Paper · Nov 5, 2026

On the anisotropic mean curvature flow in the Gaussian space

Di Zhang

math.DGWider flowsarXiv:2611.07791
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we study the anisotropic mean curvature flow in the Gaussian space endowed with a Finsler metric obtained from a Minkowski metric by a conformal deformation. We first derive the conformal transformation formulas for the fundamental tensor, the Cartan tensor, and the Chern connection of the conformally related Finsler metrics. Based on these formulas, we express the anisotropic mean curvature in the Gaussian space in terms of the corresponding anisotropic mean curvature with respect to the original Minkowski metric, and derive the associated evolution equation. Finally, we establish the short-time existence and uniqueness of smooth solutions by reformulating the geometric evolution equation as a strictly parabolic quasilinear partial differential equation.

View PDF View HTML arXiv page
Submitted
Nov 5, 2026
Subjects
math.DG
Mean curvature flowAnisotropic & nonlocal flowsStability & uniquenessHarmonic maps & DeTurck trick

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 2021, 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 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)3Cesaroni 2021 Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion (Helmut Abels, Felicitas Bürger and Harald Garcke, 2022)8Abels 2022 Short time existence and smoothness of the nonlocal mean curvature flow of graphs (Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias Weth, 2022)1Attiogbe 2022 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)5Bidabad 2023 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)2Bulanyi 2025 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)6Kubin 2025 Asymptotic behaviour of the weak inverse anisotropic mean curvature flow (Chaoqun Gao, Yong Wei and Rong Zhou, 2025)4Gao 2025 Two dimensional anisotropic mean curvature flow with contact angle condition (Can Cui and Nung Kwan Yip, 2025)7Cui 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Short time existence and smoothness of the nonlocal mean curvature flow of graphs

    Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias WethWider flows

    Topics Anisotropic & nonlocal flows, Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “evolution equation”, “time existence”

  2. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows

    Topics Anisotropic & nonlocal flows, Mean curvature flow · Shared terms “geometric evolution”

  3. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness, Mean curvature flow · Shared terms “differential equation”

  4. Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows

    Topics Anisotropic & nonlocal flows, Mean curvature flow

  5. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “Finsler”, “short time”, “time existence”

  6. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows

    Topics Anisotropic & nonlocal flows, Mean curvature flow

  7. Two dimensional anisotropic mean curvature flow with contact angle condition

    Can Cui and Nung Kwan YipWider flows

    Topics Anisotropic & nonlocal flows, Mean curvature flow · Shared terms “smooth solutions”

  8. Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows

    Topics Harmonic maps & DeTurck trick, Mean curvature flow · Shared terms “short time”, “evolution equation”, “time existence”

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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.