Questions tagged [flows]
The flows tag has no usage guidance.
92
questions
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57
views
Cocycle-conjugacy classes of flows on the C*-algebra of compact operators
A flow on a C*-algebra $A$ is a group homomorphism $\sigma $ from ${\mathbb R}$ into the group of *-automorphisms of $A$
such that the map
$$
t\in {\mathbb R}\mapsto \sigma _t(a)\in A
$$
is norm-...
- 2,183
1
vote
1
answer
96
views
Singularities of mean-convex MCF in the sphere?
Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ ...
- 4,668
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votes
0
answers
67
views
Trapped vs. nonwandering points
For a continuous flow on a topological space the concept of a nonwandering point (in forward/backward time) is well-known. Another useful concept (for example from the point of view of scattering ...
- 1,880
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votes
0
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68
views
Methods for calculating (one-parameter subgroup) actions
For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form
\begin{equation}
\mathrm{e}^{t L(z)} f(z)
\end{equation}
...
- 639
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0
answers
29
views
Singular asymptotic limits of mean-convex MCF
Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
- 4,668
4
votes
1
answer
151
views
One-sided version of the curve-shortening flow
The curve-shortening flow is
$$
\frac{\partial C}{\partial t} = \kappa n
$$
where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed ...
- 41.7k
0
votes
1
answer
122
views
Distance function to mean curvature flow
In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the ...
- 4,668
3
votes
1
answer
180
views
Vector fields $X$ and $Y$ commute on a closed set $K$. Do there exist commuting $\tilde X,\tilde Y$ with $\tilde X=X$ and $\tilde Y=Y$ on $K$?
I have a nice research idea whose proof hinges on the following question
Suppose $X_p$ and $Y_p$ are vector fields in $\mathbb{R}^3$ with $[X,Y]_p=0$ for all $p$ in some closed set $K\subset\mathbb{R}...
- 133
5
votes
0
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114
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Metric under Ricci flow on a 2-sphere can be realized by embedding
I am sorry if this is a silly question, but I am new to Ricci flows.
Let $\Sigma \subset \mathbb{R}^3$ be a smoothly embedded sphere, and denote its metric (induced by $\mathbb{R}^3$) by $g$. Suppose ...
- 2,023
2
votes
0
answers
124
views
Choosing the derivative of a flow
I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
- 121
7
votes
2
answers
379
views
Kneser theorem about the Klein bottle
I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
- 123
0
votes
0
answers
48
views
Existence of measure-preserving Lagrange flow for inhomogeneous transport equation
I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
- 173
1
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0
answers
66
views
Hyperplanes which equalize the Radon transforms of two distributions
Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
- 6,466
0
votes
0
answers
67
views
$\mathbb{R}^n$-flow, cross-section and Whitney theorem
For a $\mathbb{R}$-flow (X, $\Phi_{\mathbb{R}}$), the (local) cross-section is well defined (recall that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]}(x)=\{x\...
- 605
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68
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Is there a theory for forcing "sludge" through a network, analogous to electric current flows?
I'm familiar with the correspondence between reversible Markov chains, random walks, and electric current flows, as described in Probability on Trees and Networks by Lyons and Peres.
Is there an ...
- 33
7
votes
2
answers
304
views
Planar flow with bounded orbits and a single equilibrium point
Is there a $C^1$ flow $\varphi_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x,
$$\lim_{t\rightarrow +\infty}\varphi_t(x) = 0,$$
$$\lim_{t\rightarrow -\infty}\varphi_t(...
- 18.3k
1
vote
1
answer
109
views
Is the union of two proper flows proper?
Let $\varphi _t$ be a flow, aka. a one parameter group of homeomorphisms of the open subset $\Omega \subseteq {\mathbb R}^n$, which we assume to be continuous in the usual sense
that
$$
(t, x)\in {\...
- 2,183
7
votes
1
answer
168
views
Stability Question for Isotopies Between Compact Sets
Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$.
Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
- 409
1
vote
0
answers
93
views
Existence theory for geometric flow of space curves
Is there any existence theory applicable to general geometric flows of space curves in the following form?
$$
\partial_t \gamma = v_t t + v_n n + v_b b
$$
Here $\gamma$ is the evolving curve, $t$, $n$ ...
- 11
2
votes
0
answers
207
views
Show that the manifold interior is invariant under this flow
Let $\tau>0$, $d\in\mathbb N$, $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be continuous in the first argument with $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\right\|\le c\left\|x-y\right\|\tag1\;\...
- 91
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0
answers
77
views
Entropy of flow and time-1 map
Let $\Phi=(\phi_t)_{t\in \mathbb{R}}$ be a continuous flow on a compact metric space $X$. Let $\mu$ be a $\phi_1$-invariant measure. Then it is not hard to verify tht $\int_{0}^{1} \phi_t\mu dt$ is $\...
- 605
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votes
0
answers
125
views
Divergence as infinitesimal volume change on a Finsler manifold
Let $M$ be a smooth manifold and $Z$ a smooth vector field on it.
It generates a family of diffeomorphisms $\phi_t:M\to M$ by demanding that $\phi_0=\operatorname{id}$ and $\partial_t\phi_t(x)=Z(\...
- 7,763
3
votes
1
answer
269
views
Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold
Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
- 91
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vote
0
answers
74
views
Show that the support of the shape gradient $\nabla\mathcal F(\Omega)$ is contained in $\overline\Omega$
Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta)...
- 91
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votes
1
answer
414
views
Flow induced by differentiable velocity field is differentiable
Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...
- 91
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votes
0
answers
55
views
What is the importance of singularities of type II in the Mean Curvature Flow?
I am reading the Mean Curvature Flow and Isoperimetric Inequalities by Manuel Ritoré, Carlo Sinestrari, Vicente Miquel and Joan Porti and I am curious to know what is the importance in understand the ...
- 435
1
vote
1
answer
93
views
Explicit transitive flow on disc
$D_n\triangleq \left\{x \in \mathbb{R}^n:\, \|x\|\leq 1\right\}$ with its subspace topology. By a transitive flow on $D_n$ I mean a continuous function
$$
\phi: [0,1]\times D_n\rightarrow D_n,
$$
...
- 4,971
2
votes
0
answers
38
views
If a stochastic flow is Fréchet differentiable in the spatial parameter, does the induced transition semigroup preserve differentiability?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $X:\Omega\times[0,\infty)\times E\to E$ be $(\mathcal A\otimes\mathcal B([0,\infty))\otimes\...
- 91
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votes
1
answer
260
views
Smoothening pseudo-Anosov flows
A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...
- 225
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0
answers
35
views
Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
- 4,971
1
vote
0
answers
88
views
Convergence rate of Toda/Morse flow
Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow
\begin{align}
\frac{dA}{dt} &= \left [ C\circ A , A \right ] \\
A(0) &= A_0 \ .
\end{align}
...
- 486
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votes
0
answers
176
views
Definition of Lie derivatives of sections of natural vector bundles - product preservation needed?
Section 6.15 of Natural Operations by Kolár, Michor, and Slovak defines the Lie derivative of a section of a natural vector bundle along a vector field. Set-theoretically, the definition is clear. ...
- 10.2k
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votes
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answers
220
views
Poking into a Lie group with your finger
I consider this as a differential geometry problem. I have asked some
of my classmates who are more interested in that, and also looked into
some literature, but none of what I've found seems to help.
...
- 4,648
0
votes
0
answers
463
views
directional derivative along geodesic flow of vector field
A rather elementary question for the differential geometers. Let $M$ be a Riemannian manifold, let $X\colon M \to TM$ be a vector field, and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" of the ...
- 11
27
votes
2
answers
3k
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Formal mathematical definition of renormalization group flow
I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what ...
- 4,806
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votes
2
answers
350
views
Proof of Isoperimetric Inequality using Curve Shortening Flow
I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was ...
- 4,806
2
votes
0
answers
94
views
Solution of equation on vector field
I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like:
$$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$
with a (very well behaved) real, positive, ...
5
votes
1
answer
372
views
Modulus of continuity of flow for non-Lipschitz vector fields satisfies Osgood condition
An Osgood modulus of continuity is an increasing function $\omega:(0,1]\to(0,1]$ such that $\int_0^1\frac{dt}{\omega(t)}=\infty$.
We say a vector field $X$ satisfies Osgood condition with modulus $\...
- 731
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votes
0
answers
84
views
Geometric meaning of the extrinsic curvature neck
I'm reading by myself "Mean curvature flow with surgeries of two–convex hypersurfaces" (it's free available, it's just click on "Download PDF" on the upper right corner side) by Gerhard Huisken and ...
- 435
0
votes
0
answers
47
views
Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
- 4,971
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94
views
Ricci flow preserves locally symmetry along the flow
Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...
- 171
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votes
1
answer
88
views
Convergence of Stochastic Flow but not Flow
Suppose that $\phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a flow. Is it possible to extend $\phi$ to a family of stochastic flows $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ ...
- 4,971
4
votes
1
answer
325
views
Understanding the Hamilton's definition of $\ast$-operation
I'm studying by myself Mean Curvature Flow and I'm trying understand the definition of $\ast$-operation given by Richard Hamilton in the beginning of the section $13$ (page $40$) of his paper "Three-...
- 435
7
votes
1
answer
377
views
Gradient flows on Hilbert manifolds
I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded.
To be more precise, a ...
- 171
4
votes
0
answers
121
views
Approximation argument in geometric flows
I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following ...
- 435
1
vote
0
answers
57
views
Matching Stochastic Flows
Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the ...
- 19
3
votes
0
answers
202
views
Interior gradient estimate of mean curvature equation
I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken. I'm stuck in the ...
- 435
0
votes
2
answers
94
views
Does Max Flow produce uniform results? [closed]
I am interested in using Max Flow algorithm. I want to simulate transfer of quantity. Anyway, I am unsure of some thing.
Does Max Flow algorithm produce uniformly distributed max flow?
I have ...
- 11
2
votes
0
answers
240
views
Research trends on mean curvature flow
It's been a few months since I've been curious about mean curvature flow and now I'm reading Robert Haslhofer's lecture notes. I like the subject and would like to do research on it, but I know ...
- 139
7
votes
1
answer
779
views
Weak parabolic maximum principle on Riemannian manifolds
I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken, specifically, I'm ...
- 435