Eigenfunctions of the Laplacian on rotationally symmetric manifolds

Eigenfunctions of the Laplacian on a negatively curved, rotationally symmetric manifold are constructed explicitly under the assumption that an integral of converges. This integral is the same one which gives the existence of nonconstant harmonic functions on

     

Integrability of rotationally symmetric n-harmonic maps

A rotationally symmetric n-harmonic map is a rotationally symmetric p-harmonic map between two n-dimensional model spaces such that p=n. We show that rotationally symmetric n-harmonic maps can be integrated and are n-harmonic diffeomorphism, and apply such results to investigate the asymptotic behaviors of these maps. We also derive this integrability using Lie theory.     

Bounds on the index of rotationally symmetric self-shrinking tori

Geometriae Dedicata - A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by...     

The Ricci flow on the two ball with a rotationally symmetric metric

In this paper we study a boundary-value problem for the Ricci flow in the two-dimensional ball endowed with a rotationally symmetric metric of positive Gaussian curvature and prove short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally, we show that if the initial metric has positive Gaussian curvature and the boundary has positive geodesic curvature then the flow uniformizes the curvature along a sequence of times. The text was submitted by the author in English.     

A generalized rotationally symmetric case of the centroaffine Minkowski problem

In this paper the centroaffine Minkowski problem, a critical case of the $L_p$-Minkowski problem in the $n+1$ dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.     

Sums of Laplace eigenvalues—rotationally symmetric maximizers in the plane

The sum of the first n?1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio ³(area)/(moment of inertia) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.     

Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

We consider the area-preserving mean curvature flow with free Neumann boundaries. We show that a rotationally symmetric n-dimensional hypersurface in R~(n+1)between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.     

Uniqueness of positive solutions to the rotationally symmetric p-harmonic map equations

Let α and β be positive solutions on (0, ∞) to the rotationally symmetric p-harmonic map equation on model manifolds M(f) and M(g), where f is assumed to be sufficiently large near infinity and g″(y) ⩾ 0 for y > 0. We show that if α and β have the same limit at infinity, then α ≡ β on (0, ∞).     

Projection minima of a symmetric convex body

R ⁿ n- , : RⁿPRⁿ/ o - . —

A remark on rotationally symmetric solutions to the centroaffine Minkowski problem

In this paper we study the solvability of the rotationally symmetric centroaffine Minkowski problem. By delicate blow-up analyses, we remove a technical condition in the existence result obtained by Lu and Wang [30].     

Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds

We prove global regularity for the solution to the Cauchy problem with regular data for an equivariant harmonic map from the 2 + 1-dimensional Minkowski space into a two-dimensional, rotationally symmetric, and geodesically convex Riemannian manifold.     

Complete minimal graphs with prescribed asymptotic boundary on rotationally symmetric Hadamard surfaces

It is proved that if M is a rotationally symmetric Hadamard surface which is conformally equivalent to the hyperbolic disk then the asymptotic Dirichlet problem for the minimal surface equation is uniquely solvable for any continuous asymptotic boundary data. This result gives a partial answer of a question in Gálvez and Rosenberg (Am J Math 132:1249?C1273, 2010) about the existence of entire minimal graphs on Hadamard surfaces with sectional curvature possibly degenerating at infinity.     

Nonexistence of liouvillian solutions in the problem of motion of a rotationally symmetric ellipsoid on a perfectly rough plane

Using the Kovacic algorithm, the nonexistence of liouvillian solutions in the problem of motion of a rotationally symmetric ellipsoid on a perfectly rough horizontal plane for almost all values of parameters of the problem is proven.     

An example of a convex body without symmetric projections

Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for everyn-dimensional convex bodyK there exists a projectionP of rankk, proportional ton, such thatPK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct ann-dimensional convex bodyK such that for everyk >C√nlnn and every projectionP of rankk, the bodyPK is very far from being symmetric. In particular, our example shows that one cannot expect a formal argument extending the “symmetric” theory to the general case. This author holds a Lady Davis Fellowship.     

Parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body

Let K be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that K is contained in a centrally symmetric hexagonal prism (or a parallelepiped) of volume 4

On stability of motion of a symmetric body placed on a vibrating base

We consider the problem of stabilization of a symmetric solid body rotating about a fixed point and show that its unstable states can be stabilized by vertical vibration.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1661–1666, December, 1995.     

Group-theoretic analysis of the Navier-Stokes equations in the rotationally symmetric case and some new exact solutions

Journal of Mathematical Sciences - In the paper one investigates the group-theoretic properties of the Navier-Stokes equations in R3 in the case of rotational symmetry. One computes the fundamental...