The $G$-equivariant Kazdan--Warner problem

Technical Summary: The G-Equivariant Kazdan–Warner Problem

Problem Statement
This paper investigates the equivariant analogue of the classical Kazdan–Warner trichotomy, which classifies closed manifolds of dimension $n \ge 3$ based on the scalar curvature functions they can realize. While the classical result partitions manifolds into three mutually exclusive classes—those admitting positive scalar curvature (P), those admitting scalar-flat metrics but no positive ones (Z), and those admitting no non-negative scalar curvature metrics (N)—this work restricts the analysis to $G$-invariant data. Specifically, given a closed connected manifold $M$ with an effective isometric action by a compact connected Lie group $G$ (where the action has cohomogeneity at least one and no zero-dimensional orbits), the authors determine which $G$-invariant smooth functions can arise as the scalar curvature of a $G$-invariant Riemannian metric. The central question is whether the classical trichotomy extends verbatim to the equivariant setting or if new phenomena emerge.

Methodology
The authors employ a combination of geometric analysis, variational methods, and equivariant topology:

1. Linearization and Surjectivity: The study begins by analyzing the scalar curvature operator $F: g \mapsto \text{scal}_g$ restricted to the space of $G$-invariant metrics. By examining the formal adjoint $A^*$ of the linearization of $F$, the authors establish local surjectivity criteria. They prove that if the kernel of $A^*$ is trivial, the operator is locally surjective onto a neighborhood of the current scalar curvature.
2. Equivariant Approximation (Theorem 1.6): A critical technical tool is an extension of the Kazdan–Warner Approximation Lemma. The authors prove that $G$-invariant functions can be rearranged via $G$-equivariant diffeomorphisms to approximate other $G$-invariant functions, provided specific conditions on the orbit space $X = M/G$ are met. This relies on lifting isotopies from the orbit space (either an interval or a circle) to the manifold $M$.
3. Equivariant Yamabe Problem: The existence of constant scalar curvature metrics is addressed using variational methods. The authors utilize the Palais Principle of Symmetric Criticality and compact embedding theorems for Sobolev spaces of invariant functions (Hebey–Vaugon) to find minimizers of the Yamabe functional within the $G$-invariant class.
4. Geometric Characterization of Cohomogeneity One: For actions of cohomogeneity one, the authors leverage the structure of the orbit space and the isotropy representations. They utilize the warped product structure of invariant metrics and the properties of isotropy-irreducible principal orbits to derive integral constraints on the total scalar curvature.

Key Contributions and Results

• Discovery of a Fourth Class (Totally G-Positive): The primary result (Theorem 1.3) demonstrates that the classical trichotomy does not extend verbatim. A distinct class of pairs $(M, G)$ emerges, termed totally G-positive.

  • Definition: A pair is totally G-positive if every $G$-invariant metric on $M$ has a strictly positive total scalar curvature ($\int_M \text{scal}_g d\mu_g > 0$).
  • Consequences: Such pairs admit a $G$-invariant metric of positive constant scalar curvature but admit no $G$-invariant metric of zero or negative constant scalar curvature. Furthermore, they cannot realize any $G$-invariant function that is everywhere non-positive.
  • Characterization: Lemma 3.11 establishes that $(M, G)$ is totally G-positive if and only if the action is of cohomogeneity one with an isotropy-irreducible principal orbit. A prototypical example is $M = S^1 \times S^{n-1}$ with the $SO(n)$ action, where the isotropy representation is irreducible.

• Refined Classification for Non-Totally G-Positive Pairs: For pairs that are not totally G-positive, the classical trichotomy is recovered at the level of $G$-invariant data (Theorem 1.3, part ii). These pairs fall into exactly one of three classes:

  • $(P_G)$: Every smooth $G$-invariant function is the scalar curvature of some $G$-invariant metric.
  • $(Z_G)$): A function is realizable if and only if it is identically zero or negative somewhere.
  • $(N_G)$: A function is realizable if and only if it is negative somewhere.

• Non-Abelian Symmetry: The authors note that if the Lie algebra of $G$ is non-abelian and the pair is not totally G-positive, the pair necessarily belongs to class $(P_G)$. This relies on the Lawson–Yau result that non-abelian symmetry groups guarantee the existence of a $G$-invariant metric of positive scalar curvature. However, the paper emphasizes that non-abelian actions can still be totally G-positive (as in the $S^1 \times S^{n-1}$ example), placing them in the new fourth category rather than $(P_G)$.

• Existence of Constant Curvature Metrics: Theorem 1.4 provides a comprehensive existence result:

  • If $(M, G)$ is not totally G-positive, it admits a $G$-invariant metric of constant negative scalar curvature. If it admits a non-negative metric, it admits a scalar-flat one.
  • If it admits a non-identically-zero non-negative metric, it admits a positive constant scalar curvature metric.
  • Totally G-positive pairs admit positive constant scalar curvature metrics but no non-positive ones.

Significance
The paper claims that the intrinsic geometry of the group action, specifically the cohomogeneity and the irreducibility of the isotropy representation, governs the existence of scalar curvature metrics in the equivariant setting. The emergence of the "totally G-positive" class represents a genuine deviation from the non-equivariant theory, where the total scalar curvature can always be made negative or zero on manifolds that do not admit positive scalar curvature. The work synthesizes the Kazdan–Warner approximation techniques with the Hebey–Vaugon equivariant Yamabe theory to provide a complete classification of admissible scalar curvature functions for a broad class of symmetric manifolds.