The Li-Yau-Hamilton Estimate and the Yang-Mills Heat Equation on Manifolds with Boundary
The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold $M$ with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution $\nabla(t)$ of the Yang-Mills heat equation in a vector bundle over $M$. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply that the curvature of $\nabla(t)$ does not blow up if the dimension of $M$ is less than 4 or if the initial energy of $\nabla(t)$ is sufficiently small.
A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold
Let M be a connected complete Riemannian manifold of dimension m . We call M hyperbolic if it has a nonconstant positive superharmonic func tion. When M is not hyperbolic, then we call M parabolic. In the study of function theoretic properties of M, it is important to know whether M is hyperbolic or parabolic. We know that the behavior of the sectional curva ture tells us whether M is hyperbolic or parabolic. For example, when m=2, a theorem of Blac-Fiala-Huber [16] states that M is parabolic if the Gaussian curvature is nonnegative, and a theorem of Ahlfors says that M is hyperbolic if it is simply connected and the Gaussian curvature is bounded from above by some negative constants1). Moreover when m•†3, a theorem of Aomoto
A new method of near field analysis
The problem of the accurate determination of the total electromagnetic field produced by a current distribution on a circular disc type aperture is examined. For comparison of the near and Fresnel region solutions, an exact Maxwellian integral is used as the starting point. Aperture and space coordinates are separated by the use of Bessel function addition theorems, allowing the field to be expressed as a sum of terms containing radial and angular integrals. The radial integrals involve only the disc (aperture) distribution and coordinates while the angular integrals involve only the elevation angle. The distance from origin appears implicitly in the series coefficients. These integrals are suitable for high speed computations and have been evaluated for a number of cases. The results are compared with the small angle and general Fresnel formula values for various observation angles, distances from the aperture, and aperture distributions.