Questions tagged [flows]
The flows tag has no usage guidance.
95
questions
28
votes
2
answers
4k
views
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 ...
- 4,958
10
votes
1
answer
467
views
Optimal exponent in the Lojasiewicz-Simon gradient inequality
Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, $\gamma\...
user81607
9
votes
1
answer
564
views
$C^{k,\alpha}$ diffeomorphisms and vector fields
This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
- 14.2k
8
votes
1
answer
270
views
Extensions of Fried's Theorem: Surface bundles over Circles and Flows on 3 Manifolds
The Thurston Polytope provides a way to organize information about the embedded surfaces living in a 3-manifold.
His amazing theorem, often called the "Fibered Faces" Theorem, says that if you have ...
- 561
7
votes
2
answers
389
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
7
votes
1
answer
878
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
7
votes
1
answer
241
views
Are there such things as non-trivial entire semigroups?
I'll state the theorem I am posing up front, and then explain why I think this theorem appears to be true. I am asking if anyone can prove it, or knows references to where it is proved. Please, ...
user78249
7
votes
2
answers
309
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(...
- 18.5k
7
votes
1
answer
424
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
7
votes
1
answer
172
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
7
votes
0
answers
155
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(\...
- 8,046
6
votes
1
answer
630
views
Ricci flow + Nash embedding
I have a basic question about how geometric flows such as the Ricci flow interact with the Nash embedding theorem.
Say you have a 1-parameter family of Riemann metrics on a compact manifold $N$. If ...
- 43.1k
6
votes
1
answer
357
views
Evolving curves by Alexander Polden
I am writing a piece on curve shortening flow and lots of my sources have referenced Alexander Polden's honours thesis 'Evolving Curves' from the Australian National University. I have tried to find ...
- 235
6
votes
0
answers
221
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.
...
- 5,038
6
votes
0
answers
200
views
The geometric shape of domains of flows
Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow $\...
- 7,709
5
votes
2
answers
605
views
Flow of a nowhere vanishing complete vector field
Let X be a nowhere vanishing complete vector field on a manifold M, $\gamma: \mathbb{R} \to M$ be its flow with $\gamma(0)=p \in M$ and suppose it is not periodic. If $\gamma(\mathbb{R})$ is closed, ...
- 103
5
votes
1
answer
274
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 ...
- 235
5
votes
1
answer
428
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 $\...
- 896
5
votes
1
answer
384
views
Flow of an integer
I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS: A238729. I can't find any reference to it either. Has anyone seen it?
Here is the description:
...
5
votes
0
answers
117
views
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 ...
- 2,085
5
votes
0
answers
379
views
Gage-Grayson-Hamilton curve-shortening flow, at an angle
The Gage-Grayson-Hamilton curve-shortening flows along the normal to the curve:
&...
- 149k
5
votes
0
answers
209
views
Infinitesimal Generator of Billiard Flow
The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
- 9,591
5
votes
0
answers
278
views
Limits of $p/\ln p - q /\ln q$, $p, q$ prime
Is there any $\alpha>0$ for which there are known to exist two sequences of primes, $(p_i), (q_i)$ such that
$$\alpha = \lim_{i\to\infty} \left(p_i/\ln p_i - q_i /\ln q_i\right)\ ?$$
The ...
- 17.5k
4
votes
1
answer
331
views
Flows in word-hyperbolic groups
I was wondering if there is a good notion of flows in word-hyperbolic groups (maybe I should say flows in the Cayley graphs of word-hyperbolic groups).
More precisely, I wonder if there is an ...
- 305
4
votes
1
answer
160
views
One-sided version of the curve-shortening flow
The curve-shortening flow is
$$
\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 ...
- 42.5k
4
votes
1
answer
333
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
4
votes
0
answers
85
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
4
votes
0
answers
126
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
4
votes
0
answers
214
views
Flows associated with Killing fields
Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
- 1,368
3
votes
1
answer
273
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],\...
- 161
3
votes
1
answer
203
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
3
votes
2
answers
385
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 ...
- 4,958
3
votes
1
answer
416
views
Identity Theorem for Real-Analytic Hypersurfaces
There's an interesting statement it seems I can prove, but I can't find any references for it, which makes me suspicious of it. So, could someone verify that the statement is correct/incorrect or ...
- 343
3
votes
1
answer
617
views
Perturbed vs. unperturbed Hamiltonian system
Let's take a time-periodic Hamiltonian $H(t,x,y)$ on $\mathbb{R}^2$ and
apply an arbitrarily small time-independent perturbation to $H$ via
$$
\tilde H (t,x,y) = H(t,x,y) + \epsilon V(x,y),
$$
where $...
- 81
3
votes
0
answers
71
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 ...
- 1,890
3
votes
0
answers
58
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
3
votes
0
answers
211
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
3
votes
0
answers
129
views
Regularity of martingales with respect to spatial parameters
In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
- 161
3
votes
0
answers
122
views
PDE background for curve shortening flow
Soon I will be learning curve shortening flow and I am under the impression that a knowledge of PDEs is essential. Specifically, what from PDEs is required?
For example, I plan on brushing up using ...
- 143
3
votes
0
answers
135
views
Two semi stable limit cycles with disjoint interior
What is a precise example of a quadratic vector field on the plane with at least one semi stable limit cycles?
Furthermore, is there a quadratic polynomial vector field on the plane with two ...
- 312
3
votes
0
answers
262
views
Principal eigenvalue of Laplacian under volume preserving mean curvature flow
Consider a compact uniformly convex n-dimensional hypersurface $M_0$ without boundary , which is smoothly imbedded in $\mathbb R^{n+1}$ , and suppose that $M_0$ is represented locally by some ...
- 165
2
votes
3
answers
438
views
1-parameter group of a vector field
Let $(M,g)$ be a Riemannian manifold and $\nabla$ be the Levi-Civita connection of $g$ and let $X,Y$ be vector fields on $M$. If $\lbrace \phi _t \rbrace $ is the 1-parameter group of $X$ then what is ...
user82800
2
votes
1
answer
281
views
Properties of harmonic maps into spheres
Let $(M, g)$ be a complete, noncompact Riemannian $n$-dimensional manifold and let
$\phi \colon M \to \mathbb S^n$ be an harmonic map, where $\mathbb S^n$ is the euclidean $n$-dimensional sphere.
...
- 823
2
votes
1
answer
387
views
Network flows with shared capacities
Suppose we have a flow network, with capacity constraints on weighted sums of arc flows, such as:
$$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$
where $f(1, 2)$ denotes the flow through arc $(1, 2)$....
- 23
2
votes
0
answers
57
views
On improving the regularity of solutions to nonlinear parabolic pde
There's a common reasoning i have seen in many papers/books that work with parabolic pde's (most specifically in the context of geometric flows) but I can't fully grasp it. This reasoning is that to ...
2
votes
0
answers
135
views
Choosing the derivative of a flow
I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
- 121
2
votes
0
answers
211
views
Show that the manifold interior is invariant under this flow
Let $\tau>0$, $d\in\mathbb N$, $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be continuous in the first argument with $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\right\|\le c\left\|x-y\right\|\tag1\;\...
- 161
2
votes
0
answers
84
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 $\...
- 615
2
votes
1
answer
447
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 ...
- 161
2
votes
0
answers
40
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\...
- 161