Jacob Sturm

On the growth and stability of real-analytic functions

This paper addresses the issue of whether integrals of real-analytic functions remain finite under small deformations. An approach based on uniform estimates for certain classes of one-dimensional integrals is introduced. It is powerful enough to recover the stability properties of real integrals in two dimensions which follow from the work of Karpushkin, as well as produce new results in higher dimensions. In dimension three, the new stability results are sharp, as shown by the well-known example of Varchenko.

The Moser-Trudinger inequality on Kähler-Einstein manifolds

We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.

The Kähler–Ricci flow with positive bisectional curvature

We show that the Kähler–Ricci flow on a manifold with positive first Chern class converges to a Kähler–Einstein metric assuming positive bisectional curvature and certain stability conditions.

On the convergence of the modified Kähler–Ricci flow and solitons

We investigate the Kähler–Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler–Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy. Mathematics Subject Classification (2010). Primary 53C44; Secondary 32Q20, 32Q15.

Simple bounds for closed queueing networks

Consider a closed Jackson type network in which each queue has a single exponential server. Assume that N customers are moving among k queues. We establish simple closed form bounds on the network throughput (both lower and upper), which are sharper than those that are currently available. Numerical evaluation indicates that the improvements are significant.