[Paper Review] Singular equivariant asymptotics and the moment map II

Pablo Ramacher|ArXiv.org|Jun 12, 2009

Advanced Operator Algebra Research15 references3 citations

TL;DR

This paper establishes asymptotic expansions for oscillatory integrals involving the moment map in the presence of singular group actions, using partial resolution of singularities in the critical set. It proves that the leading-order term is given by an integral over the regular part of the critical set, with a precise formula involving the Hessian determinant, enabling stationary phase analysis even when the critical set is singular.

ABSTRACT

This is the second of a series of papers dealing with the asymptotic behavior of certain integrals occuring in the description of the spectrum of an invariant elliptic operator on a compact Riemannian manifold carrying the action of a compact, connected Lie group of isometries, and in the study of its equivariant cohomology via the moment map.

Motivation & Objective

• To address the breakdown of the stationary phase method when the critical set of the moment map phase function is singular due to non-free group actions.
• To develop a method for computing asymptotic expansions of oscillatory integrals with singular critical sets arising in equivariant spectral theory.
• To generalize previous results on orthogonal actions in Euclidean space to compact Riemannian manifolds with compact Lie group actions.
• To establish a remainder estimate in the asymptotic expansion despite the singular structure of the critical set.
• To show that the leading coefficient is independent of the choice of partial resolution, ensuring robustness of the result.

Proposed method

• The critical set of the phase function $\psi(\eta,X) = \mathbb{J}(\eta)(X)$ is analyzed, identifying it as $\mathrm{Crit}(\psi) = \{ (\eta,X) : \tilde{X}_\eta = 0 \}$, which coincides with $\Omega \times \mathfrak{g}$, where $\Omega = \mathbb{J}^{-1}(0)$.
• The singularities of the critical set are partially resolved by introducing cut-off functions $u_\varepsilon$ supported near the singular part $\mathrm{Sing}\,\Omega$, allowing the application of the generalized stationary phase theorem on the regular part.
• A partition of unity is used to construct smooth cut-off functions $u_\varepsilon$ such that $u_\varepsilon = 1$ on $\mathrm{Sing}\,\Omega \cap K$ and compactly supported in $((\mathrm{Sing}\,\Omega \cap K)_\varepsilon)$.
• The asymptotic expansion is derived by analyzing the integral $I_\varepsilon(\mu)$ with amplitude $a(1 - u_\varepsilon)$, showing convergence of the leading coefficient as $\varepsilon \to 0$.
• The leading coefficient $L_0$ is expressed as an integral over the regular part $\mathrm{Reg}\,\mathcal{C}$, weighted by the inverse square root of the Hessian determinant restricted to the normal space of $\mathrm{Reg}\,\mathcal{C}$.
• The independence of the result from the choice of resolution is proven via the Lebesgue dominated convergence theorem and Fatou's lemma, ensuring the formula is intrinsic.

Experimental results

Research questions

• RQ1 How can asymptotic expansions be derived for oscillatory integrals with singular critical sets arising from non-free group actions on compact manifolds?
• RQ2 What is the precise form of the leading-order term in the asymptotic expansion when the critical set of the moment map phase is singular?
• RQ3 Can the stationary phase method be generalized to cases where the critical set is not a smooth manifold?
• RQ4 How does the choice of partial resolution affect the leading coefficient in the asymptotic expansion?
• RQ5 Is the resulting leading coefficient independent of the resolution process, ensuring a well-defined geometric object?

Key findings

• The asymptotic expansion of $I(\mu)$ as $\mu \to 0^+$ is given by $I(\mu) = (2\pi\mu)^\kappa L_0 + O(\mu^{\kappa+1})$, where $\kappa$ is the dimension of the regular part of the critical set.
• The leading coefficient $L_0$ is given by the integral $L_0 = \int_{\mathrm{Reg}\,\mathcal{C}} \frac{a(\eta,X)}{\left|\mathrm{Hess}\,\psi(\eta,X)_{N_{(\eta,X)}\mathrm{Reg}\,\mathcal{C}}\right|^{1/2}} \, d(\mathrm{Reg}\,\mathcal{C})(\eta,X)$, which is finite and well-defined.
• The existence of the limit in the expression for $L_0$ is established via a limiting process involving cut-off functions $u_\varepsilon$, ensuring convergence as $\varepsilon \to 0$
• The formula for $L_0$ is independent of the specific partial resolution used, as shown by the Lebesgue convergence theorem and Fatou's lemma.
• The integral over $\mathrm{Reg}\,\mathcal{C}$ exists and is finite, even when the full critical set $\mathcal{C}$ is singular, due to the decay of the amplitude and the behavior of the Hessian determinant.
• The result generalizes previous work on orthogonal actions in Euclidean space to the global setting of compact Riemannian manifolds with compact Lie group actions.