Paper · May 11, 2023

Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation

Juan Carlos Fernández, Oscar Palmas, Jonatán Torres Orozco

math.AParXiv:2305.06573
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Abstract

We study the optimal partition problem for the prescribed constant -curvature equation induced by the higher order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed -curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant -curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of -operators on Ricci solitons, compute the -curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with prescribed number of nodal domains on the Koiso-Cao Ricci soliton.

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Submitted
May 11, 2023
Subjects
math.AP, math.DG
Ricci solitonsEinstein metricsScalar curvatureHomogeneous spaces

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  1. Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups

    Victor Sanmartin-Lopez

    Topics Einstein metrics, Homogeneous spaces, Scalar curvature

  2. Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds

    Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons

  3. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons

  4. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al.

    Topics Einstein metrics, Scalar curvature, Ricci solitons

  5. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics, Scalar curvature, Ricci solitons

  6. Solvable Lie algebras obtained by quivers and Einstein metrics

    Fumika Mizoguchi

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons

  7. Linear stability of compact shrinking Ricci solitons

    Huai-Dong Cao and Meng Zhu

    Topics Einstein metrics, Ricci solitons · Shared terms “Cao”

  8. Hermitian curvature flow and HKT geometry

    Beatrice Brienza and Jeffrey Streets

    Topics Einstein metrics, Homogeneous spaces, Scalar curvature

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.