A Strengthened Gromov-Bishop Inequality

I’ll discuss a spin-off from joint work with local physicists: Lenny Susskind and Adam Brown. We find an upper bound on the volume of balls in a Riemannian manifold X somewhat stronger (i.e. smaller) than that obtained by comparing to the hyperbolic space of equal dimension and Ricci quadratic from agreeing with the minimum value achieved on X. The new idea is method, “coefficient shuffling”, for studying correlated families of Jacobi equations.