Related papers: Moving Manifolds and the Poincare Conjecture
This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power…
We develop a Hamiltonian description of point particles in (2+1)-dimensions using connection and frame-field variables for general relativity. The topology of each spatial hypersurface is that of a punctured two-sphere with particles…
In this article, we study the the harmonic map heat flow from a manifold with conic singularities to a closed manifold. In particular, we have proved the short time existence and uniqueness of solutions as well as the existence of global…
This paper gives a topological characterization of Hamiltonian flows with finitely many singular points on compact surfaces, using the concept of ``demi-caract\'eristique'' in the sense of Poincar\'e. Furthermore, we describe the…
The concept of smooth deformation of Riemannian manifolds associated with the extrinsic curvature is explained and applied to the FLRW cosmology. We show that such deformation can be derived from Einstein-Hilbert-like dynamical principle…
A continuum theory is used to predict scaling laws for the morphological relaxation of crystal surfaces in two independent space dimensions. The goal is to unify previously disconnected experimental observations of decaying surface…
Surfaces with constant mean curvature (CMC) are critical points of the area with volume constraint. They serve as a mathematical model of surfaces of soap bubbles and tiny liquid drops. CMC surfaces are said to be stable if the second…
We prove an abstract structure theorem for weighted manifolds supporting a weighted $f$-Poincar\'e inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full…
A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with…
A $k$-harmonic map is a critical point of the $k$-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^{n} (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three distinct…
Motivated by the Poincare conjecture, we study properties of digital n-dimensional spheres and disks, which are digital models of their continuous counterparts. We introduce homeomorphic transformations of digital manifolds, which retain…
We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching…
We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite…
The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its…
We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric…
We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as…
We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies…
In this note we shall construct, in the framework of relativistic quantum mechanics, the Poincare-invariant motion equations with realistic mass spectra. These equations describe a system with mass spectra of the form $m^2=a^2+b^2 s(s+1)$,…
We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.
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…