All Questions
Tagged with differential-operators dg.differential-geometry
171 questions
0
votes
0
answers
141
views
The tensor product of two Fredholm operators
What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
- 356
3
votes
0
answers
71
views
Derived b-calculus and logarithmic tangent sheaves
Melrose's b-calculus provides a powerful framework for analyzing elliptic operators on manifolds with boundary. In the context of log geometry, log smooth manifolds offer a natural generalization of ...
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
4
votes
0
answers
114
views
Mean-value type property for eigenfunctions of Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
The purpose of this post is to ask for the community's opinion on a conjecture about a certain mean-value property for functions on $\operatorname{SL}(2,\mathbb{R})$. This conjecture appears at the ...
- 1,769
5
votes
0
answers
203
views
Differential equation involving Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
Throughout this post, we let $G$ denote the Lie group $\mathrm{SL}(2,\mathbb{R})$. For $t,\theta \in \mathbb{R}$ we define the following elements of $G$:
\begin{align*} A(t) &= \begin{bmatrix}e^t &...
- 1,769
15
votes
2
answers
888
views
Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?
Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms
$$
\Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
- 35.5k
3
votes
0
answers
139
views
Fixed point formula of Atiyah and Singer applied to a Dirac operator on a spin manifold
Let $G$ be a compact Lie group acting by orientation-preserving isometries on a compact even-dimensional spin manifold $X$, and assume that the $G$-action preserves the spin structure of $X$, so that ...
- 335
2
votes
0
answers
71
views
Terminology: generalized Laplacian of arbitrary signature
Let $(M,g)$ be a Riemannian manifold and $E$ any real or complex vector bundle. A linear partial differential operator $D:\Gamma(E)\to\Gamma(E)$ is called generalized Laplace operator, if its ...
- 1,171
1
vote
2
answers
157
views
Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold
I've already asked this question in: https://math.stackexchange.com/questions/4899825/question-about-the-index-of-two-elliptic-operators-over-a-4-dimensional-riemanni, and I've been suggested to ask ...
- 335
2
votes
0
answers
96
views
Differential operators and iterations of tangent bundle
Is there a relationship between higher order differential operators and higher tangent bundle viewed as bundle on the base manifold?
- 35
2
votes
1
answer
90
views
Germs of left invariant differential operators on a group
Are there germs at the identity of linear differential operators on a group which are not germs at the identity of left invariant differential operators?
I feel like the answer is no but the statement ...
user525442
4
votes
1
answer
177
views
Analogue of vector for differential operators
A differential operators of order one is a vector field which is defined pointwise . Differential operators of order greater than one are not. The closest analogue to a vector is given by a germ of a ...
- 35
5
votes
0
answers
878
views
A fourth-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$$x^3 f_{xxxt}+ f =0$$
Does anyone know if this type of PDE already appeared in the literature? ...
- 141
16
votes
1
answer
784
views
The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
- 52.3k
3
votes
1
answer
197
views
Precise definition of a linear total differential operator
In the works of A. M. Vinogradov on calculus on the infinite jet space, differential equations and "diffieties", a central notion is that of a $\mathcal C$-differential operator. If $\pi:Y\...
- 2,792
10
votes
1
answer
849
views
Hodge decomposition in elliptic complexes
EDIT: In the book "Principles of Algebraic Geometry" by Griffiths and Harris the authors prove the Hodge decomposition for the Dolbeault operator $\bar\partial$ on differential forms on a ...
- 21.8k
2
votes
1
answer
223
views
Hodge decomposition for non-elliptic complexes
It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...
- 167
3
votes
1
answer
118
views
Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$)
Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator
between the ...
- 2,533
5
votes
1
answer
543
views
The principal symbol as an element in the K-theory
This line
The symbol may naturally be thought of as an element in the K-theory
of X
appears in the nLab page on principal symbols for differential operators. What does this mean? Are they talking ...
- 1,144
4
votes
1
answer
311
views
Conormal distributions and the wave front set
Let $X$ be a smooth closed manifold and $Y$ a regular submanifold. For all conormal distributions at $Y$ on $X$, their wave front set is contained in the conormal bundle of $Y$. Is the reciprocal true?...
0
votes
0
answers
246
views
A question about second fundamental form of Riemannian isometric embedding
I have got a question unsolved for some time. I do not know whether it is trivial or not:
**I omit a very important fact: The metric at point p is second-order flat, i.e. $d_p \phi(-,v) = 0$ and $d_p^...
- 21
1
vote
0
answers
98
views
The module generated by kernel of an elliptic differential operator
Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\...
- 356
0
votes
0
answers
76
views
Linear dependence of the derivatives of a vector valued function
Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function
$$
g:\mathbb{R}^5\rightarrow\mathbb{R}^5
$$
given by
$$
g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
- 8,998
4
votes
1
answer
125
views
Question about differential operators in a completely non-integrable distribution
Say I have two integrable codistributions
$$ U = \langle du^1, \ldots, du^m \rangle, \qquad Z = \langle dz^1, \ldots, dz^N \rangle $$
on a manifold $M$, with $N >> m$. Suppose that the ...
- 818
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
5
votes
0
answers
280
views
Was an index theorem for manifold with local boundary condition proven?
I would like to ask a question on the bibliography of the index theorems on manifold with boundary.
Before my bibliographical research my understanding of the field was that for manifold with boundary,...
- 51
2
votes
1
answer
256
views
A question about Dirac operators
Let $D$ be a Dirac operator on spinor bundle $S$ over even-dimensional non-compact spin manifold $X$,
$$
\left<s_1,s_2\right>_{L_2}
= \int_X \left<s_1,s_2\right> \quad \forall s_1,s_2\in\...
- 563
3
votes
1
answer
131
views
Positivity of an operator on a compact subset of a manifold
Let $E$ and $F$ be two vector bundles over manifold $X$. Let $P:\Gamma(E)\to \Gamma(F)$ be a self-adjoint differential operator over $X$. Define inner product on the spaces $\Gamma(E)$ of smooth ...
- 563
4
votes
1
answer
377
views
Differential inequalities under which a flat function must be identically zero
Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $.
Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
- 356
4
votes
1
answer
486
views
Ricci flow for manifold learning
I know that mean curvature and diffusion-type flows are common in manifold learning because of their smoothing effects. I haven't seen Ricci flow used as much. Given that Ricci and diffusion-type ...
- 153
4
votes
0
answers
146
views
Range of divergence operator on the space of traceless symmetric $(0,2)$ tensors; conformal vector fields on an arbitrary metric on $S^2$
Let $\gamma$ be a metric on $S^2$.
I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$:
$$div_{\gamma} A = \omega$$
where $\omega$ is a 1-form.
It is known that there ...
- 969
4
votes
1
answer
241
views
Degenerate second-order Lagrangians
Let $M$ be a smooth $m$ dimensional manifold, let $\pi:E\rightarrow M$ be a smooth fibred manifold over $M$. Let us write generic fibred coordinates as $(x^i,y^\sigma)$ with $x^i$ being the base ...
- 2,792
3
votes
0
answers
198
views
Dirichlet to Neumann operator and the Riesz transform
Consider the manifold $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $\gamma$ be the induced metric on $\partial M$....
- 969
6
votes
2
answers
448
views
About the index theorems
I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (...
- 789
2
votes
0
answers
62
views
Singularity of reproducing kernel for elliptic operator
Let $(M,g)$ be a smooth compact Riemannian manifold and dimension $2$, $\Gamma$ a smooth vector bundle over $M$, and suppose $L: W^{k,2}(\Gamma)\to W^{k-2,2}(\Gamma)$ is a second order strongly ...
- 163
6
votes
0
answers
124
views
An application of Leray-Schauder degree theory for Nirenberg problem on the 2-sphere
I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case.
They prove that if $K:S^2\...
- 521
2
votes
1
answer
243
views
Elliptic operators and Leibniz rule
Let $M$ be a manifold. Does it necessarily admit an elliptic operator on $C^{\infty}(M)$ which satisfy Leibniz rule?
Let $M$ be a symplectic manifold with the standard Poisson structure on $C^{\...
- 356
5
votes
1
answer
254
views
A question on moduli space of Hitchin's equations
I am reading Hitchin's Self-Duality paper. In section 5 (page 85), he is trying to prove that $Dim H^1=12(g-1)$. In doing so, he defines an operator $d^*_2+d_1$, where $d^*_2$ and $d_1$ are given by
$...
- 111
3
votes
1
answer
366
views
When is the exterior derivation $d$ a Lie algebra morphism?
In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We ...
- 356
1
vote
0
answers
235
views
Fredholmness of elliptic operator on Hölder spaces
Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
- 163
1
vote
1
answer
197
views
Pseudo-differential operators and differential operator
I am totally new to pseudo-differential operators and I’m wondering if a differential operator is a pseudo-differential operator.
So, I want to show , using the definition of the symbol given by ...
- 97
7
votes
0
answers
253
views
Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$
The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by
$$
\eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
- 91
1
vote
0
answers
35
views
Reference request: Transverse parabolic Schauder estimates
Is there a version of the parabolic Schauder estimates for transversely parabolic linear PDE's on a manifold with a Riemannian foliation for functions that are constant on the leaves of the foliation? ...
- 11
2
votes
0
answers
191
views
Differential operators and vector fields [closed]
Let $M$ be a smooth manifold. It is well known that there is a bijective correspondance between vector fields on $M$ and differential operators of order 1. My question is: if we take a differential ...
- 21
5
votes
0
answers
191
views
Index of the Fredholm operator
I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
6
votes
0
answers
375
views
Is there an analog of the Levi–Civita connection for schemes?
Is there an analog of the Levi–Civita connection for schemes?
There exists algebraic de Rham theory, $n$-forms on vector bundles (algebraically), and familiar constructions from differential geometry....
- 327
4
votes
0
answers
242
views
Equivalent definitions of differential operator
This puzzles me from some time and is in parts connected to the questions Symmetrized derivatives version and Symmetrized derivatives version II.
For me the linear DO between vector bundles $E$ and $...
- 437
1
vote
0
answers
60
views
Global solutions for an analytic family of differential operators with initial condition
This is related to this other question question of mine.
Let $M$ be a $3$-dimensional closed (compact without boundary) strongly pseudoconvex manifold and let $\Delta_t$ be a collection of ...
- 1,409
6
votes
1
answer
351
views
Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
3
votes
3
answers
637
views
Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation
The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry,
$...
- 10.5k