The project will concentrate on the following three main topics:

A) Stability of Einstein metrics.

B) Rigidity of nearly parallel G2-structures.

C) Deformations of associative submanifolds

## Publications

We prove that the normal metric on the homogeneous space E7/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.

**Related project(s):****74**Rigidity, stability and deformations in nearly parallel G2-geometry

In this article we study the stability problem for the Einstein metrics on

Sasaki Einstein and on complete nearly parallel G2 manifolds. In the Sasaki

case we show linear instability if the second Betti number is positive.

Similarly we prove that nearly parallel G2 manifolds with positive third

Betti number are linearly unstable. Moreover, we prove linear instability

for the Berger space SO(5)/SO(3)_irr which is a 7-dimensional homology

sphere with a proper nearly parallel G2 structure.

Journal | to appear in Int. J. Math. |

Link to preprint version |

**Related project(s):****74**Rigidity, stability and deformations in nearly parallel G2-geometry

In this article we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation, and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

Journal | J. Geom. Anal. (2022) 32:137 |

Link to preprint version |

**Related project(s):****74**Rigidity, stability and deformations in nearly parallel G2-geometry

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.

**Related project(s):****74**Rigidity, stability and deformations in nearly parallel G2-geometry

We initiate a systematic study of the deformation theory of the second Einstein

metric \(g_{1/\sqrt{5}}\) respectively the proper nearly G2 structure \(\phi_{1/\sqrt{5}}\) 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 \(\phi_{1/\sqrt{5}}\). 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.

**Related project(s):****74**Rigidity, stability and deformations in nearly parallel G2-geometry

We describe the second order obstruction to deformation for nearly G_2 structures on compact manifolds. Building on work of B. Alexandrov and U. Semmelmann this allows proving rigidity under deformation for the proper nearly G_2 structure on the Aloff-Wallach space N(1,1).

Journal | J. London Math. Soc. (2) 104 (2021) 1795--1811 |

Link to preprint version |

**Related project(s):****74**Rigidity, stability and deformations in nearly parallel G2-geometry

We show that a strict, nearly Kähler 6-manifold with either second or third Betti number nonzero is linearly unstable with respect to the \(\nu\)-entropy of Perelman and hence is dynamically unstable for the Ricci flow.

Journal | Ann. Global Anal. Geom. 57 no. 1, 15-22 (2020) |

Link to preprint version |

**Related project(s):****74**Rigidity, stability and deformations in nearly parallel G2-geometry

## Team Members

**M.Sc. Paul Schwahn**

Doctoral student

Universität Stuttgart

paul.schwahn(at)mathematik.uni-stuttgart.de

**Prof. Dr. Uwe Semmelmann**

Researcher,
Project leader

Universität Stuttgart

uwe.semmelmann(at)mathematik.uni-stuttgart.de

## Former Members

**Dr. Paul-Andi Nagy**

Researcher

Universität Stuttgart