All Questions
Tagged with sp.spectral-theory dg.differential-geometry
100 questions
1
vote
0
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86
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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?
10
votes
1
answer
409
views
Can a Laplace eigenfunction have a level set with a cusp like singularity?
Let $\Omega$ be a precompact open subset of ${\mathbb R}^2$ with piecewise
smooth boundary. Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either Dirichlet or Neumann conditions on $\...
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 ...
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 ...
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
$$K(t;z,z) \leq \frac{C_M}{f_z(t)...
2
votes
0
answers
218
views
Weyl's law for minimal surfaces
I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...
1
vote
1
answer
173
views
Zero set of eigenfunction along a sub manifold
Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\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,$$ ...
5
votes
1
answer
2k
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Eigenvalues of the D'Alembertian operator
My question about the spectral theory of the D'Alembertian operator on a Lorentzian manifolds (say the spacetime $M^{3+1}$) given by $$\square = -\partial_{t}^2 + \Delta$$ for the metric $g=(-+++)$. ...
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{...
8
votes
0
answers
352
views
Two proofs of the Cheeger-Müller theorem
In the late 1970's, Cheeger and Müller independently proved the equality of analytic torsion and Reidemeister torsion for orthogonal representations, which had been conjectured by Ray-Singer. Their ...
12
votes
1
answer
1k
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Multiplicity of Laplace eigenvalues
Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher ...
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 ...
13
votes
1
answer
481
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A question on a result of Colin de Verdière
Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdière (see Construction de laplaciens dont une partie finie du spectre est ...
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 ...
10
votes
0
answers
284
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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 $-\...
8
votes
1
answer
421
views
$C^k$ one-parameter family of metrics
Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
11
votes
2
answers
1k
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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
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 $...
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
$$\...
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,...
6
votes
1
answer
928
views
Laplacian eigenfunction $L^p$ norms
Suppose I have a compact surface (possibly with boundary), and consider the eigenfunctions of the Laplacian, normalized so that their $L^2$ norms are $1.$ Is there some general result or conjecture on ...
6
votes
1
answer
503
views
Recovering Spherical Harmonics from Discrete Samples
Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...
2
votes
0
answers
171
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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. ...
14
votes
1
answer
1k
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Spectrum of Laplacian in non-compact manifolds
What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty?
What would be a ...
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 ...
1
vote
1
answer
424
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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 ...
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,\...
6
votes
1
answer
1k
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The first eigenvalue of the Schrödinger operator is simple.
Hello,
let $(M,g)$ be a compact and connected Riemannian manifold (possibly with $\partial M\neq \emptyset$). We consider the Friedrichs extension of $L=-\Delta +V: C^{\infty}(M,\mathbb{R})\subset L^...
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 ...
3
votes
2
answers
425
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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
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>...
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
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 $...
25
votes
1
answer
3k
views
Relationship between Green's function and geodesic distance?
I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and ...
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 ...
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 ...
15
votes
3
answers
3k
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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 $SU(n+1)/S(U(1)\times U(n))$?
5
votes
3
answers
662
views
Estimates for the diameter of a (nice) surface?
The Question
Let $M$ be a compact, connected, orientable surface without boundary of bounded genus smoothly embedded in $\mathbb{R}^3$; define the diameter $d_M$ of $M$ as the maximum minimal (...
7
votes
1
answer
1k
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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 ...
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 ...
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 $\...
9
votes
2
answers
543
views
Symmetric spaces, Horocycle spaces and intertwining operators
Let $G=KAN$ be an Iwasawa decomposition of a connected semisimple Lie group with finite center. Let us assume for simplicity that the associated symmetric space $G/K$ has rank 1.
Harish-Chandras ...
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}...
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 [...
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 ...
23
votes
3
answers
3k
views
Trapped rays bouncing between two convex bodies
At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
9
votes
1
answer
1k
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The smallest Laplace-Beltrami eigenvalue on hyperbolic surfaces
For $g\geq 2$, let $M_g$ be the moduli space of genus $g$ hyperbolic surfaces, and let $\lambda_1(S_x): M_g \to \mathbb{R}$ be the smallest eigenvalue of the Laplace-Beltrami operator on the surface $...
5
votes
1
answer
389
views
Is there a name for this differential operator and/or its corresponding spectrum?
Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional
$$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$
where $X_p(f)$ is the ...
10
votes
3
answers
3k
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Number Theory and Geometry/Several Complex Variables
This is a question for all you number theorists out there...based on my skimming of number theory textbooks and survey articles, it seems like most of the applications of geometry and complex ...
6
votes
2
answers
2k
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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{...