The heat kernel on symmetric spaces via integrating over the group of isometries

It is shown that the heat kernel operator for the Laplace operator on any covariantly constant curved background, i.e. in symmetric spaces, may be presented in form of an averaging over the Lie group of isometries with some nontrivial measure. Using this representation the heat kernel diagonal, i.e. the heat kernel in coinciding points is obtained. Related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed.

We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a generating function for the whole sequence of all heat invariants.

We study the low-energy approximation for calculation of the heat kernel which is determined by the strong slowly varying background fields in strongly curved quasi-homogeneous manifolds. A new covariant algebraic approach, based on taking into account a finite number of low-order covariant derivatives of the background fields and neglecting all covariant derivatives of higher orders, is proposed. It is shown that a set of covariant differential operators together with the ba...

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtained result correctly reproduces the first non-trivial heat kernel coefficient as well as the exact heat kernel diagonals on two-dimensional sphere $S^2$ and the h...

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.

We extend the uncertainty principle, the Cowling--Price theorem, on non-compact Riemannian symmetric spaces $X$. We establish a characterization of the heat kernel of the Laplace--Beltrami operator on $X$ from integral estimates of the Cowling--Price type.

We prove Beurling's theorem for rank 1 Riemmanian symmetric spaces and relate it to the characterization of the heat kernel of the symmetric space.

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De Witt-Seeley coefficient...

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-$K$-invariant $L^{1}$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-$K$-invariant case. These answer problems recently raised by J.L. V\'azquez. In the...

Let $G$ be a connected, real, semisimple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. In this paper, we derive $K$-equivariant asymptotics for heat traces with remainder estimates on compact Riemannian manifolds carrying a transitive and isometric $G$-action. In particular, we compute the leading coefficient in the Minakshishundaram-Pleijel expansion of the heat trace for Bochner-Laplace operators on homogeneous vector bundles over compact locally ...