The $\Box_b$-heat equation on quadric manifolds

TL;DR

This paper explicitly computes the heat kernel for the $ox_b$-heat equation on quadric submanifolds using Fourier analysis and Lie group representation theory, advancing understanding of heat equations in complex geometric contexts.

Contribution

It provides an explicit calculation of the heat kernel for the $ox_b$-heat equation on quadric manifolds, utilizing Fourier transform and Lie group methods, which was previously not available.

Findings

Explicit heat kernel formulas derived for quadric submanifolds.

Method employs representation theory of Lie groups and Fourier transform.

Results facilitate analysis of weighted $ar{ ext{d}}$-equations in complex analysis.

Abstract

In this article, we give an explicit calculation of the partial Fourier transform of the $\Box_b$-heat equation on quadric submanifolds of $M\subset C^n\times C^m$. As a consequence, we can also compute the heat kernel associated to the weighted dbar-equation in $C^n$ when the weight is given by $\exp(-\phi(z,z)\cdot\lambda)$ where $\phi: C^n\times C^n\to C^m$ is a quadratic, sesquilinear form and $\lambda\in R^m$. Our method involves the representation theory of the Lie group $M$ and the group Fourier transform.