Changfeng Gui

Title: New Sharp Inequalities in Analysis and Geometry

Abstract: The classical  Moser-Trudinger inequality is a borderline case of Soblolev
inequalities  and plays an important role in geometric analysis.  Aubin in
1979  showed that the best constant in the Moser-Trudinger inequality can
be improved by reducing to one half if the functions are restricted to the
complement of  a three dimensional  subspace of the  Sobolev space H¹, 
while Onofri  in 1982 discovered  an elegant optimal form of 
Moser-Trudinger inequality on sphere.  In this talk,   I will present  new
sharp inequalities  which are  variants of Aubin  and Onofri  inequalities
on the sphere with or without constraints.   The main ingredient leading
to the above inequalities is  a novel  geometric  inequality: Sphere
Covering Inequality, discovered jointly with  Amir Moradifam from UC
Riverside.
One such  inequality, for example,  incorporates  the mass center
deviation  (from  the origin) into the optimal inequality of Aubin on the
sphere  which is  for functions with  mass centered  at the origin.  In
another view point,   this  inequality also generalizes to the sphere the
Lebedev-Milin inequality and the second inequality in  the Szegö limit
theorem on the Toeplitz determinants on the circle,  which is useful in
the study of  isospectral compactness for metrics defined on compact
surfaces, among other applications.
Efforts have also  been made to show similar inequalities in higher
dimensions. Among the preliminary results,  we have improved Beckner’s
inequality for axially symmetric functions when the  dimension n = 4, 6,
8. Many questions remain open.