The dimension datum of a subgroup of a compact Lie group is a piece of spectral information about that subgroup. We find
some new invariants and phenomena of the dimension data and apply them to construct the first example of a pair of isospectral,
simply connected closed Riemannian manifolds which are of different homotopy types. We also answer questions proposed by
We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained
from M by surgery of codimension at least 3. We prove the existence of a positive constant n, depending only on the dimension n
of M, such that.
We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into IRn, n = 3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound Wn,p
depending on codimension and genus p of the Riemann surface. For tori in codimension 1, we know W3,1 = 8 .
In this paper we first introduce a transform for convex functions and use it to prove a Bernstein theorem for a Monge-Amp`ere
equation in half space.We then prove the optimal global regularity for a class of Monge-Amp`ere type equations arising in a number
of geometric problems such as Poincar´e metrics, hyperbolic affine spheres, and Minkowski type problems.
If M is a hyperbolic 3-manifold with a quasigeodesic flow, then we show that 1(M) acts in a natural way on a closed disc by
homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is M¨obius-like but not
conjugate into PSL(2,R).We conjecture that the latter possibility cannot occur.