Paper · May 1, 2018

Superforms, supercurrents, minimal manifolds and Riemannian geometry

Bo Berndtsson

math.CVWider flowsarXiv:1805.00379v2
View PDF View HTML arXiv page

As on arXiv

Abstract

Supercurrents, as introduced by Lagerberg, were mainly motivated as a way to study tropical varieties. Here we will associate a supercurrent to any smooth submanifold of . Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on and reflect the induced Riemannian structure on the submanifold. In this way we can use techniques from complex analysis to study real submanifolds. We illustrate the idea by giving area estimates of minimal manifolds and a monotinicity property of the mean curvature flow. We also illustrate the idea by a relatively short proof of Weyl's tube formula.

View PDF View HTML arXiv page
Submitted
May 1, 2018
Last revised
Jun 13, 2019 (v2)
Subjects
math.CV, math.DG

Comments: An incorrect proof in section 8 (on mean curvature flow) has been replaced; the main results are unchanged. Some additional explanations have also been added. 25 pages

Mean curvature flow

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 Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (Naoyuki Koike, 2016)4Koike 2016 Regularity of minimal submanifolds and mean curvature flows with a common free boundary (Brian Krummel, 2016)5Krummel 2016 Consequences of strong stability of minimal submanifolds (Jason D. Lotay and Felix Schulze, 2018)8Lotay 2018 Rigidity of riemannian manifolds containing an equator (Laurent Mazet, 2020)7Mazet 2020 Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds (Farnaz Ghanbari and Samreena, 2023)6Ghanbari 2023 Riemannian Geometry of G₂-type Real Flag Manifolds (Brian Grajales, Gabriel Rondón and Julieth Saavedra, 2024)3Grajales 2024 A convergence result for Mean Curvature Flow of totally real submanifolds (Tristan C. Collins, Adam Jacob and Yu-Shen Lin, 2024)1Collins 2024 A Comprehensive Review of Solitonic Inequalities in Riemannian Geometry (Bang-Yen Chen, Majid Ali Choudhary, Mohammed Nisar et al., 2024)2Chen 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A convergence result for Mean Curvature Flow of totally real submanifolds

    Tristan C. Collins, Adam Jacob and Yu-Shen LinWider flows

    Topics Mean curvature flow · Shared terms “real submanifold”

  2. A Comprehensive Review of Solitonic Inequalities in Riemannian Geometry

    Bang-Yen Chen, Majid Ali Choudhary, Mohammed Nisar et al.

    Shared terms “Riemannian geometry”

  3. Riemannian Geometry of -type Real Flag Manifolds

    Brian Grajales, Gabriel Rondón and Julieth Saavedra

    Shared terms “Riemannian geometry”

  4. Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

    Naoyuki KoikeWider flows

    Topics Mean curvature flow

  5. Regularity of minimal submanifolds and mean curvature flows with a common free boundary

    Brian KrummelWider flows

    Topics Mean curvature flow

  6. Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

    Farnaz Ghanbari and SamreenaWider flows

    Topics Mean curvature flow

  7. Rigidity of riemannian manifolds containing an equator

    Laurent MazetWider flows

    Topics Mean curvature flow

  8. Consequences of strong stability of minimal submanifolds

    Jason D. Lotay and Felix SchulzeWider flows

    Topics Mean curvature flow · Shared terms “illustrate”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984)

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.