18 research outputs found
Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold
For any bounded regular domain of a real analytic Riemannian
manifold , we denote by the -th eigenvalue of the
Dirichlet Laplacian of . In this paper, we consider and as
a functional upon the set of domains of fixed volume in . 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 . 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