Finite group actions on compact negatively curved manifolds

We show that some diameters of orbits of a non-trivial finite group G acting effectively on a compact connected negatively curved manifold M have a universal bound depending only on the dimension n of the underlying manifold. It can also be viewed as a rigidity result.     

Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

In the study of the collapsed manifolds with bounded sectional curvature,the following two results provide basic tools:a(singular)fibration theorem by K.Fukaya[J.Differential Gcom.,1987.25(1):139156]and J.Cheeger,K.Fukaya,and M.Gromov[J.Airier.Math.Soc.,1992.5(2):327372],and the stability for isometric compact Lie group actions on manifolds by R.S.Palais[Bull.Amer.Math.Soc.,1961,67(4):362364]and K.Grove and H.Karcher[Math.Z..1973,132:1120].The main results in this paper(partially)generalize the two results to manifolds with local bounded Ricci covering geometry.     

On compact symplectic manifolds with Lie group symmetries

In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.

     

Compact lie group actions with isotropy subgroups of maximal rank

When a compact connected Lie group acts smoothly on a manifold X with only connected isotropy subgroups of maximal rank, the action is completeley determined by the corresponding action of its Weylgroup WG on the fixed space X^T of the maximal torus. Isotropy subgroups of such actions are determined.     

Equivariant deformations of meromorphic actions on compact complex manifolds

Meromorphicity is the most basic property for holomorphic -actions on compact complex manifolds. We prove that the meromorphicity of -actions on compact complex manifolds are not necessarily preserved by small deformations, if the complex dimension of complex manifolds is greater than two. In contrast, we also show that the meromorphicity of -actions on compact complex surface depends only on the topology (the first Betti number) of the surface. We construct such examples of dimension greater than two by studying an equivariant deformation of certain complex threefold, so called a twistor space. Received January 25, 2000 / Published online October 30, 2000     

Morse index of some min-max critical points. I. Application to multiplicity results

We give here a general result on lower bounds for Morse indices of critical points obtained by some min-max principles. Combining this information with a semi-classical inequality yields sharp estimates on the growth of some critical values, from which we deduce new multiplicity results for solutions of semi-linear second-order elliptic equations.     

Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds

We study some basic analytical problems for nonlinear Dirac equations involving critical Sobolev exponents on compact spin manifolds. Their solutions are obtained as critical points of certain strongly indefinite functionals defined on H1/2-spinors with critical growth. We prove the existence of a non-trivial solution for the Brezis-Nirenberg type problem when the dimension m of the manifold is larger than 3. We also prove a global compactness result for the associated Palais-Smale sequences and the regularity of -weak solutions.     

On based-free actions of compact lie groups on the hilbert cube

It is proved that a based-free action α of a given compact Lie groupG on the Hilbert cubeQ is equivalent to the standard based-free action σ if and only if the orbit spaceQ ₀/α of the free partQ ₀=Q* is aQ-manifold having the proper homotopy type of the orbit spaceQ ₀/σ. The existence of an equivariant retraction (Q ₀, σ)→(Q ₀, α) is established. It is proved that for any TikhonovG-spaceX the family of all equivariant mapsX→ conG separates the points and the closed sets inX. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 163–174, February, 1999.     

Unbounded derivations commuting with compact group actions. II

Let ($\text{A}$, G, α) be a $\text{C}{}^1$ dynamical system and let δ be a closed ¹ derivation in $\text{A}$ which commutes with α and satisfies $\text{A}$^G ? ker(δ). If $\text{A}$ is a separable Type I $\text{C}{}^1$ algebra and G is a second countable compact group, then δ generates a strongly continuous one parameter group of ¹ automorphisms of $\text{A}$.