On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists for all time on a toric variety then it minimizes the Calabi functional. In this case the infimum of the Calabi functional is given by the supremum of the normalized Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson.

Global geometrical optics method is a new semi-classical approach for the high frequency linear waves proposed by the author in [13]. In this paper, we rederive it in a more concise way. It is shown that the right candidate of solution ansatz for the high frequency wave equations is the extended WKB function, other than the WKB function used in the classical geometrical optics approximation. A new and main contribution of this paper is an interface analysis for the Helmholtz equation when the incident wave is of extended WKB-type. We derive asymptotic expressions for the reflected and/or transmitted propagating waves in the general case. These expressions are valid even when the incident rays include caustic points.

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii).

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.

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