Almost Complex Research Articles

Some Integration Problems in Almost-Complex and Complex Manifolds

The theory of almost-complex manifolds leads to a number of problems which properly belong to the field of differential equations. This paper deals with three such problems. The first concerns differential equations in the exterior operator d (frequently also denoted 8) on differential forms; the other two deal with holomorphic coordinates and holomorphic curves. The latter are first stated in differential-geometric terms, and then re-stated as problems in differential equations. The equivalence of these formulations is established in a separate section. The differential equations problems (Theorems I, II', III') and their proofs can be read independently. Two appendices contain underlying facts that are either not available in the form required, or can be obtained only from widely scattered sources.

Errata: “A remark on the Nijenhuis tensor”

Errata: “A remark on the Nijenhuis tensor”

Non-existence of almost-complex structures on quaternionic projective spaces

Non-existence of almost-complex structures on quaternionic projective spaces

On the almost-complex structure of tangent bundles of Riemannian spaces

On the almost-complex structure of tangent bundles of Riemannian spaces

Theory of vector-valued differential forms part II.: Almost-Complex Structures

Theory of vector-valued differential forms part II.: Almost-Complex Structures

Theory of vector-valued differential forms: Part II: Almost-Complex Structures

Theory of vector-valued differential forms: Part II: Almost-Complex Structures

On the geometrical meaning of the vanishing of the nijenhuis tensor in an X2n with an almost complex structure

On the geometrical meaning of the vanishing of the nijenhuis tensor in an X2n with an almost complex structure

Some Problems on Differentiable and Complex Manifolds

A conference with the title Fiber bundles and differential geometry was held at Cornell University from May 3 to May 7, 1953.* It was supported by a grant from the National Science Foundation. The purpose of the present paper is to record those problems presented at the conference which concern differentiable, almost-complex, complex, and algebraic manifolds. Several of the problems have been solved since the conference. These solutions gave rise to new problems which we will discuss. This report is divided into three sections: differentiable manifolds, almostcomplex manifolds, complex manifolds. The last part contains also the problems on algebraic varieties. Since some problems of the first two sections were motivated by theorems on algebraic varieties, there are occasional references in Sections 1 and 2 to Section 3.1. The author wishes to thank Professors K. Kodaira and D. C. Spencer heartily for many discussions and valuable suggestions during the period of preparation of this report.

A general Schwarz lemma for almost-Hermitian manifolds

We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bisectional curvature bounded between two negative constants that satisfies some additional mild assumptions.

Some Remarks on Circle Action on Manifolds

This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold $M^{2mn}$(resp. a smooth manifold $N^{4mn}$), if there exists a partition $\lambda=(\lambda_{1},...,\lambda_{u})$ of weight $m$ such that the Chern number $(c_{\lambda_{1}}... c_{\lambda_{u}})^{n}[M]$ (resp. Pontrjagin number $(p_{\lambda_{1}}... p_{\lambda_{u}})^{n}[N]$) is nonzero, then \emph{any} circle action on $M^{2mn}$ (resp. $N^{4mn}$) has at least $n+1$ fixed points. When an even-dimensional smooth manifold $N^{2n}$ admits a semi-free action with isolated fixed points, we show that $N^{2n}$ bounds, which generalizes a well-known fact in the free case. We also provide a topological obstruction, in terms of the first Chern class, to the existence of semi-free circle action with \emph{nonempty} isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Bott's residue formula and rigidity theorem.

The infimum of the Nijenhuis energy

We prove that on any symplectic manifold whose symplectic form represents a rational cohomology class there exists a sequence of compatible almost complex structures whose Nijenhuis energy (the L²-norm of the Nijenhuis tensor) tends to zero. The sequence is obtained by stretching the neck around a Donaldson hypersurface.

Stability under deformations of extremal almost-Kähler metrics in dimension $4.$

Given a path of almost-K\ahler metrics compatible with a fixed symplectic form on a compact $4$-manifold such that at time zero the almost-K\ahler metric is an extremal K\ahler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-K\ahler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.