Paper · Feb 1, 2018

Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

Felix Schulze

math.DGWider flowsarXiv:1802.00226v3
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Abstract

Let be simply connected, complete, with non-positive sectional curvatures, and a 2-dimensional closed integral current (or flat chain mod 2) with compact support in . Let be an area minimising integral 3-current (resp. flat chain mod 2) such that . We use a weak mean curvature flow, obtained via elliptic regularisation, starting from , to show that S satisfies the optimal Euclidean isoperimetric inequality: . We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in .

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Submitted
Feb 1, 2018
Last revised
Feb 4, 2020 (v3)
Subjects
math.DG, math.AP

Comments: 29 pages, final version, to appear in GAFA

Mean curvature flowSurfacesSphere theoremsWeak & singular flowsGeometric inequalities

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    Mohammad Ghomi and Joel SpruckWider flows

    Topics Geometric inequalities · Shared terms “Cartan Hadamard”, “Hadamard manifolds”, “isoperimetric inequality”

  2. Isometry theorem of Cartan-Hadamard manifold

    Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal et al.

    Topics Sphere theorems · Shared terms “Hadamard manifolds”, “Cartan Hadamard”, “positive sectional”

  3. Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces

    Fang HongWider flows

    Topics Geometric inequalities · Shared terms “Cartan Hadamard”

  4. A priori bounds for geodesic diameter. Part III. A Sobolev-Poincaré inequality and applications to a variety of geometric variational problems

    Ulrich Menne and Christian ScharrerWider flows

    Topics Geometric inequalities, Weak & singular flows · Shared terms “integral current”, “minimising integral”, “area minimising”

  5. Sharp dynamical isoperimetric principle for mean curvature flow

    Beomjun Choi, Wenkui Du, Seung Chul Park et al.Wider flows

    Topics Geometric inequalities, Weak & singular flows · Shared terms “isoperimetric”, “equality”

  6. On the Hamilton-Lott conjecture in higher dimensions

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Sphere theorems, Weak & singular flows · Same author Felix Schulze · Shared terms “Euclidean”

  7. Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows

    Topics Geometric inequalities, Sphere theorems · Shared terms “positive sectional”, “sectional curvature”, “equality”

  8. Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds

    Fabio Cavalletti and Andrea Mondino

    Topics Geometric inequalities, Weak & singular flows · Shared terms “isoperimetric inequality”, “Euclidean isoperimetric”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.