[Paper Review] On the Laplace-Beltrami operator on compact complex spaces

Francesco Bei|arXiv (Cornell University)|Jun 16, 2017

Geometry and complex manifolds30 references3 citations

TL;DR

This paper establishes the discrete spectrum and trace-class heat kernel properties for the Laplace-Beltrami and Hodge-Kodaira Laplacians on compact, irreducible Hermitian complex spaces of complex dimension $v > 1$. Using Sobolev embedding and spectral estimates, it proves eigenvalue growth bounds $\lambda_k k^{-1/v} \to \infty$ and heat trace decay $\operatorname{Tr}(e^{-t\Delta^\mathcal{F}}) \leq C t^{-v}$, extending results to $L^2$-cohomology and $\overline{\partial}$-operators on complex surfaces.

ABSTRACT

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$.

Motivation & Objective

• To establish the existence of a self-adjoint Friedrichs extension with discrete spectrum for the Laplace-Beltrami operator on compact, irreducible Hermitian complex spaces of complex dimension $v > 1$.
• To derive asymptotic estimates for the eigenvalue growth of the Laplace-Beltrami and Hodge-Kodaira Laplacians on such spaces.
• To prove that the associated heat operators are trace-class, with explicit decay estimates in terms of volume and dimension.
• To extend spectral results to the $\overline{\partial}$-Laplacian and $L^2$-cohomology on complex surfaces with singularities.

Proposed method

• Proving a Sobolev embedding theorem: $W^{1,2}(\operatorname{reg}(X),h) \hookrightarrow L^{2v/(v-1)}(\operatorname{reg}(X),h)$ for $v > 1$.
• Using the Friedrichs extension to construct a self-adjoint realization of the Laplace-Beltrami operator on $L^2(\operatorname{reg}(X),h)$.
• Establishing the eigenvalue growth estimate $\liminf \lambda_k k^{-1/v} > 0$ via Sobolev and spectral theory.
• Applying the heat kernel trace estimate $\operatorname{Tr}(e^{-t\Delta^\mathcal{F}}) \leq C \cdot \operatorname{vol}_h(\operatorname{reg}(X)) \cdot t^{-v}$ for $0 < t \leq 1$.
• Using duality and $\overline{\partial}$-complex theory to extend results to $\overline{\partial}$-Laplacians on $(0,q)$-forms.
• Deriving trace-class estimates for heat operators on $L^2\Omega^{0,q}(\operatorname{reg}(X),h)$ with decay $t^{-2}$ for complex dimension 2.

Experimental results

Research questions

• RQ1 Does the Laplace-Beltrami operator on a compact, irreducible Hermitian complex space of dimension $v > 1$ admit a self-adjoint extension with discrete spectrum?
• RQ2 What is the asymptotic growth rate of the eigenvalues of the Laplace-Beltrami and Hodge-Kodaira Laplacians on such spaces?
• RQ3 Are the associated heat operators trace-class, and can their traces be uniformly bounded in terms of $t$ and volume?
• RQ4 How do spectral properties of the $\overline{\partial}$-Laplacian on $(0,q)$-forms relate to those on functions?
• RQ5 Can the McKean-Singer formula be extended to singular complex surfaces via $L^2$-cohomology and heat kernel traces?

Key findings

• The Friedrichs extension of the Laplace-Beltrami operator on $\operatorname{reg}(X)$ has discrete spectrum for compact, irreducible Hermitian complex spaces of complex dimension $v > 1$.
• The eigenvalues $\lambda_k$ of the Laplace-Beltrami operator satisfy $\liminf \lambda_k k^{-1/v} > 0$ as $k \to \infty$.
• The heat operator $e^{-t\Delta^\mathcal{F}}$ is trace-class, with trace satisfying $\operatorname{Tr}(e^{-t\Delta^\mathcal{F}}) \leq C \cdot \operatorname{vol}_h(\operatorname{reg}(X)) \cdot t^{-v}$ for $0 < t \leq 1$.
• For the Hodge-Kodaira Laplacian on functions, the Friedrichs extension has discrete spectrum and satisfies the same eigenvalue growth and trace estimates.
• The $\overline{\partial}$-Laplacians on $(0,q)$-forms for $q = 0,1,2$ on complex surfaces have discrete spectrum and trace-class heat kernels with $\operatorname{Tr}(e^{-t\Delta_{\overline{\partial},0,q}^\mathcal{F}}) \leq C_q t^{-2}$ for $t \in (0,1]$.
• The McKean-Singer formula holds: $\chi(M,\mathcal{O}_M) = \sum_{q=0}^2 (-1)^q \operatorname{Tr}(e^{-t\Delta_{\overline{\partial},0,q,\operatorname{rel}}})$, linking Euler characteristic to heat trace on resolutions.