All Questions
Tagged with sp.spectral-theory riemannian-geometry
26 questions with no upvoted or accepted answers
10
votes
0
answers
284
views
Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
- 109
8
votes
0
answers
220
views
Regularilty of Commutative Spectral Triples
In Connes' approach to non-commutative geometry, the notion of a spectral triple is said to generalize compact Riemannian manifolds to the non-commutative setting. Motivating classical examples ...
- 173
7
votes
0
answers
89
views
Eigenvalue lower bounds for manifold with positive Ricci curvature
For closed $n$-manifold with Ricci curvature $\ge (n-1)$, it is known that the first eigenvalue $\lambda_1\ge n$ with equality holds if and only if $M$ is isometric to the Euclidean sphere $S^n$. My ...
- 481
7
votes
0
answers
189
views
Geometrically-explicit upper bound for on-diagonal heat kernel
Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form
$$K(t;z,z) \leq \frac{C_M}{f_z(t)...
6
votes
0
answers
393
views
Steklov eigenvalue problem for a planar region bounded by ellipse
The Steklov problem for a compact planar region $\Omega$ is
\begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$},
\end{...
- 141
4
votes
0
answers
137
views
Eigenvalues of Schrödinger operator with Robin condition on the boundary
Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
- 2,095
4
votes
0
answers
161
views
Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
- 5,230
4
votes
0
answers
82
views
On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
- 4,145
4
votes
0
answers
178
views
Products of eigenfunctions on compact Riemann surfaces
Let $M$ be a compact Riemann surface with genus $g\geq 2$, endowed with the Riemannian metric with constant sectional curvature $-1$. Let $f_1, f_2$ be two (global) eigenfunctions for the Laplace-...
- 323
4
votes
0
answers
151
views
Eigenvalues of Laplacian and eigenvalues of curvature operator
Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
- 651
4
votes
0
answers
240
views
Does the zeta regularized Laplacian determinant measure the volume of some parameter space? How many "spanning trees" on a manifold?
Let $(M,g)$ be a Riemannian manifold, with Laplacian $\Delta$. If $\lambda_i$ are the nonzero eigenvalues of $\Delta$, we can define the zeta function $\zeta(s) = \Sigma \lambda_i^{-s}$. By analytic ...
- 1,462
3
votes
0
answers
112
views
Is the square root of curl^2-1/2 a natural (Dirac-)operator?
In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...
- 1,942
3
votes
0
answers
615
views
Estimates of eigenvalues of elliptic operators on compact manifolds
The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula
$$\...
- 21.8k
2
votes
0
answers
70
views
Laplace spectrum on $U(n)$
Consider $\psi:SU(n)\times U(1)\to U(n)$, $(w,z)\mapsto \bar{z}\cdot w$. One can show that $\psi$ serves as a projection and $SU(n)\times U(1)$ is a principal $\mathbb Z_n$-bundle over $U(n)$.
Suppose ...
2
votes
0
answers
94
views
Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics
Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
- 2,095
2
votes
0
answers
71
views
Examples of elementary group of isometries of the ideal boundary of hyperbolic plane
A Riemannian manifold $(X,g)$ is called Hadamard manifold if it is complete and simply connected and has everywhere non-positive sectional curvature. An example of such a manifold would be the ...
- 41
2
votes
0
answers
56
views
Weyl asymptotic in towers
Let $M$ be a compact Riemannian manifold. The Weyl law gives the asymptotic of the counting function $N(T)$ of Laplace eigenvalues $|\lambda|\le T$ as $T\to\infty$.
Now suppose you are given a tower ...
user473423
2
votes
0
answers
85
views
Proving an eigenvalue bound without resorting to Weyl's law
Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
- 4,145
2
votes
0
answers
77
views
Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator
I would appreciate any answers or even references for the following problem.
Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
- 4,145
2
votes
0
answers
53
views
A question about the choice of a special harmonc spinor
Let $X$ be a complete Riemannian manifold and $H$ be the kernel of generalized Dirac operator $D$ on $L(S)$, where $S$ is the Dirac bundle. Let $K$ be a compact subset of $X$ and $K\subset \Omega$ be ...
- 563
2
votes
0
answers
400
views
Spectrum of the Witten Laplacian on compact Riemannian manifolds
Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$
How generally is it true that this ${\rm ...
- 2,246
2
votes
0
answers
123
views
Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds
Consider the elliptic operator $Lu = - \Delta u + \langle \nabla u , X \rangle + c \, u $ acting on functions on a closed Riemannian manifold $M$. Here $\Delta$ denotes the Laplace-Beltrami operator, $...
- 386
2
votes
0
answers
149
views
Off-diagonal estimates for Poisson kernels on manifolds
Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
- 1,090
1
vote
0
answers
62
views
Gradient estimate of the eigenfunction of Laplacian on hyperbolic space
I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...
1
vote
0
answers
100
views
Question about Dirac operator
Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that
$$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$
for $\...
- 563
1
vote
0
answers
46
views
The Morse Index of a $T$- periodic geodesics is a integer number?
It is well known that compact Riemannian manifolds $(M, g)$ with
periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum
of $ \sqrt{ - \Delta}$, the square root of ...
- 245