Skip to main content

From geodesic rays in spaces of Kähler metrics to the Hele-Shaw flow

Objective

Very recently Dr. Julius Ross at the Univ. of Cambridge and I found a striking connection between geodesic rays in spaces of Kähler metrics and the Hele-Shaw flow (Laplacian growth). By this connection both have a similar interpretation as certain families of embedded holomorphic curves attached along their boundaries to a Lagrangian submanifold.

The first objective is to develop the regularity theory of the Hele-Shaw flow (Laplacian growth) using techniques from the theory of moduli spaces of embedded holomorphic curves. These are powerful techniques used with great success in e.g. Gromov-Witten Theory and various Floer theories in symplectic topology. I thus hope to extend the short-time regularity result of Kufarev and Vinogradov, and also gain new insights as to how and when singularities occur.

The second objective is to develop the regularity theory for (weak) geodesic rays in spaces of (cohomologically equivalent) Kähler metrics, using the Hele-Shaw flow as a one (complex) dimensional model case. Donaldson, and later Chen and Tian, have successfully applied techniques from the theory of moduli spaces of embedded holomorphic curves to a related problem connected to the regularity of geodesic segments rather than rays. Dr. Ross and I have a preliminary method to adapt some of these techniques to the setting of geodesic rays.

Field of science

  • /natural sciences/mathematics/pure mathematics/topology

Call for proposal

FP7-PEOPLE-2012-IEF
See other projects for this call

Funding Scheme

MC-IEF - Intra-European Fellowships (IEF)

Coordinator

THE CHANCELLOR MASTERS AND SCHOLARSOF THE UNIVERSITY OF CAMBRIDGE
Address
Trinity Lane The Old Schools
CB2 1TN Cambridge
United Kingdom
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 231 283,20
Administrative Contact
Renata Schaeffer (Ms.)