U(1) invariant Membranes and Singularities

A formulation of U(1) - symmetric classical membrane motions (preserving one rotational symmetry) is given, and reductions to systems of ODE's, as well as some ideas concerning singularities and integrability.Comment: 6 page

Quasi-Static BMN Solutions

Classical solutions of membrane equations that were recently identified as limits of matrix-solutions are looked upon from another angl

Membranes and Matrix Models

Section I contains introductory remarks about surface motions. Section II gives a detailed derivation of $H=-\Delta-Tr\sum_{i<j}[X_i,X_j]^2$ as describing a quantized discrete analogue of relativistically invariant membrane dynamics. Section III concerns the question of zero-energy bound-states in SU(N)-invariant supersymmetric matrix models. Section IV discusses the space of solutions of some differential matrix equations on $(-\infty,+\infty)$, interpolating between different representations of $su(2)$. Some exercises are added, and one remark/conjecture concerning 5-commutators

Some Classical Solutions of Membrane Matrix Model Equations

Some exact solutions to the classical matrix model equations that arise in the context of M(embrane) theory are given, and their topological nature is identified.Comment: 5 pages, LATEX fil

On M-Algebras, the Quantisation of Nambu-Mechanics, and Volume Preserving Diffeomorphisms

M-branes are related to theories on function spaces $\cal{A}$ involving M-linear non-commutative maps from $\cal{A} \times \cdots \times \cal{A}$ to $\cal{A}$. While the Lie-symmetry-algebra of volume preserving diffeomorphisms of $T^M$ cannot be deformed when M>2, the arising M-algebras naturally relate to Nambu's generalisation of Hamiltonian mechanics, e.g. by providing a representation of the canonical M-commutation relations, $[J_1,\cdots, J_M]=i\hbar$. Concerning multidimensional integrability, an important generalisation of Lax-pairs is given.Comment: 16 pages, LaTe

New integrable systems and a curious realisation of SO(N)

A multiparameter class of integrable systems is introduced.Comment: 5 page

Conservation Laws and Formation of Singularities in Relativistic Theories of Extended Objects

The dynamics of an M-dimensional extended object whose M+1 dimensional world volume in M+2 dimensional space-time has vanishing mean curvature is formulated in term of geometrical variables (the first and second fundamental form of the time-dependent surface $\sum_M$), and simple relations involving the rate of change of the total area of $\sum_M$, the enclosed volume as well as the spatial mean -- and intrinsic scalar curvature, integrated over $\sum_M$, are derived. It is shown that the non-linear equations of motion for $\sum_M(t)$ can be viewed as consistency conditions of an associated linear system that gives rise to the existence of non-local conserved quantities (involving the Christoffel-symbols of the flat M+1 dimensional euclidean submanifold swept out in ${\Bbb R}^{M+1}$). For M=1 one can show that all motions are necessarily singular (the curvature of a closed string in the plane can not be everywhere regular at all times) and for M=2, an explicit solution in terms of elliptic functions is exhibited, which is neither rotationally nor axially symmetric. As a by-product, 3-fold-periodic spacelike maximal hypersurfaces in ${\Bbb R}^{1,3}$ are found.Comment: 14 pages, LaTe

Asymptotic Zero Energy States for SU(N greater or equal 3)

Some ideas are presented concerning the question which of the harmonic wavefunctions constructed in [hep-th/9909191] may be annihilated by all supercharges.Comment: LaTex, 4 page

On the Deformation of Time Harmonic Flows

It is shown that time-harmonic motions of spherical and toroidal surfaces can be deformed non-locally without loosing the existence of infinitely many constants of the motion.Comment: 8 pages, LaTex, Talk presented at the Centro Stefano Franscin

On Zero-Mass Bound States in Super-Membrane Models

For the simplest case of a supermembrane matrix model, various symmetry reductions are given, with the fermionic contribution(s) (to an effective Schr\"odinger equation) corresponding to an attractive $\delta$-function potential (towards zero-area configurations). The differential equations are real, and are shown not to admit square-integrable real solutions (even when allowing non-vanishing boundary conditions at infinity). Complex solutions, however, are not excluded by this argument.Comment: 6 pages Late