Geometric Analysis and General Relativity

Geometric analysis can be said to originate in the 19’th century work of Weierstrass, Riemann, Schwarz and others on minimal surfaces, a problem whose history can be traced at least as far back as the work of Meusnier and Lagrange in the 18’th century. The experiments performed by Plateau in the mid-19’th century, on soap films spanning wire contours, served as an important inspiration for this work, and let to the formulation of the Plateau problem, which concerns the existence and regularity of area minimizing surfaces in R spanning a given boundary contour. The Plateau problem for area minimizing disks spanning a curve in R was solved by Jesse Douglas (who shared the first Fields medal with Lars V. Ahlfors) and Tibor Rado in the 1930’s. Generalizations of Plateau’s problem have been an important driving force behind the development of modern geometric analysis. Geometric analysis can be viewed broadly as the study of partial differential equations arising in geometry, and includes many areas of the calculus of variations, as well as the theory of geometric evolution equations. The Einstein equation, which is the central object of general relativity, is one of the most widely studied geometric partial differential equations, and plays an important role in its Riemannian as well as in its Lorentzian form, the Lorentzian being most relevant for general relativity. The Einstein equation is the Euler-Lagrange equation of a Lagrangian with gauge symmetry and thus in the Lorentzian case it, like the Yang-Mills equation, can be viewed as a system of evolution equations with constraints. After imposing suitable gauge conditions, the Einstein equation becomes a hyperbolic system, in particular using spacetime harmonic coordinates (also known as wave coordinates), the Einstein equation becomes a quasilinear system of wave equations. The constraint equations implied by the Einstein equations can be viewed as a system of elliptic equations in terms of suitably chosen variables. Thus the Einstein equation leads to both elliptic and hyperbolic problems, arising from the constraint equations and the Cauchy problem, respectively. The groundwork for the mathematical study of the Einstein equation and the global nature of spacetimes was laid by, among others, Choquet-Bruhat who proved local well-posedness for the Cauchy problem, Lichnerowicz, and later York who provided the basic ideas for the analysis of the constraint equations, and Leray who formalized the notion of