On singular foliations on the solid torus

We study smooth foliations on the solid torus S¹×D²$S^1\times D^2$ having S¹×{0}$S^1\times \{\mathbf{0}\}$ and S¹×∂D²$S^1\times \partial D^2$ as the only compact leaves and S¹×{0}$S^1\times \{\mathbf{0}\}$ as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.     

Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature

We prove that if for the curved n-body problem the masses are given, the minimum distance between the point masses of a specific type of relative equilibrium solution to that problem has a universal lower bound that is not equal to zero. We furthermore prove that the set of all such relative equilibria is compact. This class of relative equilibria includes all relative equilibria of the curved n  -body problem in H²${\mathbb{H}}^2$ and a significant subset of the relative equilibria for S²${\mathbb{S}}^2$, S³${\mathbb{S}}^3$ and H³${\mathbb{H}}^3$.     

Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2-sphere in the connected sum of two closed 3-dimensional manifolds _M1≠S³$M_1\ne S^3$ and _M2≠S³,RP³$M_2\ne S^3,\mathbb{R}{\mathbb{P}}^3$, which must develop finite time singularity.     

Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations

The paper revisits the concept of S^p$S^p$-pseudo-almost periodicity recently introduced by the author. In particular, we study the existence of pseudo-almost periodic solutions to some nonautonomous differential equations in the case when the semilinear forcing term is both continuous and S^p$S^p$-pseudo-almost periodic for p>1$p>1$. Applications include the existence of pseudo-almost periodic solutions to the heat equation with a forcing term, which is S^p$S^p$-pseudo-almost periodic and jointly continuous.     

On diagonal resolvable spheres

In this paper we give the complete answer to a question posed by A. Arhangel?skii and prove that the sphere Sⁿ$S^n$ is diagonal resolvable if and only if Sⁿ$S^n$ is an H  -space if and only if n∈{0,1,3,7}$n\in \{0,1,3,7\}$. Moreover, we prove that any upper half even dimensional Q$\mathbb{Q}$-sphere cannot be diagonal resolvable.     

Hypersurfaces in hyperbolic space with support function

Based on [19], we develop a global correspondence between immersed hypersurfaces ?:Mⁿ→Hⁿ⁺¹$?:{\mathrm{M}}^n\to {\mathbb{H}}^{n+1}$ satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e^2ρ_gSn$e^{2\rho }{g}_{{\mathbb{S}}^n}$ on domains Ω in the boundary Sⁿ${\mathbb{S}}^n$ at infinity of Hⁿ⁺¹${\mathbb{H}}^{n+1}$, where ρ   is the horospherical support function, _∂∞?(Mⁿ)=∂Ω${\partial }_{\infty }?({\mathrm{M}}^n)=\partial \mathrm{\Omega }$, and Ω is the image of the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$. To do so we first establish results on when the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$ is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and conformal metrics on domains in Sⁿ${\mathbb{S}}^n$. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ of constant mean curvature.     

Minimal genus in 4-manifolds with a free circle action

Let N be a closed irreducible 3-manifold and assume N   is not a graph manifold. We improve for all but finitely many S¹$S^1$-bundles M over N   the adjunction inequality for the minimal complexity of embedded surfaces. This allows us to completely determine the minimal complexity of embedded surfaces in all but finitely many S¹$S^1$-bundles over a large class of 3-manifolds.     

Periodic solutions of asymptotically linear autonomous Newtonian systems with resonance

In this paper, we study the existence, continuation and bifurcation from infinity of2π$2\pi$-periodic solutions of autonomous Newtonian systems. We underline that the resonant case is considered. To prove the results, we apply the degree for S¹$S^1$-equivariant gradient maps defined by Rybicki (1994) in [15] and the angle condition introduced by Bartsch and Li (1997) in [16].     

RIGIDITY THEOREMS OF CLIFFORD TORUS

In this article, we prove that the Clifford torus S1(1-r2~(1/2)) × Sn-1(r) is the only closed hypersurface in the unit sphere Sn+1(1) with infinite fundamental group, which satisfy r2 ≥ (n-1)/n, RicM ≤ C-(H), and S ≤ S+(H). Moreover, we give a characterization of Clifford torus S1(1-r2~(1/2)) × Sn-1(r) with r2 = 2(n-1)+nH2±|H| n2H2+4(n-1)(1/2)/2n(1+H2) .     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.