EPSRC Reference: 
EP/J014206/1 
Title: 
Gluing, Rigidity and Uniqueness Questions in Geometric Analysis 
Principal Investigator: 
Lotay, Professor J 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
UCL 
Scheme: 
Standard Research 
Starts: 
01 May 2012 
Ends: 
31 July 2012 
Value (£): 
9,634

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
We want to study two different but related types of geometric object called special Lagrangian (SL) submanifolds and Lagrangian selfexpanders. SL submanifolds have the attractive property that they are volumeminimizing, so can be thought of as like soap films. We also wish to consider another related type of volumeminimizing objects called Cayley 4folds. Mathematicians have studied the equations governing soap films for over two hundred years and many widely applicable mathematical techniques were first developed to study the soap film equations (nonlinear elliptic equations). These techniques are now used by mathematicians, physicists and engineers in a whole range of problems completely unrelated to soap films. While much is now known about soap films themselves, their study is still an active area of research with several recent important breakthroughs. For generalised soap films like SL submanifolds, much less is known; some completely new phenomena occur which we are only just beginning to understand.
The study of Lagrangian selfexpanders is motivated by a distinguished and natural way to move geometric objects which live inside larger spaces called Mean Curvature Flow (MCF). Under MCF, a sphere will simply shrink, whereas Lagrangian selfexpanders grow. MCF tries to move a given surface so that its area shrinks as rapidly as possible. MCF has a strong smoothing effect in which local irregularities tend to get smoothed out very rapidly, in much the same way that heat spreads out from a heat source. For this reason it has been used by many engineers as a robust way to remove noise from empirical data, e.g. images of brains from various types of scanners. The engineers often rely on tools developed primarily by mathematicians. One difficulty with MCF is that over long time periods its smoothing effects may be overwhelmed by nonlinear feedback and thus singularities may develop in the flow. This project will contribute to our understanding of how singularities can form in a special type of MCF called Lagrangian Mean Curvature Flow.
Our project is to study SL submanifolds, Lagrangian selfexpanders and Cayley 4folds with "ends". We want to show that in certain circumstances knowing only the "ends" of a geometric object completely determines its global structure. This is important because it says if we understand how an object looks only at a very largescale then we can infer how it looks at all scales. If we draw the curve xy=1 for positive x and y, we see that it has two "ends": one which gets closer to the xaxis and the other which gets closer to the yaxis. Overall the curve xy=1 approaches a pair of straight lines (the axes) which intersect at just one point (the origin). If we consider the same equation xy=1, but now where x and y are complex numbers, we get a surface with two ends each of which is asymptotic to a plane. Moreover, the two asymptotic planes meet at just one point, so we call them transverse. We want to study objects with the same property: they have two ends that each approach a plane and the pair of asymptotic planes are transverse.
Our aim is to show that if we have a pair of transverse planes then either there is no SL submanifold or Lagrangian selfexpander with two ends asymptotic to them, or there is just one (possibly satisfying some extra conditions to make it unique). We also hope to explore some of the consequences of these structural results; hopefully this will eventually lead to the solution of the important and difficult problems of finding SL submanifolds using Lagrangian Mean Curvature Flow and defining invariants using Cayley 4folds. There are also connections to more diverse areas including the study of generalised soap films and soap bubbles, "gluing" problems, Homological Mirror Symmetry (which was inspired by ideas from theoretical physics in String Theory and MTheory), and the study of nonlinear partial differential equations.

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