In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence the internal geometry of M. For example, if the scalar curvature of M is greater than certain constant related to this boundary effect, no incompressible surface of higher genus can exist.

For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a zeroth order pseudodifferential operator by mimicking Szego’s definition of this determinant for the operator: multiplication by a bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local contribution to this determinant can be computed in terms of a much simpler “zeta-regularized” determinant.

Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n.1 in Cn+1. Tanaka introduced a spec- tral sequence E(p,q) r (X) with E(p,q) 1 (X) being the Kohn-Rossi cohomology group and E(k,0) 2 (X) being the holomorphic De Rham cohomology denoted by Hk h(X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated sin- gularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has re- mained unsolved for over a quarter of a century. In this paper, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dimX = 3 and H2 h(X) = 0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.

Continuing the program of [DS] and [U1], we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Taubes invariant is easy to define, our invariants agree with those analogues. We also prove a vanishing result for some of the invariants that count nodal curves.

All continuous GL(n) covariant valuations that map convex bodies to convex bodies are completely classified. This establishes a characterization of moment and projection operators and shows that the Holmes-Thompson area is the unique Minkowski area that is also a bivaluation.