All Questions
Tagged with dg.differential-geometry sp.spectral-theory
100 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
5
votes
1
answer
186
views
Reference for Weyl's law for higher order operators on closed Riemannian manifolds
I am looking at page 32 (beginning of Chapter 5) here. We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading ...
- 153
5
votes
1
answer
2k
views
Eigenvalues of the D'Alembertian operator
My question about the spectral theory of the D'Alembertian operator on a Lorentzian manifolds (say the spacetime $M^{3+1}$) given by $$\square = -\partial_{t}^2 + \Delta$$ for the metric $g=(-+++)$. ...
- 67
5
votes
1
answer
345
views
Convergence of Riemannian metrics spectra
Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
- 51
5
votes
0
answers
227
views
Relations between two Schwartz kernels in dimensions $n$ and $n+1$
Let $(M,g)$ be an $n$-dimensional pseudo-Riemannian manifold and $\Box_g$ be the Laplace-Beltrami operator on $M$. Consider $z \in \mathbb{C}$ such that $\mathrm{ Im}(z)>0$, and we define $P_0 := \...
- 319
5
votes
0
answers
167
views
Eta invariants of fiber bundles
The general question is: What is known about the eta invariants of fiber bundles?
The particular case I am interested in is the following. The fiber bundle is a bundle $S$ of even-dimensional round ...
- 1,582
4
votes
1
answer
3k
views
Laplace spectrum of the $2$-Sphere [closed]
The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$ acting on functions. The eigenvalues of this ...
- 643
4
votes
1
answer
212
views
Existence of eigen basis for elliptic operator on compact manifold
Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting ...
- 21.8k
4
votes
1
answer
501
views
Estimating laplace-beltrami spectra for a graph surface in $R^3$
Consider a surface $\Gamma$ in $R^3$. The surface $\Gamma$ is a graph, i.e. $\Gamma = (x,y, h(x,y))$, for $x \in R^2$ and some smooth function $h$, where $h$ and all its derivatives are periodic on [...
- 617
4
votes
0
answers
97
views
Eigenvalues of the Hodge-Laplace for 2-forms on $S^3$
Consider the Laplace operator $\Delta = d^{*} d + d d^{*}$ on $\Omega^2(S^3)$. What is the minimal eigenvalue of $\Delta$?
(My computations showed that the answer is 4; the eigenforms correspond to ...
- 41
4
votes
0
answers
199
views
Spectral problems with the wrong sign on the Poincaré disk
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
- 2,816
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
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
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
3
votes
3
answers
243
views
Compact surfaces with arbitrary gaps in spectrum
Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...
- 31
3
votes
1
answer
136
views
$L^\infty$-bound on Laplace-eigenfunctions
Suppose we are on a closed Riemannian manifold $M$. Any function $f\in C^\infty(M)$ may be decomposed as $$f = \sum_{j = 0}^\infty f_j\phi_j,$$ where $\phi_j\in C^\infty(M)$ are the Laplace ...
3
votes
1
answer
214
views
Convergence of spectrum
Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$.
Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
- 1,211
3
votes
2
answers
483
views
regularity of eigenfunctions of Schrödinger Operator
Hello,
I consider a compact and connected (smooth) Riemannian manofold $(M,g)$. I'm interested in the eigenfunctions of the Schrödinger Operator $L=-\Delta+ V$ acting on (smooth) functions. Do you ...
- 675
3
votes
1
answer
373
views
Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
3
votes
2
answers
425
views
Gap between first two nonzero Laplacian eigenvalues on closed compact surface?
Much has been said about bounds on Laplacian eigenvalues, and the literature can be tough to sort through! I am specifically interested in the case where the domain is a closed compact surface, and am ...
- 3,059
3
votes
1
answer
190
views
Laplace eigenfunction on a polygonal domain symmetric about an axis
Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
- 33
3
votes
1
answer
734
views
eigenvalue problem on the geodesic ball of sphere
I have a question about eigenvalue problem on the geodesic ball in $n$-dimensional sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}$.
Consider the eigenvalue problem in the geodesic ball $\Omega=\{x_{n+1}...
- 834
3
votes
1
answer
410
views
First eigenvalue of $\Delta$ on Kaehler manifold with $Ricci\ge k$.
Let $M$ be a Kaehler manifold of complex dimension $n$. Let $\Delta$ be the real Laplacian of the underline Riemannian manifold. Let's assume the Ricci curvature of $M$ satisfies $\text {Ric}\ge k>...
- 283
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
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
3
votes
1
answer
223
views
structure of metrics on a compact manifold
is there a reference on the structure of the space of metrics on a compact manifold that induce a given measure $\mu $?
i have a given manifold $M$, a given measure $\mu$ with an everywhere positive $...
- 167
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
$$\...
- 21.8k
3
votes
0
answers
318
views
Controlling the Second Eigenvalue of a Schrödinger Operator
Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$.
Let $L$ be the operator
$$
L=\Delta+V
$$
where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
- 2,299
2
votes
2
answers
172
views
Eigenfunctions of Schrödinger Operators on the boundary
Hello,
let's consider a compact and connected Riemannian manifold with the Schrödinger Operator $L=-\Delta +V:dom(H)\subset L^2(M)\rightarrow L^2(M)$ whereas $dom(L):=\lbrace f\in C^{\infty}(M,\...
- 675
2
votes
1
answer
1k
views
Global Lichnerowicz Formula Proof (in the Kahler case)?
For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...
- 3,399
2
votes
0
answers
70
views
Laplace spectrum on $U(n)$
Consider $\psi:SU(n)\times U(1)\to U(n)$, $(w,z)\mapsto \bar{z}\cdot w$. One can show that $\psi$ serves as a projection and $SU(n)\times U(1)$ is a principal $\mathbb Z_n$-bundle over $U(n)$.
Suppose ...
2
votes
0
answers
67
views
Regularity and decay of Fourier-like series on a manifold
Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
- 1,903
2
votes
0
answers
245
views
Convergence of metric and eigenvalues on a tubular neighbourhood
Background:
Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
- 537
2
votes
0
answers
100
views
Existence and uniqueness for $\Delta f + \lambda f = g$ on $S^2$ for $\lambda>0$ [closed]
Consider the PDE
$$\Delta f + \lambda f = g$$
on $S^2$, where $\Delta$ is with respect to the round metric, $g \in L^2(S^2)$ and $\lambda>0$. I wish to study the existence and uniqueness of this ...
- 969
2
votes
0
answers
53
views
A question about the choice of a special harmonc spinor
Let $X$ be a complete Riemannian manifold and $H$ be the kernel of generalized Dirac operator $D$ on $L(S)$, where $S$ is the Dirac bundle. Let $K$ be a compact subset of $X$ and $K\subset \Omega$ be ...
- 563
2
votes
0
answers
218
views
Weyl's law for minimal surfaces
I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...
- 1,616
2
votes
0
answers
302
views
Log of heat kernel for positive time
A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and $p(t,x,...
- 705
2
votes
0
answers
171
views
Laplacian on manifolds with corners
So far I've studied smooth riemannian manifolds and the Laplace-Beltrami operator associated to it. Therefore I know basic theorems like Stoke's theorem, divergence theorem, green's identities etc. ...
1
vote
1
answer
242
views
Is the trace of the heat kernel always finite?
consider any smooth Riemannian manifold $(N,g)$, an open subset $U\subset N$ and the Dirichlet heat kernel $p(t;x,y)$ for $U$. I am wondering, if it is true that
$\int_U p(t;x,x)dx <\infty$ for any ...
- 107
1
vote
1
answer
144
views
An application of min-max characterization of eigenvalues
Let $(M,g_0)$ be a $n$-dimensional closed Riemannian manifold with a Riemannian covering $(\widetilde{M},\widetilde{g}_0)$. Let
$$
\mathcal{V}_{ab}=\{g\colon a^2 g_0\leq g\leq b^2 g_0\}, \quad \text{...
- 563
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
1
vote
1
answer
424
views
Asymptotic of the heat kernel
This is the same question I asked in stackexchange:
https://math.stackexchange.com/questions/519152/asymptotic-of-the-heat-kernel
I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian ...
- 98
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
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 $\...
1
vote
0
answers
184
views
One question about Schrodinger Semigroups-(B. Simon)
This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
- 1,915
1
vote
0
answers
71
views
Nodal domains on a surface
What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?
In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
- 823
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
1
vote
0
answers
221
views
Distance between probability amplitude functions
Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$:
The Wasserstein distance
For $...
- 705
0
votes
0
answers
79
views
Convergence of metric implies convergence of eigenvalues?
Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions:
Does $g_\varepsilon$ converge to the flat metric on ...
- 537
0
votes
0
answers
126
views
Hessian estimates of eigenfunctions without Bochner
let $\Omega$ be a bounded domain in a Riemannian manifold $(M,g)$. Consider the Dirichlet eigenvalues and eigenfunctions of Laplacian on $\Omega$, that are, the $\lambda_i>0$ and $\phi_i\in H^{1}_0(...
- 193