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.

In this paper, we develop a three-dimensional parallel solver using the fifth order high-resolution weighted essentially non-oscillatory (WENO) finite difference scheme to perform extensive simulation for three-dimensional gaseousdetonations. A careful study is conducted for the propagation modes of three-dimensional gaseous detonation wave-front structures in a long square duct. The numerical results indicate that, the instability of detonation, overdrive factor, transverse dimension and initial perturbation are the main factors that influence the appearance of spinning detonation.

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.

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The superradiant scattering of a scalar field with frequency and angular momentum (ω, m) by a near-extreme Kerr black hole with mass and spin (M, J) was derived in the seventies by Starobinsky, Churilov, Press and Teukolsky. In this paper we show that for frequencies scaled to the superradiant bound the full functional dependence on (ω, m, M, J) of the scattering amplitudes is precisely reproduced by a dual two-dimensional conformal field theory in which the black hole corresponds to a specific thermal state and the scalar field to a specific operator. This striking agreement corroborates a conjectured Kerr/CFT correspondence.