25.04.2024, 16:15
– Raum 1.10
Forschungsseminar Differentialgeometrie
Atiyah-Singer-Indexsatz
Christian Bär (UP)
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 $g \ge 2$. 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.