Linked Questions
15 questions linked to/from Elliptic operators corresponds to non vanishing vector fields
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Proposals for polymath projects
Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
18
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3
answers
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Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators
EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
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28
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2
answers
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Codimension of the range of certain linear operators
Added:8/15/2024 What about holomorphic or real analytic version? Please see the comment discussions on this post.
Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We ...
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6
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2
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On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
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3
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Limit cycles as closed geodesics (in negatively or positively curved space)
Updated 1/25/2023 I just added a related post below:
Jacobi fields, Conjugate points and limit cycle theory
EDIT: Here is a related post which concern quadratic vector fields rather than Van ...
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4
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2
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How to compute the index of such operator?
Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (Edit: which are sub-bundles of $\...
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6
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1
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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 ...
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1
answer
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The "Rolle theorem" for sections of a vector bundle
1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
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A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
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2
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0
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Lifting a quadratic system to a non-vanishing vector field on $S^{3}$ or $T^{1} S^{2}$
Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non-vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
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Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
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2
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1
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Keeping track of limit cycles via certain second order differential operator
Inspired by the two posts which are linked bellow we ask the following question:
Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
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3
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0
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When a PDE add a Laplacian term
I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
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Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE
A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...
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A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field
Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...
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