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In [8], Hitchin proved that a compact K/ihlerian twistor space is in fact a projective algebraic manifold. Moreover, it is the twistor space associated to either the Euclidean 4-sphere S 4 or the complex projective plane CF 2 with Fubini-Study metric. The twistor spaces are 3 and the flag of lines in~ p2 respectively. In [10, 11], the author proved the existence of self-dual metric with positive scalar curvature on the connected-sums of two or three copies of the complex projective planes. Their twistor spaces are Moish6zon spaces. In fact, they are the small resolution of the intersection of two quadrics in~ ps with four nodes and the double covering of CP 3 branched along a quartic with thirteen nodes. Recently, the joint work of Donaldson and Friedman [3] produced a general procedure to construct new twistor spaces and hence self-dual manifolds. In this article, we shall follow the spirit of Hitchin's work and prove the

Given a compact set of real numbers, a random $C^{m + \alpha}$ -diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$ , almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$ . This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$ -equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha )$ . In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^{m}$ -diffeomorphisms for any $m$ .

In this article, we continue the work in Guan-Li and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.

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