## Publications of SPP2026

On this site you find preprints and publications produced within the projects and with the support of the DFG priority programme „Geometry at Infinity“.

We show that the well-known family of $2$-lobed Delaunay tori $\;f^b\;$ in $\;S^3,\;$ parametrized by $\;b \in \mathbb R_{\geq1},\;$ uniquely minimizes the Willmore energy among all immersions from tori into $3$-space of conformal class $\;(a, b)\;$. As a corollary we obtain an alternate proof of the Willmore conjecture in $3$-space. This new strategy can be generalized to arbitrary codimensions provided a classification of isothermic constrained Willmore tori is possible and all $\;f^b\;$ remain stable in all codimensions.

**Related project(s):****16**Minimizer of the Willmore energy with prescribed rectangular conformal class

Let *N* be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold *M*. In this paper, we study the extent to which *N* admits as much symmetry as *M*. Our main results are examples of *N* that exhibit two extremes of behavior. On the one hand, we find *N* with maximal symmetry, i.e. Isom(*M*) acts on *N* by isometries with respect to some negatively curved metric on *N*. For these examples, Isom(*M*) can be made arbitrarily large. On the other hand, we find *N* with little symmetry, i.e. no subgroup of Isom(*M*) of "small" index acts by diffeomorphisms of *N*. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This survey is succinct rather than comprehensive, and its aim is to inspire geometers and analysts alike to study these methods so that they can be adapted and potentially applied more widely.

**Related project(s):****5**Index theory on Lorentzian manifolds

The purpose of this article is to define and study new invariants of topological spaces: the *p*-adic Betti numbers and the *p*-adic torsion. These invariants take values in the *p*-adic numbers and are constructed from a virtual pro-*p* completion of the fundamental group. The key result of the article is an approximation theorem which shows that the *p*-adic invariants are limits of their classical analogues. This is reminiscent of Lück's approximation theorem for *L*2-Betti numbers.

After an investigation of basic properties and examples we discuss the *p*-adic analog of the Atiyah conjecture: When do the *p*-adic Betti numbers take integer values? We establish this property for a class of spaces and discuss applications to cohomology growth.

**Related project(s):****18**Analytic L2-invariants of non-positively curved spaces

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting.

**Related project(s):****28**Rigidity, deformations and limits of maximal representations

Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$.

We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \geqslant 6$) and of $\mathrm{SL}_n(\mathbb Z)$ (with $n \geqslant 3$) with respect to natural generating sets.

In the latter case, these bounds improve upon previously known lower bounds whenever $n> 6$.

**Related project(s):****8**Parabolics and invariants

We show a Heinz-Kato inequality in Banach spaces for sectorial operators having bounded imaginary powers.

**Related project(s):****30**Nonlinear evolution equations on singular manifolds

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.

Journal | Int. Math. Res. Not. IMRN |

Volume | 18 |

Pages | 5786--5822 |

Link to preprint version | |

Link to published version |

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected 5-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension 4 are connected sums of *\(S^3\)*-bundles over \(S^2\). Furthermore, the Betti numbers of the 5-manifolds and of the quotient 4-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free \(S^3\) actions on simply connected 8-manifolds with quotient a 5-manifold and show, in particular, that the Pontrjagin classes, the \(\hat A\) -genus and the signature of the 8-manifold must all necessarily vanish.

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces

Journal | Geom. Topol. |

Volume | 22 |

Pages | 591-644 |

Link to preprint version |

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

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.

Journal | Acta Math. |

Volume | 221 |

Pages | 159-202 |

Link to preprint version | |

Link to published version |

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

We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.

Journal | Math. Ann. |

Volume | Online First |

Link to preprint version |

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

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

Journal | Geom. Funct. Anal. |

Volume | To appear |

Link to preprint version |

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

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.

**Related project(s):****27**Invariants and boundaries of spaces

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

**Related project(s):****5**Index theory on Lorentzian manifolds

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of Higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball, and we compute its dimension explicitly. For example, the Hitchin component of the right-angled hyperbolic \(\ell\)-polygon reflection group into \(\mathrm{PGL}(2m,\mathbb{R})\), resp. \(\mathrm{PGL}(2m+1,\mathbb{R})\), is homeomorphic to an open ball of dimension \((\ell-4)m^2+1\), resp. \((\ell-4)(m^2+m)\). We also give applications to the study of the pressure metric and the deformation theory of real projective structures on 3-manifolds.

**Related project(s):****1**Hitchin components for orbifolds

Geometric structures on manifolds became popular when Thurston used them in his work on the Geometrization Conjecture. They were studied by many people and they play an important role in Higher Teichmüller Theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Thesis that this technique can be used to construct geometric structures in interesting cases. Here we will survey some recent results in this direction, which are joint work with Qiongling Li.

**Related project(s):****1**Hitchin components for orbifolds

Let X be a compact manifold, D a real elliptic operator on X, G a Lie group, P a principal G-bundle on X, and B_P the infinite-dimensional moduli space of all connections on P modulo gauge, as a topological stack. For each connection \nabla_P, we can consider the twisted elliptic operator on X. This is a continuous family of elliptic operators over the base B_P, and so has an orientation bundle O^D_P over B_P, a principal Z_2-bundle parametrizing orientations of KerD^\nabla_Ad(P) + CokerD^\nabla_Ad(P) at each \nabla_P. An orientation on (B_P,D) is a trivialization of O^D_P.

In gauge theory one studies moduli spaces M of connections \nabla_P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X, g). Under good conditions M is a smooth manifold, and orientations on (B_P,D) pull back to

orientations on M in the usual sense of differential geometry.

This is important in areas such as Donaldson theory, where one needs an orientation on M

to define enumerative invariants.

We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (B_P,D), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds,

instantons, the Kapustin-Witten equations, and the Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds.

**Related project(s):****33**Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

Suppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator D. Let G be SU(m) or U(m), and E be a rank m complex bundle with G-structure on X. Write B_E for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z_2-bundle O^D_E on B_E parametrizing orientations of det D_Ad A for twisted elliptic operators D_Ad A at each [A] in B_E. A theorem of Walpuski shows O^D_E is trivializable.

We prove that if we choose an orientation for det D, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of O^D_E for all such bundles E on X, satisfying natural compatibilities.

Now let (X,\varphi,g) be a compact G_2-manifold, with d(*\varphi)=0. Then we can consider moduli spaces M_E^G_2 of G_2-instantons on E over X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general. The restriction of O^D_E to M_E^G_2 is the Z_2-bundle of orientations on M_E^G_2. Thus, our theorem induces canonical orientations on all such G_2-instanton moduli spaces M_E^G_2.

This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of G_2-manifolds (X,\varphi,g) by counting moduli spaces M_E^G_2, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6.

**Related project(s):****33**Gerbes in renormalization and quantization of infinite-dimensional moduli spaces