Xianzhe Dai

η‐invariants and determinant lines

The η‐invariant of an odd dimensional manifold with boundary is investigated. The natural boundary condition for this problem requires a trivialization of the kernel of the Dirac operator on the boundary. The dependence of the η‐invariant on this trivialization is best encoded by the statement that the exponential of the η‐invariant lives in the determinant line of the boundary. Our main results are a variational formula and a gluing law for this invariant. These results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of Dirac operators, also known as the ‘‘global anomaly formula.’’ The ideas developed here fit naturally with recent work in topological quantum field theory, in which gluing (which is a characteristic formal property of the path integral and the classical action) is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary.

Smoothing Riemannian Metrics with Ricci Curvature Bounds

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.

Splitting of the Family Index

We establish a general splitting formula for index bundles of families of Dirac type operators. Among the applications, our result provides a positive answer to a question of Bismut and Cheeger [BC2].

A Positive Mass Theorem for Spaces with Asymptotic SUSY Compactification

We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-Horowitz-Maeda [HHM].

APS boundary conditions, eta invariants and adiabatic limits

We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-PatodiSinger boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.

Circle bundles and the Kreck-Stolz invariant

We present a direct analytic calculation of the s-invariant of KreckStolz for circle bundles, by evaluating the adiabatic limits of I invariants. We believe that this method should have wider applications.

Negative Ricci curvature and isometry group

We show that for n-dimensional manifolds with Ricci curvature bounded between two negative constants the order of their isometry groups is uniformly bounded by the Ricci curvature bounds, the volume, and the injectivity radius. We also show that the degree of symmetry (seex2 for denition) is lower semicontinuous in the Gromov-Hausdor topology.

Adiabatic limit, Bismut-Freed connection, and the real analytic torsion form

Abstract For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang in [Math. Ann. 340: 569–624, 340]. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally under this procedure.

A note on positive energy theorem for spaces with asymptotic SUSY compactification

We extend the higher dimensional positive mass theorem in [Dai, X., Commun. Math. Phys. 244, 335–345 (2004)] to the Lorentzian setting. This includes the original higher dimensional positive energy theorem whose spinor proof is given in [Witten, E., Commun. Math. Phys. 80, 381–402 (1981)] and [Parker, T., and Taubes, C., Commun. Math. Phys. 84, 223–238 (1982)] for dimension 4 and in [Zhang, X., J. Math. Phys. 40, 3540–3552 (1999)] for dimension 5.

Comparison between two analytic torsions on orbifolds

In this paper, we establish an equality between the analytic torsion introduced by Dar (Math Z 194(2): 193–216, 1987) and the orbifold analytic torsion defined by Ma (Trans Am Math Soc 357(6): 2205–2233, 2005) on an even dimensional manifold with isolated conical singularities which in addition has an orbifold structure. We assume the orbifold flat vector bundle is an honest vector bundle, although the metric on the flat bundle may not be flat.