All Questions
19 questions
5
votes
1
answer
371
views
Lower bound on the first eigenvalue of the Lichnerowicz Laplacian on positive Einstein manifolds
Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless ...
- 891
20
votes
1
answer
545
views
Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?
Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...
- 363
4
votes
0
answers
151
views
Eigenvalues of Laplacian and eigenvalues of curvature operator
Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
- 651
15
votes
3
answers
3k
views
The first eigenvalue of the laplacian for complex projective space
What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?
- 173
7
votes
2
answers
920
views
Exotic spectrum of Laplace operator
Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
- 101
4
votes
0
answers
98
views
Spectrum of Laplace-Beltrami with piecewise constant coefficients
By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...
- 61
8
votes
1
answer
2k
views
Growth of Laplacian eigenvalues on a compact domain?
Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct ...
- 3,059
11
votes
2
answers
1k
views
Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold
It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the (...
- 3,223
19
votes
4
answers
2k
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 $\...
- 18.7k
22
votes
1
answer
966
views
Avoiding integers in the spectrum of the Laplacian of a Riemann surface
Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...
- 5,891
4
votes
0
answers
73
views
Tight bound on spectral gap of compact homogeneous manifold?
This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter $d$:
$$ \lambda_1 \ge c/d^2, $$
where $c=\pi^2/4$. Can this bound be ...
- 361
1
vote
0
answers
52
views
Laplacian eigenvalue problem for systems coupled along the boundary
I am looking for references on eigenvalue problems for systems of the following type:
Let $\Omega$ be the region enclosed by a right triangle with legs $\Gamma_1$, $\Gamma_2$, and hypotenuse $\...
7
votes
1
answer
355
views
Eigenfunctions restricted on closed geodesics
Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l_1\mathbb{Z}\oplus l_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial x^2}-\frac{\partial^...
1
vote
0
answers
86
views
introduction textbook to the Laplacian on a circle bundle
I am looking for an introduction to spectral theory of $\Delta$ on a circle bundle over a compact M. Is there an analog of Selberg trace formula?
- 19
6
votes
2
answers
2k
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
$$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{...
- 15.1k
1
vote
1
answer
173
views
Zero set of eigenfunction along a sub manifold
Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...
- 113
7
votes
1
answer
1k
views
How to construct a scalar differential operator having the same spectrum as a non-scalar differential operator exploiting symmetries?
I am interested in eigenvalue problems for differential operators acting on one forms on closed two-dimensional manifolds and how they relate to eigenvalue problems of associated operators acting on ...
3
votes
0
answers
146
views
Spectral theory of differential forms over a circle bundle
Here is the set up : I consider the unitary tangent bundle of a surface $(S,g)$ endowed with the Sasakian metric ; $(T^1S, g_s)$, in fact we have the following fibration :
\begin{equation*}
\mathbb{...
user50806
2
votes
1
answer
1k
views
Global Lichnerowicz Formula Proof (in the Kahler case)?
For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...
- 3,399