Paper · Mar 18, 2019

Eigenvaluues monotonicity of Witten-Laplacian along the mean curvature flow

Shahroud Azami

math.DGWider flowsarXiv:1903.09091
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Abstract

In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic quantities under the mean curvature flow.

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Submitted
Mar 18, 2019
Subjects
math.DG
Mean curvature flowEigenvalues & spectrum

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 Eigenvalues under the Ricci flow of model geometries (Songbo Hou, 2016)2Hou 2016 Isoperimetric inequalities for eigenvalues by inverse mean curvature flow (Fangcheng Guo, Guanghan Li and Chuanxi Wu, 2016)1Guo 2016 Eigenvalues of Weighted-Laplacian under the extended Ricci flow (Abimbola Abolarinwa, 2016)3Abolarinwa 2016 Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds (Songzi Li and Xiang-Dong Li, 2017)5Li 2017 Convexity Estimates for High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2020)6Lynch 2020 Eigenvalue Inequalities for the Clamped Plate Problem of L²_ν Operator (Lingzhong Zeng, 2021)4Zeng 2021 Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds (Songbo Hou and Shusen Yang, 2021)8Hou 2021 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)7He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Isoperimetric inequalities for eigenvalues by inverse mean curvature flow

    Fangcheng Guo, Guanghan Li and Chuanxi WuWider flows

    Topics Eigenvalues & spectrum · Shared terms “Laplace operator”

  2. Eigenvalues under the Ricci flow of model geometries

    Songbo Hou

    Topics Eigenvalues & spectrum · Shared terms “monotonic quantities”, “Laplace”

  3. Eigenvalues of Weighted-Laplacian under the extended Ricci flow

    Abimbola Abolarinwa

    Topics Eigenvalues & spectrum · Shared terms “Laplacian”, “Laplace”

  4. Eigenvalue Inequalities for the Clamped Plate Problem of Operator

    Lingzhong ZengWider flows

    Topics Eigenvalues & spectrum

  5. Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifolds

    Songzi Li and Xiang-Dong Li

    Shared terms “Witten Laplacian”

  6. Convexity Estimates for High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Eigenvalues & spectrum

  7. Spectral bounds for -Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers

    Fei He

    Topics Eigenvalues & spectrum · Shared terms “Laplace”, “Laplacian”

  8. Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds

    Songbo Hou and Shusen Yang

    Topics Eigenvalues & spectrum · Shared terms “Laplace operator”

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) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

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.