We study the heat kernel of a regular symmetric Dirichlet form on a metric space with doubling measure, in particular, a connection between the properties of the jump measure and the long time behaviour of the heat kernel. Under appropriate optimal hypotheses, we obtain the Hölder regularity and lower estimates of the heat kernel.
We prove the diagonal upper bound of heat kernels for regular Dirichlet forms on metric measure spaces with volume doubling condition. As hypotheses, we use the Faber-Krahn inequality, the generalized capacity condition and an upper bound for the integrated tail of the jump kernel. The proof goes though a parabolic mean value inequality for subcaloric functions.
Prarit AgarwalQueen Mary University of London, Mile End Road, London E1 4NS, UK; Elaitra LtdDongmin GangDepartment of Physics and Astronomy & Center for Theoretical Physics, Seoul National University, 1 Gwanak-ro, Seoul 08826, Korea; Asia Pacific Center for Theoretical Physics (APCTP), Pohang 37673, KoreaSangmin LeeCollege of Liberal Studies, Seoul National University, Seoul 08826, Korea; Department of Physics and Astronomy & Center for Theoretical Physics, Seoul National University, 1 Gwanak-ro, Seoul 08826, KoreaMauricio Andrés Romo JorqueraYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
Geometric Analysis and Geometric TopologyarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.15001
We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement S^3∖K. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of SL(2,C) Chern-Simons theory, one can define perturbative invariants of knot K in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of K and K. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.