, more generally on λk(HA,q). Some of them are sharp. Besides the dimension and the volume of the man- ifold, the geometric quantities which plays an important role in these estimates are: the first eigenvalue λ′′ (M) of the Hodge-de 1,1 Rham Laplacian acting on co-exact 1-forms, the mean value of the scalar potential q, the L2-norm of the magnetic field B = dA, and the distance, taken in L2, between the harmonic component of A and the subspace of all closed 1-forms whose cohomology class is in- tegral (that is, having integral flux around any loop). In particular, this distance is zero when the first cohomology group H1(M,R) is trivial. Many other important estimates are obtained in terms of the conformal volume, the mean curvature and the genus (in dimension 2). Finally, we also obtain estimates for sum of eigen- values (in the spirit of Kro ̈ger estimates) and for the trace of the heat kernel

Laplacian eigenvalues functionals and metric deformations on compact manifolds

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues