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This paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω⊂$R$^{$n$}be a bounded domain with a$C$^{2}boundary and μ a measure compactly supported in Ω. Then we say ∂Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution. $$\Delta u = - \mu in \Omega ,u = 0 and \frac{{\partial u}}{{\partial v}} = - 1 on \partial \Omega .$$

We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by$c$_{0}- and$l$_{1}-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

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This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow-ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature Kähler metrics.