All Questions
Tagged with dg.differential-geometry sp.spectral-theory
100 questions
64
votes
5
answers
15k
views
Intuitively, what does a graph Laplacian represent?
Recently I saw an MO post Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? that got me interested. ...
- 290
28
votes
6
answers
3k
views
Why is there no symplectic version of spectral geometry?
First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta_g:C^\infty(M)\to C^\infty(M)$ is defined as
$$
\Delta_g=\mathrm{div}_g\circ\mathrm{grad}_g,
$$
where the ...
- 1,942
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 ...
- 3,059
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. ...
- 9,007
23
votes
1
answer
1k
views
Eigenvalues of Laplace operator
Assume that $(M,g)$ is a Riemannian manifold.
Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of ...
- 356
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
20
votes
2
answers
2k
views
Can one hear the (topological) shape of a drum?
Let $(M,g)$ be a (say closed) Riemannian manifold. One can try to understand the geometry/topology of $(M,g)$ by studying the eigenvalues of the Laplacian (this I guess has two versions: when ...
- 953
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
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
18
votes
2
answers
2k
views
Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold
Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $\Delta_g$ have finite multiplicity and tend to ...
- 993
17
votes
3
answers
770
views
Does a spectral gap lift to covering spaces?
Let $M$ be a complete Riemannian manifold. Denote $\Delta_M\ge0$ the unique self-adjoint extension of the Laplace-Beltrami operator in $L^2(M)$ and $\sigma(\Delta_M)\subset [0,\infty)$ its spectrum. ...
- 779
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
14
votes
1
answer
739
views
Eigenfunctions of the laplacian on $\mathbb{CP}^n$
I want to find explicit formulas for the eigenfunctions of the Laplacian on $\mathbb{CP}^n$ endowed with the Fubini Study metric.
For the first eigenvalue $\lambda_1 = 4(n+1)$, the eigenfunctions ...
- 320
14
votes
1
answer
1k
views
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 ...
- 13.5k
14
votes
1
answer
668
views
Why are we interested in spectral gaps for Laplacian operators
Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
- 185
13
votes
1
answer
481
views
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 ...
- 1,407
12
votes
1
answer
1k
views
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 ...
- 121
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
10
votes
3
answers
3k
views
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 ...
- 1,665
10
votes
2
answers
938
views
Weyl law for (non-semiclassical) Schrodinger operator
The Weyl law for a semiclassical Schrodinger operator
$$ A_h\ := \ -h^2\Delta+V(x) $$
on an $d$-dimensional complete Riemannian manifold $M$
says that the number $N(A_h,1)$ of eigenvalues of $A_h$ ...
- 658
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 $\...
- 494
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
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 ...
- 261
9
votes
1
answer
1k
views
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 $...
- 13.1k
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
8
votes
2
answers
263
views
Is the $n/2$-th heat kernel coefficient topological?
I have asked the same question on math.SE, without much success so I'm trying my luck here too.
Let $M$ be an $n$-dimensional manifold, with $n$ even and consider the heat kernel of the Laplacian on $...
- 191
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 $g_t$ ...
- 1,407
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 ...
- 266
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
7
votes
1
answer
930
views
Why is the length spectrum called a spectrum?
Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$.
Question: is $\mathcal{L}(X)$ a ...
- 1,095
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^...
7
votes
1
answer
1k
views
On eigenfunctions of the Laplace Beltrami operator [closed]
How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
- 91
7
votes
2
answers
582
views
Exponential convergence of Ricci flow
I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...
- 6,001
7
votes
1
answer
281
views
Harmonic functions on $(M,g)$ closed, induce an embedding in Euclidean space
In Hajime Urakawa's monograph The Spectral Geometry of the Laplacian on page 41, we make an assumption that I can't quite justify on my own. The following is our setup:
Let $(M^n,g)$ be a closed ...
- 1,247
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 ...
7
votes
1
answer
372
views
Spectral gaps for spin manifold Laplace spectrum
For a (compact) spin manifold, we know that the eigenvalues $\lambda_n$ of the Dirac operator are countable, with finite multiplicity, and satisfy
$$
|\lambda_n| \to \infty, ~~~ \text{ as } n \to \...
7
votes
1
answer
327
views
First eigenvalue of the spherical cap
Let $S$ be the round $n$-sphere of radius $R$ in Euclidean space, and let $r$ be the intrinsic distance from the north pole. Further, let $U(r)$ be the spherical cap of intrinsic radius r. (So $U(0)$ ...
7
votes
0
answers
89
views
Eigenvalue lower bounds for manifold with positive Ricci curvature
For closed $n$-manifold with Ricci curvature $\ge (n-1)$, it is known that the first eigenvalue $\lambda_1\ge n$ with equality holds if and only if $M$ is isometric to the Euclidean sphere $S^n$. My ...
- 481
7
votes
0
answers
189
views
Geometrically-explicit upper bound for on-diagonal heat kernel
Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form
$$K(t;z,z) \leq \frac{C_M}{f_z(t)...
6
votes
1
answer
388
views
A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues
Let $(M^n,g)$ be a closed $n$ dimensional Riemannian manifold with $\mathrm{Ric}_g\ge -K$, $(K\ge 0)$. Weyl's law(along with Karamata Tauberian Theorem) asserts that the eigenvalue $\lambda_i$ of $-\...
- 193
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
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 ...
- 7,237
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 ...
- 96.4k
6
votes
1
answer
1k
views
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^...
- 675
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 (...
- 3,059
5
votes
1
answer
224
views
Spectral theory of infinite volume hyperbolic manifolds
I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
- 1,407
5
votes
2
answers
458
views
Question about Neumann eigenvalues on manifolds
Question:
Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the ...
- 537
5
votes
2
answers
322
views
Multiplicity of Laplace eigenvalues and symmetry
Let $S$ be a smooth closed connected hyperbolic surface. On $S$ we have the Laplace operator $\Delta$, whose eigenvalues form a discrete sequence
\begin{equation}
0=\lambda_0<\lambda_1\leq \...
- 218
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 ...
- 3,059
5
votes
1
answer
147
views
Stable region of minimal hypersurfaces with finite Morse index
In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1):
Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...
- 823