# Multidimensional surfaces in curved spaces

New Max Planck Research Group takes up its work - a joint project of the Albert Einstein Institute and the University of Potsdam

Geometric measure theory (GMT) is a mathematical language. It is an essential prerequisite for a better understanding of multidimensional surfaces and curved spaces, for example, or the connection between the two. GMT can be used to describe complicated curves, surfaces and bodies and to investigate their properties. "Geometric measure theory provides the tools to understand and mathematically describe quite different phenomena," says Menne. "It can be used to study problems as diverse as black hole horizons in relativity, crystal growth and image reconstruction in image processing."

Menne's Max Planck research group at the AEI will work particularly closely with the Department of Geometric Analysis and Gravity. Its head and AEI Managing Director, Prof. Dr. Gerhard Huisken, says: "With Ulrich Menne, a branch fundamental to the interaction between mathematics and physics - geometric measure theory - will be significantly strengthened. In addition, Menne will be active in teaching at the University of Potsdam and will thus play a bridging role in the cooperation between the Max Planck Institute and the university for the next five years".

At the University of Potsdam, the Partial Differential Equations group of Prof. Dr. Jan Metzger will be particularly closely working together with Menne's new research group. "The joint appointment of Prof. Dr. Ulrich Menne will further strengthen the field of geometric analysis already established at the University of Potsdam in research and teaching," said Metzger. "The profile of our master's program will thus be expanded to include geometric measure theory."

## Geometric measure theory: multidimensional surfaces in curved spaces

A classical mathematical problem is the determination of special surfaces at a given boundary curve. For example, one can use GMT to find out what the smallest possible area enclosed by a given boundary curve looks like. For two-dimensional surfaces in an unbent ("flat") three-dimensional space this is already largely understood: in addition to objects whose shapes can be illustrated by the deforming and stretching of rubber, there are also complicated configurations that are made up of many surface pieces. It becomes particularly complicated when surfaces with more than three dimensions have to be taken into account. In these cases, mathematicians arrive at a solution in two steps: firstly, they show on a more abstract level that a solution is possible at all. Only then is it proven that the solution has the geometric properties that are expected, for example, from a physical perspective. GMT is so flexible that it can also be used in the curved space of Einstein's theory of relativity, for example, to study the properties of black holes.

Prof. Dr. Ulrich Menne (born 1980 in Frankfurt am Main) was awarded a scholarship to study mathematics with in Erlangen-Nuremberg and Bonn. He completed his PhD in Tübingen in 2008. After his postdoctoral research stays at the Albert Einstein Institute and the ETH Zurich, Menne begins setting up his research group at the AEI in April 2012.

## Conference on GMT

A conference on geometric measure theory will take place at the Albert Einstein Institute on July 2-4, 2012. Scientists from all over the world, who have made important contributions to GMT in recent years, are expected to attend the expert meeting. About 50 mathematicians will participate.