Heat operator and $ \zeta $-function estimates for surfaces
Using Kato's comparison principle for heat semi-groups we derive estimates for the trace of the heat operator on surfaces with variable curvature. This estimate is from above for positively curved surfaces of genus 0 and from below for genus g ≥ 2. It is shown that the estimates are asymptotically sharp for small time and in the case of positive curvature also for large time. As a consequence we can estimate the corresponding \( \zeta \)-function by the Riemann \( \zeta \)-function.
KeywordsKato Large Time Function Estimate Small Time Curve Surface
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