Harmonic maps and the Einstein equation

Abstract

In 1917, very shortly after Einstein had announced his master equation on the general relativity and Schwarzschild had discovered the first explicit solution to the equation, H. Weyl characterized the Schwarzschild metric, by a harmonic function. Since then, the solutions to the Einstein equation with a certain set of symmetries are identified with elliptic variational problems, in particular the harmonic map equation. In collaboration with Marcus Khuri, Yukio Matsumoto and Gilbert Weinstein, we constructed a new set of stationary solutions to the 5-dimensional vacuum Einstein equation, which contains non-spherical event horizons. The higher dimensional spacetime exhibit a wider range of topological structures, compared to our 4-dimensional physical spacetime, and those stationary solutions are thus geometrically interesting.