# Flows in Flatland: A Romance of Few Dimensions

@article{Katz2015FlowsIF, title={Flows in Flatland: A Romance of Few Dimensions}, author={Gabriel Katz}, journal={Arnold Mathematical Journal}, year={2015}, volume={3}, pages={281-317} }

This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on n-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central…

## 9 Citations

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### Causal Holography of Traversing Flows

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We study smooth traversing vector fields v on compact manifolds X with boundary. A traversing v admits a Lyapunov function $$f: X \rightarrow \mathbb R$$ f : X → R such that $$df(v) > 0$$ d f ( v ) >…

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## 33 References

### A Morse complex on manifolds with boundary

- Mathematics
- 2010

Given a compact smooth manifold M with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology…

### Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary

- Mathematics
- 2014

As has been observed by Morse [1], any generic vector field v on a compact smooth manifold X with boundary gives rise to a stratification of the boundary by compact submanifolds , where . Our main…

### Using simplicial volume to count multi-tangent trajectories of traversing vector fields

- Mathematics, Computer Science
- 2015

A lower bound on the number of trajectories may be tangent to the boundary with reduced multiplicity n is proved by adapting methods of Gromov by adapting his “amenable reduction lemma” to vector fields on hyperbolic manifolds.

### CONVEXITY OF MORSE STRATIFICATIONS AND GRADIENT SPINES OF 3-MANIFOLDS

- Mathematics
- 2006

We notice that a generic nonsingular gradient field v = ∇f on a compact 3-fold X with boundary canonically generates a simple spine K(f, v) of X. We study the transformations of K(f, v) that are…

### Seifert fibered spaces in 3-manifolds

- Mathematics
- 1979

Publisher Summary This chapter describes Seifert Fibered Spaces in 3-Manifolds. There exist finitely many disjoint, non-contractible, pairwise non-parallel, embedded 2-spheres in M, whose homotopy…

### The Morse Complex for a Morse Function on a Manifold with Corners

- Mathematics
- 2004

A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse…

### Complements of Discriminants of Smooth Maps: Topology and Applications

- Mathematics
- 1992

Introduction Cohomology of braid groups and configuration spaces Applications: Complexity of algorithms, superpositions of algebraic functions and interpolation theory Topology of spaces of real…

### Seifert fibered spaces in irreducible, sufficiently-large 3-manifolds

- Mathematics
- 1976

In [2], F. Waldhausen announced theorems about singular annuli and tori in a bounded, orientable, irreducible 3-manifold M, analogous to the Dehn LemmaLoop Theorem for singular disks and the Sphere…

### The Stratified Spaces of Real Polynomials & Trajectory Spaces of Traversing Flows

- Mathematics
- 2014

This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from…

### Complexity of Shadows & Traversing Flows in Terms of the Simplicial Volume

- Mathematics
- 2015

We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate…