Prof. Dr. Uwe Semmelmann
Researcher, Project leader

Universität Stuttgart
E-mail: uwe.semmelmann(at)mathematik.uni-stuttgart.de
Homepage: https://www.igt.uni-stuttgart.de/en/team…
Project
74Rigidity, stability and deformations in nearly parallel G2-geometry
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds
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):
74Rigidity, 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):
74Rigidity, 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):
74Rigidity, 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):
74Rigidity, 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):
74Rigidity, 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):
74Rigidity, 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):
74Rigidity, stability and deformations in nearly parallel G2-geometry