We study the motion of an n-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power p of the mean curvature. We prove that for any p ∈ (0, 1], the flow exists for all time when the Ricci tensor of the ambient space is bounded from below on the set of timelike unit vectors. Moreover, if we ass...

Let X = P × P embedded with O(1, 2). We prove that its (n + 1)-secant variety σn+1(X) is a hypersurface, while it is expected that it fills the ambient space. The equation of σn+1(X) is the symmetric analog of the Strassen equation. When n = 4 the determinantal map takes σ5(X) to the hypersurface of Lüroth quartics, which is the image of the Barth map studied by LePotier and Tikhomirov. This hi...

Our main object of study is X ⊂ P5F2 , where X is the degree 5 Fermat hypersurface, which we are looking at in P over F2. X is the zero locus {X 0 +X 1 + ...+X 5 = 0}. The question is, “what kind of rational curves lie on X?” Since F2 only has one invertible element, P(F2) = F 2 \ 0, which suggests that the Fermat quintic has the same zero locus as the linear hypersurface X0 + X1 + ... + X5 = 0...

Let M be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S(1) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if M is the real projective space RP = Sn+1(1)/{±...

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

Abstract The following aspects of multidimensional space are considered in the article. parametrizations hypersurface second order considered, which allow visualizing them process discretization. development principle vector modeling for displaying objects on a two-dimensional plane made it possible to visually display solution problem linear approximation second-order hypersurface. proposed me...

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter subgroup of the isometry group of the complex space form, there is an essent...

Given a smooth positive function F∈C∞(Sn) such that the square of its 1-homogeneous extension on Rn+1∖{0} is uniformly convex, Wulff shape WF convex body in Euclidean space Rn+1 with F being support boundary ∂WF. In this paper, we introduce fully nonlinear locally constrained anisotropic curvature flow ∂∂tX=(1−Ek1/kσF)νF,k=2,…,nin space, where Ek denotes normalized kth mean respect to WF, σF an...

For a hypersurface in complex projective space, X ⊂ Pn, we investigate the singularities and Kodaira dimension of the Kontsevich moduli spacesM0,0(X, e) parametrizing rational curves of degree e on X. If d + e ≤ n and X is a general hypersurface of degree d, we prove thatM0,0(X, e) has only canonical singularities and we conjecture the same is true for the coarse moduli space M0,0(X, e). We pro...