B.S., University of Michigan, 1986
Ph.D., California Institute of Technology, 1991
Analysis and Partial Differential Equations, Differential Geometry, Partial Differential Equations
Differential Geometry and Partial Differential Equations
Broadly speaking, my research is in geometric analysis: conformal geometry, partial differential equations, the calculus of variations, and global differential geometry.
Some of my recent work has focused on fully nonlinear elliptic equations in conformal geometry. The solvability of these equations and estimates of solutions are strongly influenced by the topology and geometry of the underlying manifold, especially in low dimensions. This connection has resulted in some interesting applications to the geometry of four-manifolds.
- M.J. Gursky, The Weyl functional, de Rham cohomology, and K\"ahler—Einstein metrics, Annals of Math. 148 (1998), 315—337.
- M.J. Gursky and C. LeBrun, Yamabe constants and spin$^c$ structures, GAFA 8 (1998), 965—977.
- M.J. Gursky, The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Comm. Math. Phys. 207 (1999), 131—143.
- M.J. Gursky, Four—manifolds with $\delta W^+=0$ and Einstein constants of the sphere, Math. Annalen 318 (2000), 417—431.
- M.J. Gursky and J. Viacolvsky, A new variational characterization of three-dimensional space forms, Invent. Math. 145 (2001), 251—278.
- S.-Y.A. Chang, M.J. Gursky, and P. Yang, An equation of Monge- Ampere type in conformal geometry, and four manifolds of positive Ricci curvature, Annals of Math. 155 (2002), 711—789.