We estimate the heat conducted by a cluster of many small cavities. We show that the dominating heat is a sum, over the number of cavities, of the heat generated by each cavity after interacting with each other. This interaction is described through densities computable as solutions of a closed, and invertible, system of time domain integral equations of second kind. As an application of these expansions, we derive the effective heat conductivity which generates approximately the same heat as the cluster of cavities, distributed in a three-dimensional bounded domain, with explicit error estimates in terms of that cluster. At the analysis level, we use time domain integral equations. Doing that, we have two choices. First, we can favor the space variable by reducing the heat potentials to the ones related to the Laplace operator (avoiding Laplace transform). Second, we can favor the time variable by reducing the representation to the Abel integral operator. As the model under investigation has time-independent parameters, we follow the first approach here.

We show that Norbury’s McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of π1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.

On a compact K¨ahler manifold we introduce a cohomological obstruction to the solvability of the constant scalar curvature (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. As a special case we find an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain “adiabatic” classes. We apply this to find new examples of general type threefolds with classes which do not admit a cscK representative. When the twist vanishes our obstruction extends the slope stability of Ross-Thomas to effective divisors on a K¨ahler manifold. Thus we find examples of non-projective slope unstable manifolds.

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