Torus manifolds with non-abelian symmetries

Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) with (G)-action then the action of a finite covering group of (G) factors through (\tilde{G}=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod SO(2l_i)\times T^{l_0}). The action of (\tilde{G}) on (M) restricts to an action of (\tilde{G}'=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod U(l_i)\times T^{l_0}) which has the same orbits as the (\tilde{G})-action. We define invariants of torus manifolds with (G)-action which determine their (\tilde{G}')-equivariant diffeomorphism type. We call these invariants admissible 5-tuples. A simply connected torus manifold with (G)-action is determined by its admissible 5-tuple up to (\tilde{G})-equivariant diffeomorphism. Furthermore we prove that all admissible 5-tuples may be realised by torus manifolds with (\tilde{G}")-action where (\tilde{G}") is a finite covering group of (\tilde{G}').Comment: 56 pages; a mistake in section 6 corrected; accepted for publication in Trans. Am. Math. So

Non-abelian symmetries of quasitoric manifolds

A quasitoric manifold $M$ is a $2n$-dimensional manifold which admits an action of an $n$-dimensional torus which has some nice properties. We determine the isomorphism type of a maximal compact connected Lie-subgroup $G$ of $\text{Homeo}(M)$ which contains the torus. Moreover, we show that this group is unique up to conjugation.Comment: 14 pages; presentation improved; accepted for publication in Muenster J. of Mat

Circle actions and scalar curvature

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metrics of positive scalar curvature on many manifolds with circle actions. Results from equivariant bordism allow us to show that there is an invariant metric of positive scalar curvature on the connected sum of two copies of a simply connected semi-free $S^1$-manifold $M$ of dimension at least six provided that $M$ is not $\text{spin}$ or that $M$ is $\text{spin}$ and the $S^1$-action is of odd type. If $M$ is spin and the $S^1$-action of even type then there is a $k>0$ such that the equivariant connected sum of $2^k$ copies of $M$ admits an invariant metric of positive scalar curvature if and only if a generalized $\hat{A}$-genus of $M/S^1$ vanishes.Comment: 25 pages; several changes according to comments of a referee made; to appear in Trans. Am. Math. So

Positively curved GKM-manifolds

Let T be a torus of dimension at least k and M a T-manifold. M is a GKM_k-manifold if the action is equivariantly formal, has only isolated fixed points, and any k weights of the isotropy representation in the fixed points are linearly independent. In this paper we compute the cohomology rings with real and integer coefficients of GKM_3- and GKM_4-manifolds which admit invariant metrics of positive sectional curvature.Comment: 19 pages, 7 figures. Final version, to appear in IMR

Differentiable stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.Comment: 13 pages; final version; accepted for publication in Comm. Anal. Geo