We consider the flow of closed convex hypersurfaces in Euclidean space R^{n+1} with speed given by a power of the k-th mean curvature E_k plus a global term chosen to impose a constraint involving the enclosed volume V_{n+1} and the mixed volume V_{n+1−k} of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves S 1[0, T) 2 which satisfies the following evolution equation 2 F t 2 (u, t)= k (u, t) N(u, t)- (u, t),(u, t) S 1[0, T) with the initial data F (u, 0)= F 0 (u) a n d F t (u, 0)= f (u) N 0, where k is the mean curvature and is the unit inner normal vector of the plane curve F (u, t), f (u) and are the initial velocity and the unit inner normal vector of the initial convex closed curve Fo, respectively, and is given by ( 2 F s t, F t) T, in which stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere

What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving explicitly this inverse potential problem.

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