Heat Kernel on Warped Products

Abstract We study the spectral properties of the scalar Laplacian on a n -dimensional warped product manifold $$M=\Sigma \times _f N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Σ</mml:mi> <mml:msub> <mml:mo>×</mml:mo> <mml:mi>f</mml:mi> </mml:msub> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> with a $$(n-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional compact manifold N without boundary, a one dimensional manifold $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> without boundary and a warping function $$f\in C^\infty (\Sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We consider two cases: $$\Sigma =S^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Σ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> when the manifold M is compact, and $$\Sigma =\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Σ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> when the manifold M is non-compact. In the latter case we assume that the warping function f is such that the manifold M has two cusps with a finite volume. In particular, we study the case of the warping function $$f(y)=[\cosh (y/b)]^{-2\nu /(n-1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>cosh</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>/</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>ν</mml:mi> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> in detail, where $$y\in \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> and b and $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> are some positive parameters. We study the properties of the spectrum of the Laplacian in detail and show that it has both the discrete and the continuous spectrum. We compute the resolvent, the eigenvalues, the scattering matrix, the heat kernel and the regularized heat trace. We compute the asymptotics of the regularized heat trace of the Laplacian on the warped manifold M and show that some of its coefficients are global in nature expressed in terms of the zeta function on the manifold N .