We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M, g) that are C0-asymptotic to Schwarzschild
of mass m > 0. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild.
We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered
coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and
S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that
enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volumepreserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.
We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its
geodesics. We identify the kernel of this transform acting on symmetric tensor fields.
In the first paper of this series, “Electrodynamics and the Gauss Linking Integral on the 3-sphere and in Hyperbolic 3-space,” we
developed a steady-state version of classical electrodynamics in these two spaces, including explicit formulas for the vector-valued
Green’s operator, explicit formulas of Biot-Savart type for the magnetic field, and a corresponding Amp`ere’s Law contained in
Maxwell’s equations, and then used these to obtain explicit inte- gral formulas for the linking number of two disjoint closed curves.
In this second paper, we obtain integral formulas for twisting, writhing, and helicity, and prove that link = twist + writhe on
the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower
bounds for the first eigenvalue of the curl operator on subdomains of these two spaces.
An announcement of these results, and a hint of their proofs, can be found in the Math ArXiv, math.GT/0406276, while an
expanded version of the first paper, with full proofs, can be found at math.GT/0510388.
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.