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 K\"ahler). This is motivated by the very recent work of Hertog-Horowitz-Maeda.Comment: To appear in CM

Hitchin-Thorpe Inequality for Noncompact Einstein 4-Manifolds

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifolds come with such a geometry at infinity.Comment: 17 page

An Index Theorem for Toeplitz Operators on Odd Dimensional Manifolds with Boundary

We establish an index theorem for Toeplitz operators on odd dimensional spin manifolds with boundary. It may be thought of as an odd dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of $\eta$ type associated to $K^1$ representatives on even dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.Comment: Completely revised. A gap in the proof fixed. To appear in JF

The Intersection R-Torsion for Finite Cone

We prove a formula for the intersection R-torsion of a finite cone and use it to introduce a family of spectral invariants which is closely related to Cheeger's half torsion.Comment: introduction revise