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Questions tagged [flows]

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2
votes
1answer
153 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-...
5
votes
0answers
64 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 ...
2
votes
0answers
71 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
vote
0answers
45 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 ...
2
votes
0answers
137 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
85 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
84 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
218 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
96 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,...
0
votes
0answers
67 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
votes
0answers
91 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
127 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
45 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
79 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
votes
0answers
32 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
92 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 $X(0,\cdot)=id$ and satisfying the semigroup property $X(t,X(...
1
vote
0answers
138 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
178 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
vote
0answers
22 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 ...
20
votes
3answers
1k views

Hermann Weyl's work on combinatorial topology and Kirchhoff's current law in Spanish

Hermann Weyl was one of the pioneers in the use of early algebraic/combinatorial topological methods in the problem of electrical currents on graphs and combinatorial complexes. The ...
8
votes
1answer
159 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
242 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
174 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 ...
5
votes
1answer
219 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
269 views

Derivation of the volume preserving mean curvature flow

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Picture above is from Huisken, Gerhard, The volume preserving ...
3
votes
0answers
158 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
vote
0answers
244 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 ...
0
votes
0answers
60 views

Maximum flow value problem

Let $G=(V,E)$ be a digraph. We call it "cyclable" if there exists $k$ $\geq$ 1 circuits in $G$,$C_1$,...$C_k$, of which sets of vertices, V($C_1$),...,V($C_k$) , makes a partition of $V(G)$. I must ...
2
votes
0answers
121 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}...
4
votes
0answers
220 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
102 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,...
7
votes
1answer
193 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
vote
0answers
75 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
170 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
55 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
votes
0answers
154 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
299 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 ...
11
votes
1answer
320 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\...
2
votes
0answers
63 views

When do positively invariant subset contain a given set?

Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't ...
0
votes
0answers
237 views

Existence of Solution steady navier stokes with do nothing outflow condition

We consider the stationary navier stokes equation with mixed boundary conditions $$ \begin{align} -\nu\Delta u +(u\cdot\nabla) u + \nabla p &= 0\ \textrm{in}\ \Omega \\ div\ u&=0\ \textrm{in}\ ...
1
vote
0answers
250 views

Are back edges mandatory in Ford Fulkerson algorithm?

Consider the algorithm of Ford Fulkerson where, for each iteration, you add flow along a path equal to the maximum residual capacity along this path. Does it exist, for every network, a choice of ...
2
votes
0answers
76 views

How analyze the following fully nonlinear equation

Now I want to consider the following pde $u_t(x,t)=\sigma(x,t)(1+|D_xu(x)|^2)^{1/2}$, with initial condition $u(x,0)=g(x)$ which is analytic, and on domain $D\times \mathbf{R}^{+}$, $D\subset \mathbf{...
2
votes
1answer
189 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 $(1, 2)$....
4
votes
0answers
168 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 $S_t u(\xi)$, ...
0
votes
0answers
476 views

Parallelism degree of a DAG

Let me first give a motivation. Suppose a connected DAG G with one source X and one sink Y. The goal is to find some "bottleneck" node between X and Y, i.e. node through which every path from X to Y ...
6
votes
0answers
181 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 $\...
5
votes
1answer
373 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: ...
4
votes
1answer
275 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 ...
5
votes
0answers
259 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 ...