Extremal eigenvalues of the Conformal Laplacian under Sire–Xu normalization
نویسندگان
چکیده
Let (Mn,g) be a closed Riemannian manifold of dimension n≥3. We study the variational properties kth eigenvalue functional g̃∈[g]↦λk(Lg̃) under non-volume normalization proposed by Sire–Xu. discuss necessary conditions for existence extremal eigenvalues such normalization. Also, we general problem when k=1.
منابع مشابه
Extremal eigenvalues of the Laplacian in a conformal class of metrics : the ” conformal spectrum ”
Let M be a compact connected manifold of dimension n endowed with a conformal class C of Riemannian metrics of volume one. For any integer k ≥ 0, we consider the conformal invariant λk(C) defined as the supremum of the k-th eigenvalue λk(g) of the Laplace-Beltrami operator ∆g, where g runs over C. First, we give a sharp universal lower bound for λk(C) extending to all k a result obtained by Fri...
متن کاملEigenvalues of the Conformal Laplacian
We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.
متن کاملSmall Eigenvalues of the Conformal Laplacian
We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.
متن کاملEigenvalues of the normalized Laplacian
A graph can be associated with a matrix in several ways. For instance, by associating the vertices of the graph to the rows/columns and then using 1 to indicate an edge and 0 otherwise we get the adjacency matrix A. The combinatorial Laplacian matrix is defined by L = D − A where D is a diagonal matrix with diagonal entries the degrees and A is again the adjacency matrix. Both of these matrices...
متن کاملExtremal G-invariant Eigenvalues of the Laplacian of G-invariant Metrics
The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S endowed with S-invariant metrics, we consider the subsequence λ k of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dim...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Nonlinear Analysis-theory Methods & Applications
سال: 2021
ISSN: ['1873-5215', '0362-546X']
DOI: https://doi.org/10.1016/j.na.2021.112308