We prove that if a manifold of nonnegative scalar curvature contains a two-sided hypersurface which is locally of least area and admits no metric of positive scalar curvature, then it splits isometrically in a neighborhood of the hypersurface.

The hyperbolic Gauss map $G$ of a complete constant mean curvature one surface $M$ in hyperbolic 3-space, is a holomorphic map from $M$ to the Riemann sphere. When $M$ has finite total curvature, we prove $G$ can miss at most three points unless $G$ is constant. We also prove that if $M$ is a properly embedded mean curvature one surface of finite topology, then $G$ is surjective unless $M$ is a horosphere or catenoid cousin.

The spectrum of the Laplacian on hyperbolic space is a proper subset of the positive reals. We study eigenfunctions, defined on the complements of compact sets, whose eigenvalues lie below the bottom of the spectrum. Such eigenfunctions may arise by perturbing the metric on compact subsets of the space. One divides the eigenfunctions by normalizing factors, so that the quotients have analytic boundary values on the ideal boundary at infinity. The renormalized eigenfunctions are approximated by special polynomials, in nontangential approach regions to the ideal boundary.

This is the second part of a series of articles explaining applications of Floer homology to mirror symmetry and D-brane. This article is independent of part I [Fu9]. We will associate an $A_{\infty}$ category to a symplectic manifold. This is an improved version of previous ones [Fu1], [Fu4] in which there were some flaw. The correction is based on a book [FOOO] written jointly with Oh, Ohta, Ono. While correcting the flaw, we find various interesting new phenomena which are related to mirror symmetry.

We also discuss homological algebra of $A_{\infty}$ category in this article.

This article is a survey article. So most of the material written here are minor modifications of the results which are already known to somebody. However it is rather hard to find a reference of them.

If a convex surface, such as an egg shell, is illuminated from any given direction, then the corresponding shadow cast on the surface forms a connected subset. The shadow problem, first studied by H. Wente in 1978, asks whether a converse of this phenomenon is true as well. In this report it is shown that the answer is yes provided that each shadow is simply connected; otherwise, the answer is no. Further, the motivations behind this problem, and some ramifications of its solution for studying constant mean curvature surfaces in 3-space (soap bubbles) are discussed.

We prove several fundamental properties of 4-dimensional CR-submanifolds of a nearly Kähler 6-dimensional sphere and construct explicit examples of such submanifolds.

We investigate a certain class of minimal surfaces in Euclidean space, which are constructed from a generalization of the Weierstrass formula. We also show a characterization of the catenoid.

The topology of hyperKähler quotients of quaternionic vector spaces by tori is studied. We discuss the relation between their topology and a combinatorial property of some polyhedral complexes. As its simple application we compute their Chern classes.

We study hypersurfaces in Euclidean, hyperbolic or Lorentz-Minkowski space with the property that it is foliated by a one-parameter family of round spheres. We describe partially such hypersurfaces with some assumption on its curvature. In general, we shall consider the situation that the mean curvature or the Gaussian curvature is constant.

In this note we give a generalized form of the well-known height estimate for constant mean curvature graphs due to J. Serrin. An application is also given that effectively relates the global rate of convergence of a family of constant mean curvature surfaces recently considered by A. Ros and H. Rosenberg to their convergence behavior on the boundary. Some information concerning boundary behavior is also obtained by applying reflection techniques and corner comparison in particular. This latter analysis allows one to construct a counterexample to one part of a conjecture of Rosenberg.

Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete $m$-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete and satisfies Weyl's asymptotic formula.

Using relevant properties of the Seiberg-Witten monopole invariants, a non-existence theorem of metrics of positive scalar curvature on certain series of Brieskorn 3-manifolds is proved. This gives a short proof without the help of Gromov-Lawson's theorem.

The space of geodesics on a Zoll manifold, i.e., a Riemannian manifold all of whose geodesics are closed with the same minimal period, carries a natural symplectic structure. In this note, it is shown that the space of geodesics on a Zoll three-sphere is symplectomorphic to the product of two copies of two-spheres with the same area.

Surfaces of constant mean curvature one in hyperbolic 3-space have quite similar properties to minimal surfaces in Euclidean 3-space. We shall list the possibilities of constant mean curvature one surfaces in hyperbolic 3-space with low total absolute curvature, or low dual total absolute curvature, and compare them with the known classification of minimal surfaces with low total curvature. Complete proofs of the new results will be published in two forthcoming papers (listed in the bibliography).

We present an altenative proof of the volume formula of the moduli space of spatial polygons, which was given by Kamiyama-Tezuka. Our method is based on the commutativity of geometric quantization and symplectic reduction, originating from a conjecture of Guillemin-Sternberg.