We prove a surgery formula for the smooth Yamabe invariant $\sigma(M)$ 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 $\Lambda_n$, depending only on the dimension n of $M$, such that \[ \sigma(N) \ge \mathrm{min}\{\sigma(M),\Lambda_n\}.\]

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 Langlands.

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 Ampère’s Law contained in Maxwell’s equations, and then used these to obtain explicit integral 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.

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.

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds $(M, g)$ that are $\mathcal{C}^0$-asymptotic to Schwarzschild of mass $m \gt 0$. 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 volume-preserving 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.