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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 ...
B K's user avatar
  • 1,942
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 ...
Ali Taghavi's user avatar
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 ...
Renato G. Bettiol's user avatar
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)...
Elchanan Solomon's user avatar
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 ...
Max Schattman's user avatar
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. ...
Jan Bohr's user avatar
  • 779
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 ...
freidtchy's user avatar
  • 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 ...
alvarezpaiva's user avatar
  • 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 ...
Sven Mortenson's user avatar
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 ...
SMS's user avatar
  • 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 ...
user82132's user avatar
  • 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 (...
Raziel's user avatar
  • 3,223
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 $-\...
noname's user avatar
  • 109
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 $g_t$ ...
SMS's user avatar
  • 1,407
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 ...
Gabe K's user avatar
  • 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 ...
Dominic Wynter's user avatar
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 ...
user60933's user avatar
  • 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)...
Giovanni De Gaetano's user avatar
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{...
Ilya Nikokoshev's user avatar
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 ...
Student's user avatar
  • 537
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 ...
Zhiqiang's user avatar
  • 891
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 ...
student's user avatar
  • 51
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 (...
GradStudent's user avatar
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 ...
anonymos's user avatar
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 ...
Mathematics enthusiast's user avatar
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 ...
Hammerhead's user avatar
  • 1,211
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>...
Ralph's user avatar
  • 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 ...
B K's user avatar
  • 1,942
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 $...
jesus's user avatar
  • 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 $$\...
asv's user avatar
  • 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 ...
Mathematics enthusiast's user avatar
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 ...
Radeha Longa's user avatar
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{...
Radeha Longa's user avatar
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 $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...
Hasti Musti's user avatar
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 $\...
Radeha Longa's user avatar