Publications

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.

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.

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

We study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillopé and Zworski, we establish a factorization formula for the twisted scattering determinant and describe the behavior of the scattering matrix in a neighborhood of \(1/2\).

Related project(s):
70Spectral theory with non-unitary twists

Using non-Abelian Hodge theory for parabolic Higgs bundles,

we construct infinitely many non-congruent  hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in ${\bf SL}_3(\Z)$. These give rise to non-isometric semi-flat Calabi--Yau metrics on special Lagrangian torus bundles over an open ball in $\R^{3}$  with a Y-vertex deleted, thereby answering a question raised by Loftin, Yau, and Zaslow.

Related project(s):
55New hyperkähler spaces from the the self-duality equations

The Lawson surfaces $\xi_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon $\Gamma_t$ along its boundary, where $t= \tfrac{1}{2g+2}$ is an angle of  $\Gamma_t$.  In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential $\eta_t$ for every $f_t$ with $t\in(0, \tfrac{1}{4}]$. Moreover, the coefficients of $\eta_t$ are shown to depend real analytically on $t$. This implies that the Taylor approximation of the DPW potential $\eta_t$ and of the area obtained at $t=0$ found in \cite{HHT2} determines these quantities for  all $\xi_{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces $\xi_{1,g}$.

Related project(s):
55New hyperkähler spaces from the the self-duality equations56Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp(n,R), G2(2), E8(C), F4(C), and G2(C) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

Related project(s):
58Profinite perspectives on l2-cohomology

This article presents a method for proving upper bounds for the first \(\ell^2\)-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first \(\ell^2\)-Betti number.

Our approach extends to generalizations of \(\ell^2\)-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

Related project(s):
58Profinite perspectives on l2-cohomology

We provide a posteriori error estimates in the energy norm for temporal semi-discretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak-strong stability estimates which we combine with suitable reconstructions of the numerical solution. We present time-adaptive numerical simulations based on the a posteriori error estimators for solutions involving blow-up.

Related project(s):
67Asymptotics of singularities and deformations

This is the second in a two part series of papers concerning Morse quasiflats - higher dimensional analogs of Morse quasigeodesics. Our focus here is on their asymptotic structure. In metric spaces with convex geodesic bicombings, we prove asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity for Morse quasiflats. Moreover, we provide some immediate consequences.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We analyze weak convergence on CAT(0) spaces and the existence and properties of corresponding weak topologies.

Related project(s):
66Minimal surfaces in metric spaces II

We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.

Related project(s):
66Minimal surfaces in metric spaces II

We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.

Related project(s):
66Minimal surfaces in metric spaces II

We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

Related project(s):
66Minimal surfaces in metric spaces II

We prove that a topological 4-manifold of globally non-positive curvature is homeomorphic to Euclidean space.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-diffeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-diffeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

Let (N,h) be a time- and space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of N with induced Riemannian metric g and second fundamental form K. If (N,h) satisfies the dominant energy condition in a strict sense, then the Dirac-Witten operator of M⊂ N is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial data pairs on $M$ satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's α-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac-Witten operator may be non-invertible, and we will study the kernel of this operator in this case.

We will show that the kernel may only be non-trivial if π1(M) is virtually solvable of derived length at most 2. This allows to extend the index theoretical methods to spaces of initial data pairs, satisfying the dominant energy condition in the weak sense.

We will show further that a spinor φ is in the kernel of the Dirac--Witten operator on (M,g,K) if and only if (M,g,K,φ) admits an extension to a Lorentzian manifold (N,h) with parallel spinor ψ such that M is a Cauchy hypersurface of (N,h), such that g and K are the induced metric and second fundamental form of M, respectively, and φ is the restriction of ψ to M.

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Related project(s):
35Geometric operators on singular domains

We provide a direct connection between the Zmax (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank 2.

We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the Zmax-JSJ decomposition of such groups.

Related project(s):
60Property (T)

We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise, and are intricately related to, groups with no Baumslag-Solitar subgroups. Indeed, for groups of cohomological dimension 2 we prove that algebraic hyperbolicity is equivalent to containing no Baumslag-Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension 2.

We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.

Related project(s):
60Property (T)