Paper · Jun 5, 2026

Solutions to Mean Curvature Flow with Uniform Bounds on the Mean Curvature and Its Gradient

Priyamvada Vishwamitra

math.DGWider flowsarXiv:2606.07162
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Abstract

In the setting of a complete, smooth properly immersed mean curvature flow, we assume uniformly bounded and on and some bounded initial geometry to get local spatial estimates for the second fundamental form with . For , this leads us to a local space time bound for the second fundamental form which allows us to smoothly extend the flow past the singular time for a short time.

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Submitted
Jun 5, 2026
Subjects
math.DG, math.AP
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 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 A local estimate for the mean curvature flow (Zhen Wang, 2021)1Wang 2021 Type II smoothing in mean curvature flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2021)2Angenent 2021 Uniqueness of entire graphs evolving by Mean Curvature flow (Panagiota Daskalopoulos and Mariel Saez, 2021)8Daskalopoulo 2021 A note on Alexandrov immersed mean curvature flow (Ben Lambert and Elena Mäder-Baumdicker, 2022)3Lambert 2022 Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature (Zichang Liu, 2024)6Liu 2024 The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow (Yiqi Huang and Wenshuai Jiang, 2025)4Huang 2025 Half-space theorems for translating solitons of the r-mean curvature flow (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2026)5Alencar 2026 Well-posedness of mean curvature flow (Yongheng Han, 2026)7Han 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A local estimate for the mean curvature flow

    Zhen WangWider flows

    Topics Mean curvature flow · Shared terms “initial geometry”

  2. Type II smoothing in mean curvature flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Mean curvature flow · Shared terms “uniformly bounded”, “fundamental form”, “space time”

  3. A note on Alexandrov immersed mean curvature flow

    Ben Lambert and Elena Mäder-BaumdickerWider flows

    Topics Mean curvature flow · Shared terms “immersed mean”

  4. The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

    Yiqi Huang and Wenshuai JiangWider flows

    Topics Mean curvature flow · Shared terms “uniformly bounded”, “fundamental form”, “space time”

  5. Half-space theorems for translating solitons of the r-mean curvature flow

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows

    Topics Mean curvature flow · Shared terms “properly immersed”, “fundamental form”

  6. Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

    Zichang LiuWider flows

    Topics Mean curvature flow · Shared terms “fundamental form”, “singular time”

  7. Well-posedness of mean curvature flow

    Yongheng HanWider flows

    Topics Mean curvature flow · Shared terms “fundamental form”

  8. Uniqueness of entire graphs evolving by Mean Curvature flow

    Panagiota Daskalopoulos and Mariel SaezWider flows

    Topics Mean curvature flow · Shared terms “fundamental form”

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. 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.