Paper · Mar 14, 2024

Novel Boundary Conditions for the Ricci Flow

Rasmus Jouttijärvi

math.DGarXiv:2403.09169
View PDF View HTML arXiv page

As on arXiv

Abstract

If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. We would like these conditions to reflect the geometric nature of the flow and allow for a variety of initial data. Importantly, the conditions should be compatible with the expected evolution of Einstein metrics. We propose it is natural to choose those conditions, for which the first variation of certain functionals, such as the Einstein-Hilbert action and Perelmans lambda-functional, does not admit a boundary term. We provide a proof of the short term existence of solutions of the initial boundary value problem, under these conditions.

View PDF View HTML arXiv page
Submitted
Mar 14, 2024
Subjects
math.DG, math.AP
Ricci flowEinstein metricsBoundaries & non-compact

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 Ricci Flow method in the existence problem of the K"ahler-Einstein metrics (Liu Chao, 2022)3Chao 2022 Linear stability of compact shrinking Ricci solitons (Huai-Dong Cao and Meng Zhu, 2023)8Cao 2023 Dynamical Stability and Instability of Poincaré–Einstein Manifolds (Klaus Kroencke and Louis Yudowitz, 2023)2Kroencke 2023 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)4Kotschwar 2023 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)5Kroencke 2025 Ricci solitons as critical points of quadratic curvature functionals (Atreyee Bhattacharya and Sayoojya Prakash, 2026)7Bhattacharya 2026 On the Stability of Einstein Manifolds with Boundary (Rasmus Jouttijärvi, 2026)1Jouttijärvi 2026 The rigity of spaces with cyclic parallel Ricci tensor (Fengyuan Zhang, 2026)6Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Stability of Einstein Manifolds with Boundary

    Rasmus Jouttijärvi

    Topics Einstein metrics, Boundaries & non-compact · Same author Rasmus Jouttijärvi · Shared terms “boundary condition”, “Hilbert action”, “Einstein Hilbert”

  2. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz

    Topics Einstein metrics, Boundaries & non-compact

  3. Ricci Flow method in the existence problem of the K"ahler-Einstein metrics

    Liu Chao

    Topics Einstein metrics

  4. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Einstein metrics, Boundaries & non-compact

  5. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Einstein metrics, Boundaries & non-compact

  6. The rigity of spaces with cyclic parallel Ricci tensor

    Fengyuan Zhang

    Topics Einstein metrics, Boundaries & non-compact · Shared terms “Riemannian manifolds”

  7. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics

  8. Linear stability of compact shrinking Ricci solitons

    Huai-Dong Cao and Meng Zhu

    Topics Einstein metrics · Shared terms “variation”, “Perelman”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.