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Search: MSC category 58J35 ( Heat and other parabolic equation methods )

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1. CJM 2005 (vol 57 pp. 506)

Gross, Leonard; Grothaus, Martin
Reverse Hypercontractivity for Subharmonic Functions
Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, $e^{-tA}$, can be bounded {\it below} from $L^p$ to $L^q$ when $p,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.

Keywords:Reverse hypercontractivity, subharmonic
Categories:58J35, 47D03, 47D07, 32Q99, 60J35

2. CJM 2002 (vol 54 pp. 1086)

Polterovich, Iosif
Combinatorics of the Heat Trace on Spheres
We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D.~Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in [P].

Categories:05A19, 58J35

3. CJM 2000 (vol 52 pp. 695)

Carey, A.; Farber, M.; Mathai, V.
Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion
Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von~Neumann algebras as developed in \cite{CFM}. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic $L^2$ torsion, which shows that it is {\it not\/} in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic $L^2$ torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.

Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost Kähler manifolds, von~Neumann algebras, determinant lines
Categories:58J52, 58J35, 58J20

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