Questions tagged [flows]

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3
votes
0answers
52 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 ...
4
votes
0answers
33 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 $$ ...
2
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0answers
57 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} ...
2
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0answers
69 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. ...
5
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0answers
185 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. ...
0
votes
0answers
187 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 ...
20
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2answers
1k 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 ...
3
votes
2answers
163 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 ...
2
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0answers
60 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, ...
2
votes
1answer
129 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 $\...
4
votes
0answers
61 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 ...
0
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0answers
40 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:\...
1
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0answers
81 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 ...
0
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1answer
71 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 $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ ...
2
votes
1answer
293 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-...
6
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0answers
148 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 ...
4
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0answers
103 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 ...
1
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0answers
49 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 ...
3
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0answers
171 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 theorem $2.3$. ...
0
votes
2answers
88 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 ...
2
votes
0answers
168 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 ...
7
votes
1answer
425 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 reading the ...
1
vote
1answer
127 views

Slow and fast forming singularities of the mean curvature flow

Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form. We have a type I singularity if $$ \max_{p \in M} |A(p,...
1
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0answers
105 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
3
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0answers
102 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 ...
1
vote
1answer
140 views

A change of parameters used on Curve Shortening Flow

I'm trying understand the article "Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem" by Ben Andrews and Paul Bryan and they stated on the final ...
1
vote
0answers
50 views

On the defect of a flow network

This problem in graph theory was actually motivated by some problems in Theory of Fractals. To formulate the problem I need to recall some definitions related to flow network. A flow network is a ...
3
votes
0answers
100 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 ...
2
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0answers
34 views

2,3-Flows on 4-edge-connected graphs

I just stumbled across the following question concerning special Nowhere-Zero-Flows on 4-edge-connected graphs: Does every 4-edge-connected graph admit a flow (on some and thus all orientations) ...
1
vote
1answer
111 views

Separation property for non-injective flows

Let us consider a non-injective flow $X$ on $\mathbb{R}^d$, i.e. a continuous map $X:\mathbb{R}_+\times \mathbb{R}^d \to \mathbb{R}^d$ with $X(0,\cdot)=\mathrm{id}$ and satisfying the semigroup ...
1
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0answers
167 views

An application of Implicit Function Theorem for Curve Shortening Flow on plane

I'm trying understand the article "Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem" by Ben Andrews and Paul Bryan and they stated that ...
2
votes
1answer
196 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. ...
1
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0answers
24 views

Conservative flows on infinite volume manifolds

I am interested in the following type of problem: $M$ is a open manifold and $\mu$ an infinite measure on $M$ that is absolutely continuous with respect to the Lebesgue measure. I consider a complete ...
8
votes
1answer
190 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 ...
5
votes
2answers
369 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, ...
3
votes
1answer
274 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 ...
6
votes
1answer
249 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 ...
1
vote
2answers
350 views

Derivation of the volume preserving mean curvature flow

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Picture above is from Huisken, Gerhard, The volume preserving ...
7
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1answer
211 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, ...
3
votes
0answers
185 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 ...
1
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0answers
249 views

Fill up a circle with spaghetti

Consider the unit disk $C \subseteq \mathbb R^2$, and imagine the following process: Repeat $N$ times: Take a strand of spaghetti (the length is fixed!) and randomly place it in the disk so that it ...
2
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0answers
127 views

Geometric flow of the total tension functional

I apologize if this question is silly or confusing, I am completely new to this subject. Let $(M,g)$ be a Riemannian manifold. Denote by $\nabla$ the Levi-Civita connection of $(M,g)$. Now, let $S^{n}...
5
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0answers
271 views

Gage-Grayson-Hamilton curve-shortening flow, at an angle

The Gage-Grayson-Hamilton curve-shortening flows along the normal to the curve:                     &...
1
vote
1answer
106 views

Solvability of $u_t=\nabla_{u_\theta} u_\theta -\frac{1}{2}\nabla R +\frac{1}{2\tau}u_\theta + 2 Ric(u_\theta,\cdot)$

I ask a question in math.stackexchange, but nobody answer it. So, I ask here. In Cao and Zhu's paper about Ricci flow, the $\mathcal{L}$-geodesic is defined as picture below.In fact, $\mathrm{Ric}(X,...
9
votes
1answer
284 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 $C^k$...
1
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0answers
87 views

Laplace eigenvalue and floer theory

I'm trying to study the spectrum operator via Floer Theory. The idea is to consider the Rayleigh quotient as an operator on the Hilbert space $W^{1,2}$ and to study it's negative gradient flow. Any ...
2
votes
0answers
187 views

Intuitive understanding of the mean curvature flow [closed]

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = (...
0
votes
0answers
67 views

Obtaining Hessian of the embedding from an induced metric

Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...
4
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0answers
188 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 ...
2
votes
3answers
348 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 ...