PDE seminar Characteristic varieties of linearized isometric embedding

2011-2-7 16:30 - 2011-2-7 17:30
Faculty of Science Building #3 Room 202
Qing Han (Univ. of Notre Dame)
It is a classical question in geometry whether Riemannian
manifolds admit an isometric embedding in Euclidean space. The isometric
embedding for n dimensional manifold can be expressed by a nonlinear
partial differential system and the number of equations is in the square
order of n. The linearized system is highly degenerate and its
characteristic variety consists of the entire phase space. By introducing
appropriate parameters, this partial differential system can be reduced to
a smaller one of only n equations. The main task now is to analyze the
characteristic variety (defined as a subset of unit sphere in phase
space). We will prove that the characteristic variety is not smooth when
the dimension is at least 5 for parameters are small. This proves a
conjecture due to Bryant, Griffiths and Yang in a special case.