School of Mathematics Professor Mohammad Ghomi and Joel Spruck of Johns Hopkins University this week posted in ArXiv their proposed solution to the Cartan-Hadamard Conjecture, a long-standing problem in Analysis and Geometry, which is concerned with a generalization of the classical isoperimetric inequality.
In Euclidean space, to enclose a given volume with the smallest possible perimeter, one should use a ball. Comparing the volume of the ball to its surface area yields a numerical measure of the efficiency of this enclosure which lies at the base of a range of important inequalities in mathematics and physics. There are more general models of space than the Euclidean framework, however. Some of the most natural and attractive ones are those with nonpositive curvature, which are known as Cartan-Hadamard manifolds. The simplest examples of Cartan-Hadamard manifolds include potato chips and coral reefs. Cartan-Hadamard manifolds aim to include more area in less space, like waves of a petunia flower.
The exploration of the isoperimetric inequality in Cartan-Hadamard manifolds can be traced back to work of Andre Weil, a student of Hadamard at the time, who addressed the two dimensional case in 1926. Around fifty years later, Thierry Aubin, Misha Gromov, Yuri Burago and Victor Zalgaller conjectured that, in all dimensions, Cartan-Hadamard manifolds should satisfy the same isoperimetric inequality as in Euclidean space. Prior to Ghomi and Spruck’s work, the only other known cases where in dimensions 3 and 4, due to Bruce Kleiner in 1992 and Chris Croke in 1984 respectively.
The proposed proof settles the conjecture in all dimensions. As a result several other well-known inequalities in Analysis and Mathematical Physics may now be generalized to spaces of nonpositive curvature as well. These include the Sobolev inequalities, which are fundamental in theory of functions, and the Faber-Krahn inequality, which dates back to a conjecture of Lord Rayleigh in 1877 concerning the spectrum of the Laplace operator, or the theory of sound.
Mohammad Ghomi completed his Ph.D. in 1998 at Johns Hopkins University under the direction of Spruck. As a geometer, Ghomi works on classical problems involving curves and surfaces in Euclidean space and, more generally, Riemannian submanifolds. He is interested in all aspects of mathematics that touch upon these objects, ranging from differential geometry and topology to real algebraic geometry and combinatorics.