[Paper Review] The spectral theory of generalized Laplacians associated to integrable metrics on compact Riemann surfaces

Mounir Hajli|arXiv (Cornell University)|Jan 9, 2013

Geometry and complex manifolds7 references4 citations

TL;DR

This paper extends the spectral theory of generalized Laplacians to integrable metrics on compact Riemann surfaces, constructing a singular Laplacian Δ_{X,∞} with discrete, positive spectrum and proving its zeta function admits holomorphic continuation at s=0. The authors define holomorphic analytic torsion via ζ′_{Δ_{X,∞}}(0), and show it agrees with a limit of torsions from smooth approximating metrics, thereby generalizing Ray-Singer analytic torsion to singular metrics in Arakelov geometry.

ABSTRACT

We extend the spectral theory of generalized Laplacians to integrable metrics on compact Riemann surfaces. As a consequence, we attach in a direct way, a holomorphic analytic torsion to any integrable metrics. We also provide a different approach to define the holomorphic analytic torsion. We prove that both approaches agree.

Motivation & Objective

• To extend the classical spectral theory of Laplacians to integrable metrics on compact Riemann surfaces, which are singular and not covered by standard smooth metric assumptions.
• To define holomorphic analytic torsion directly for integrable metrics using the zeta function of the generalized Laplacian Δ_{X,∞}.
• To establish a limiting equivalence between the analytic torsion defined via Δ_{X,∞} and the limit of torsions from smooth metric approximations.
• To provide a new, direct construction of analytic torsion in Arakelov geometry that applies to canonical metrics on P^1 as a toric variety.

Proposed method

• Construct a singular Laplacian Δ_{X,∞} on smooth functions A^{(0,0)}(X) associated to an integrable metric h_{X,∞} on a compact Riemann surface X.
• Prove that Δ_{X,∞} is densely defined, symmetric, and admits a maximal positive selfadjoint extension to a Hilbert space H_2(X).
• Define the zeta function ζ_{Δ_{X,∞}}(s) = ∑ λ_n^{-s} for Re(s) > 1 and show it admits a holomorphic continuation to C, analytic at s=0.
• Define the holomorphic analytic torsion as T = ζ′_{Δ_{X,∞}}(0), providing a direct definition for integrable metrics.
• Construct a sequence of smooth metrics (h_{X,u})_u converging uniformly to h_{X,∞}, and prove that the corresponding torsions converge to T.
• Use Cheeger's isoperimetric inequality to bound the first nonzero eigenvalue uniformly from below, ensuring spectral stability under approximation.

Experimental results

Research questions

• RQ1 Can the spectral theory of Laplacians be extended to integrable metrics on compact Riemann surfaces, which are singular and not smooth?
• RQ2 Does the zeta function of the generalized Laplacian Δ_{X,∞} associated to an integrable metric admit a holomorphic continuation at s=0?
• RQ3 Is the holomorphic analytic torsion defined via ζ′_{Δ_{X,∞}}(0) equivalent to the limit of torsions computed from smooth approximating metrics?
• RQ4 Can this generalized analytic torsion be applied to canonical metrics on P^1 viewed as a toric variety in Arakelov geometry?

Key findings

• The generalized Laplacian Δ_{X,∞} associated to an integrable metric h_{X,∞} has a discrete, positive, and infinite spectrum on A^{(0,0)}(X).
• The zeta function ζ_{Δ_{X,∞}}(s) converges absolutely for Re(s) > 1 and admits a holomorphic continuation to the entire complex plane, analytic at s=0.
• The holomorphic analytic torsion T((X,ω_{X,∞}),(O,h_O)) is well-defined as ζ′_{Δ_{X,∞}}(0), providing a direct construction for integrable metrics.
• For any sequence of smooth metrics (h_{X,p}) converging uniformly to h_{X,∞}, the corresponding analytic torsions converge to T((X,ω_{X,∞}),(O,h_O)).
• The first nonzero eigenvalue λ_{p,1} of the Laplacian on (X,h_{X,p}) is uniformly bounded from below by a positive constant κ independent of p, ensuring spectral stability.
• The construction confirms the results and computations in references [10] and [11] by providing a rigorous spectral-theoretic foundation for analytic torsion in the singular setting.