The tag has no usage guidance.

learn more… | top users | synonyms

8
votes
2answers
879 views

Characterize where the Dirichlet Problem for the Laplacian is always solvable

Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...
16
votes
4answers
1k views

High multiplicity eigenvalue implies symmetry?

It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions $\...
1
vote
1answer
505 views

Harmonic forms for Complex Projective Space.

For complex projective space with the Fubini-Study metric and associated Laplace-de Rham operator $dd^\ast+d^\ast d$. How does one find a concrete description of the space of harmonic forms? That is, ...
3
votes
1answer
520 views

Relation between the de Rham and Hodge Laplacians on the Exterior Algebra

For a Kahler manifold $M$ we have two well-known Laplacians: the de Rham Laplacian $\Delta_{\text{d}} = ($d$ + $d$^\ast)^2$, and the Dolbeault Laplacian $\Delta_{\overline{\partial}} = (\overline{\...
4
votes
4answers
1k views

Boundaries of the eigenvalues of a symmetric matrix (or of its Lapacian)

Given the adjacency matrix $A_{ij}$ of a graph with $N$ vertices and $M$ links (or any binary symmetric matrix of size $N \times N$), is it possible to establish lower and upper boundaries of its ...
3
votes
2answers
758 views

Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature

A friend in physics asked this question, and I didn't know the answer. Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...
5
votes
2answers
1k views

Eigenvalues of Laplacian

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{...