arXiv subjects
Publications and source records attributed to Uwe Semmelmann.
We show that a simply connected compact symmetric space of dimension $n\ge 3$ admits a conformal Killing $2$-form which is not Killing if and only if it is homothetic to one of $\mathbb{S}^n$, $\mathbb{CP}^m$ or $\mathbb{HP}^q$.
This survey deals with two closely connected topics: first, the stability of Einstein metrics under the Einstein-Hilbert functional, and second, their deformation theory and the study of the moduli space of Einstein metrics on a compact manifold. To first order, both problems reduce to studying the spectrum and eigentensors of the Lichnerowicz Laplacian. We give an introduction to the classical theory and survey recent results and advances.
We calculate the index and nullity of the three orientable focal manifolds of isoparametric hypersurfaces in spheres with three distinct principal curvatures. It turns out that the index is equal to the dimension of the ambient Euclidean space and the nullity is completely determined by the normal part of Killing vector fields of the ambient sphere. In that sense, the Veronese embeddings of the projective planes are as stable as possible for non totally geodesic submanifolds of the sphere.
We classify $7$-dimensional Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion whose holonomy is contained in $\mathrm{G}_2$, up to naturally reductive homogeneous spaces and nearly parallel $\mathrm{G}_2$-structures. This extends and completes the classification initiated by Th. Friedrich in the cocalibrated case. Incidentally, we also obtain the list of $\mathrm{SU}(3)$ geometries with parallel skew-symmetric torsion, up to naturally reductive homogeneous spaces and nearly Kähler manifolds.
In this article we study the stability problem for positive quaternion-Kähler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of symmetric 2-tensors. We also give improved eigenvalue estimates for the Hodge-Laplacian on 2-forms. On the parallel subbundle Sym^2 E of the 2-form bundle we prove a sharp lower bound for the first non-zero eigenvalue.
In this short note, we prove a global Pestov identity on the (orthonormal) frame bundle of a Riemannian manifold and deduce similar identities on associated homogeneous fibrations. As a particular example, this provides a concise proof of the Pestov identity on the unit tangent bundle of the manifold.
In this note, we characterise the existence of non-trivial invariant spinors on maximal flag manifolds associated to complex simple Lie algebras. This characterisation is based on the combinatorial properties of their set of positive roots. We also give some bounds for the dimension of the space of invariant spinors in each case.
Compact Hermitian symmetric spaces are Kähler manifolds with constant scalar curvature and non-negative sectional curvature. A famous result by A. Gray states that, conversely, a compact simply connected Kähler manifold with constant scalar curvature and non-negative sectional curvature is a Hermitian symmetric space. The aim of the present article is to transpose Gray's result to the quaternion-Kähler setting. In order to achieve this, we introduce the quaternionic sectional curvature of quaternion-Kähler manifolds, we show that every Wolf space has non-negative quaternionic sectional curvature, and we prove that, conversely, every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space. The proof makes crucial use of the nearly Kähler twistor spaces of positive quaternion-Kähler manifolds.
We study the deformability of the symmetric Einstein metrics on the spaces $\mathrm{SU}(n)/\mathrm{SO}(n)$ and $\mathrm{SU}(2n)/\mathrm{Sp}(n)$, thereby concluding the problem to second order for all irreducible symmetric spaces. The obstruction integrals are calculated from invariant polynomials on certain Lie algebra representations. To aid the computation, we develop so-called sandwich operators for compact Lie algebras and relate them to quadratic Casimir operators. We also explain the source of the infinitesimal Einstein deformations on irreducible symmetric spaces, except for the complex Grassmannians, by exploring their relation to simple Jordan algebras. As an application we prove the nonlinear instability of most of the infinitesimally deformable irreducible compact symmetric spaces.
We show that every quaternion-Kähler manifold of negative scalar curvature is stable as an Einstein manifold and therefore scalar curvature rigid. In particular, this implies that every irreducible nonpositive Einstein manifold of special holonomy is stable. In contrast, we demonstrate that there exist quaternion-Kähler manifolds of positive scalar curvature which are not scalar curvature rigid even though they are semi-stable.
It is known that the frame flow on a closed $n$-dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ is odd and $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n \neq 8, 134$, we show that: if $n \equiv 2$ mod $4$ or $n=4$, the frame flow is ergodic if the manifold is $\sim 0.3$-pinched, if $n \equiv 0$ mod $4$, it is ergodic if the manifold is $\sim 0.6$-pinched. In the three dimensions $n=7,8,134$, the respective pinching bounds that we need in order to prove ergodicity are $0.4962...$, $0.6212...$, and $0.5788...$. This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that $0.25$-pinched even-dimensional manifolds have an ergodic frame flow.
In this article we study the concept of quaternionic bisectional curvature introduced by B. Chow and D. Yang for quaternion-Kähler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space. We also show that all symmetric quaternion-Kähler manifolds different from the quaternionic projective space admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact Kähler manifolds of non-negative sectional curvature.
We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension $3$ turns out to be purely algebraic. One of our main results is the complete and explicit description of infinitesimal Einstein deformation integrable to second order on the complex $2$-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $ \mathrm{Gr}_2(\bbC^{n+2})$ for $n$ odd is rigid.
We study the integrability to second order of the infinitesimal Einstein deformations of the symmetric metric $g$ on the complex Grassmannian of $k$-planes inside $\mathbb{C}^n$. By showing the nonvanishing of Koiso's obstruction polynomial, we characterize the infinitesimal deformations that are integrable to second order as an explicit variety inside $\mathfrak{su}(n)$. In particular we show that $g$ is isolated in the moduli space of Einstein metrics if $n$ is odd.
The homogeneous minimal hypersurfaces in $S^n$ have $g = 1,2,3,4$, or $6$ distinct (constant) principal curvatures. While the Morse index and nullity have been calculated for all such hypersurfaces having $g = 1,2,3$, it has remained an open problem to compute these quantities for any of those with $g = 4$ or $6$. In this paper, we calculate the Morse index and nullity of two homogeneous minimal hypersurfaces in $S^n$ with $g = 4$. Moreover, we observe that their Laplace spectra contain irrational eigenvalues that are not expressible in radicals.
We initiate a systematic study of the deformation theory of the second Einstein metric $g_{1/\sqrt{5}}$ respectively the proper nearly $G_2$ structure $φ_{1/\sqrt{5}}$ of a $3$-Sasaki manifold $(M^7,g)$. We show that infinitesimal Einstein deformations for $g_{1/\sqrt{5}}$ coincide with infinitesimal $G_2$ deformations for $φ_{1/\sqrt{5}}$. The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of $g$, with eigenvalue twice the Einstein constant of the base $4$-dimensional orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal $G_2$ deformations which are unobstructed to second order.
We prove that the normal metric on the homogeneous space $E_7/\mathrm{PSO}(8)$ is stable with respect to the Einstein-Hilbert action, thereby exhibiting the first known example of a non-symmetric metric of positive scalar curvature with this property.
We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving Lichnerowicz-Obata type estimates by Ivanov et al. The limiting eigenspace is fully decribed in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz-Matsushima estimate for Kähler-Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun, in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.