Biomedical subjects
Publications and source records attributed to V Guillemin.
Using ideas suggested by some recent developments in string theory, we give here an elementary demonstration of one of the key steps in Douglas' celebrated proof of the existence of solutions of the Plateau problem in n dimensions.
We state a theorem that relates the theory of dimensional reduction in Hamiltonian mechanics to the spectral properties of elliptic operators with symmetries on compact manifolds. As an application, we show that the spectrum of the Schrödinger operator, -[unk]hDelta + V, as [unk]h --> 0, contains geometric information about the closed trajectories of a classical particle with Hamiltonian p(2) + V(q). More generally, we show that this is true for particles with internal degrees of freedom and subject to an external Yang-Mills field, the classical limit being the Wong-Sternberg-Weinstein system for such particles.
New results are announced linking properties of the symbol module and characteristic variety of a differential complex with test estimates near the characteristic variety of the type considered by Hörmander ((1/2)-estimate). The first result is the invariance of the test estimates under pseudo-differential change of coordinates, and this leads to the introduction of a normal form for the complex in the neighborhood of a Cohen-MacCauley point of the symbol module. If the characteristic variety V is a manifold near the Cohen-MacCauley point (x(0),zeta(0)) with parametrizing functions p(1),...,p(q), where q is the codimension of the characteristic variety in the complexified contangent bundle, the matrix [Formula: see text] of Poisson brackets defines invariantly a Hermitian form Q on the normal space to V at (x(0),zeta(0)) when the dp(zeta)(x(0),zeta(0)) are used as basis, and the test estimates are satisfied at the ith stage of the complex if sig. Q (signature of Q) is >/= n - i + 1 (n the dimension of the base manifold) or rank Q - sig. Q >/= i + 1. Finally, conditions are given in order that, on a manifold with smooth boundary, the associated boundary complexes satisfy the (1/2)-estimate.
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