Paper · Mar 25, 2021

Some New Results in Geometric Analysis

Matei P. Coiculescu

math.DGWider flowsarXiv:2103.15594v4
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Abstract

This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preserving flow on space curves. Lastly, we present an abridgment of our work on a family of three-dimensional Lie groups, which, when equipped with canonical left-invariant metrics, interpolate between Sol and hyperbolic space.

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Submitted
Mar 25, 2021
Last revised
Jul 21, 2021 (v4)
Subjects
math.DG
DOI
10.26300/6fjz-ax05

Comments: Undergraduate thesis, Brown University, 72 pages. Further clarifications made in Chapter 2. arXiv admin note: text overlap with arXiv:2005.06430

Curve shortening3-manifoldsHomogeneous spaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)7Cortissoz 2017 The curve shortening flow with density of a spherical curve in codimension two (Francisco Viñado-Lereu, 2019)8Viñado-Lereu 2019 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)4Woolgar 2021 Soliton Solutions to the Curve Shortening Flow on the 2-dimensional hyperbolic plane (Fabio Nunes da Silva and Keti Tenenblat, 2021)2da Silva 2021 The Affine Shape of a Figure-Eight under the Curve Shortening Flow (Matei P. Coiculescu and Richard Evan Schwartz, 2021)1Coiculescu 2021 Which shapes can appear in a Curve Shortening Flow Singularity? (Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al., 2024)6Angenent 2024 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)3Sun 2024 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)5Krznarić 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Affine Shape of a Figure-Eight under the Curve Shortening Flow

    Matei P. Coiculescu and Richard Evan SchwartzWider flows

    Topics Curve shortening · Same author Matei P. Coiculescu · Shared terms “figure eight”, “eight curves”

  2. Soliton Solutions to the Curve Shortening Flow on the 2-dimensional hyperbolic plane

    Fabio Nunes da Silva and Keti TenenblatWider flows

    Topics Curve shortening, 3-manifolds · Shared terms “hyperbolic space”

  3. Curve Shortening Flow of Space Curves with Convex Projections

    Qi SunWider flows

    Topics Curve shortening · Shared terms “space curves”

  4. Self-similar curve shortening flow in hyperbolic 2-space

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

  5. Geometric aspects of the curve shortening flow in the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening · Shared terms “Lastly”

  6. Which shapes can appear in a Curve Shortening Flow Singularity?

    Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al.Wider flows

    Topics Curve shortening · Shared terms “eight curves”, “figure eight”

  7. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  8. The curve shortening flow with density of a spherical curve in codimension two

    Francisco Viñado-LereuWider flows

    Topics Curve shortening, 3-manifolds

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.

  3. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.