## 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 discuss a method to construct Dirac-harmonic maps developed by J. Jost, X. Mo and M. Zhu. The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in the publication by Jost, Mo and Zhu were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions. In several cases we show the existence of solutions.

Journal | Lett. Math. Phys. |

Link to preprint version | |

Link to published version |

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f from a manifold X to the product of the k-sphere with the n-k dimensional torus, with k=1,2,3. When X is a closed oriented manifold endowed with a metric g of positive scalar curvature and the map f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g and the contracting factor of the map f.

Journal | Proc. AMS |

Publisher | AMS |

Link to preprint version | |

Link to published version |

**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature

In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger, and Yu. To take the fundamental groups of the manifold and its boundary into account requires working with maximal C* completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions.

**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature

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. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on 3-manifolds.

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

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds can not be established.

Journal | to appear in Proceedings of the American Mathematical Society |

Link to preprint version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor'yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so called refinements.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this expository paper we answer two fundamental questions concerning discrete magnetic Schrödinger operator associated with weighted graphs. We discuss when formal expressions of such operators give rise to self-adjoint operators, i.e., when they have self-adjoint restrictions. If such self-adjoint restrictions exist, we explore when they are unique.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We study pairs of Dirichlet forms related by an intertwining order

isomorphisms between the associated \(L^2\)-spaces. We consider the

measurable, the topological and the geometric setting respectively.

In the measurable setting, we deal with arbitrary (irreducible)

Dirichlet forms and show that any intertwining order isomorphism is

necessarily unitary (up to a constant). In the topological setting

we deal with quasi-regular forms and show that any intertwining

order isomorphism induces a quasi-homeomorphism between the

underlying spaces. In the geometric setting we deal with both

regular Dirichlet forms as well as resistance forms and essentially

show that the geometry defined by these forms is preserved by

intertwining order isomorphisms. In particular, we prove in the

strongly local regular case that intertwining order isomorphisms

induce isometries with respect to the intrinsic metrics between the

underlying spaces under fairly mild assumptions. This applies to a

wide variety of metric measure spaces including

\(\mathrm{RCD}(K,N)\)-spaces, complete weighted Riemannian manifolds

and complete quantum graphs. In the non-local regular case our

results cover in particular graphs as well as fractional Laplacians

as arising in the treatment of \(\alpha\)-stable Lévy processes. For

resistance forms we show that intertwining order isomorphisms are

isometries with respect to the resistance metrics.

Our results can can be understood as saying that diffusion always

determines the Hilbert space, and -- under natural compatibility

assumptions -- the topology and the geometry respectively. As special

instances they cover earlier results for manifolds and graphs.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We study topological Poincaré type inequalities on generalgraphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory ofLaplacians once a probability measure on the graph is chosen. More specifically,we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We show that a properly convex projective structure \(\mathfrak{p}\) on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if \(\mathfrak{p}\) is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that \(\mathfrak{p}\) admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable \(L^2\)-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

**Related project(s):****26**Projective surfaces, Segre structures and the Hitchin component for PSL(n,R)

We construct $\eta$- and $\rho$-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

Journal | Math. Proc. Camb. Philos. Soc. |

Publisher | Cambridge University Press |

Volume | August 2018 |

Link to preprint version | |

Link to published version |

**Related project(s):****4**Secondary invariants for foliations

For a closed, connected direct product Riemannian manifold (M,g)=(M1×⋯×Ml,g1⊕⋯⊕gl), we define its multiconformal class [[g]] as the totality {f12g1⊕⋯⊕fl2gl} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a function fi2>0 on the total space M. A multiconformal class [[g]] contains not only all warped product type deformations of g but also the whole conformal class [g~] of every g~∈[[g]]. In this article, we prove that [[g]] carries a metric of positive scalar curvature if and only if the conformal class of some factor (Mi,gi) does, under the technical assumption dimMi≥2. We also show that, even in the case where every factor (Mi,gi) has positive scalar curvature, [[g]] carries a metric of scalar curvature constantly equal to −1 and with arbitrarily large volume, provided l≥2 and dimM≥3. In this case, such negative scalar curvature metrics within [[g]] for l=2 cannot be of any warped product type.

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

We study the behavior of the spectrum of the Dirac operator together with a symmetric W1,∞-potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or pin− structure on the limit space N, then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric W1,∞-potential. In the case of an orientable limit space N, D is the spin Dirac operator DN on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then D=DN⊕−DN.

Journal | Manuscripta Mathematica |

Publisher | Springer |

Pages | 1-24 |

Link to preprint version | |

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

We consider pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators, which are defined on the product manifold $M_1 \times M_2$ for closed manifolds $M_1$ and $M_2$. We prove a topological index theorem for Fredholm operators of tensor product type. To this end we construct a suitable double deformation groupoid and prove a Poincaré duality type result in relative $K$-theory.

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

We extend two known existence results to simply connected manifolds with

positive sectional curvature: we show that there exist pairs of simply

connected positively-curved manifolds that are tangentially homotopy equivalent

but not homeomorphic, and we deduce that an open manifold may admit a pair of

non-homeomorphic simply connected and positively-curved souls. Examples of such

pairs are given by explicit pairs of Eschenburg spaces. To deduce the second

statement from the first, we extend our earlier work on the stable converse

soul question and show that it has a positive answer for a class of spaces that

includes all Eschenburg spaces.

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Link to preprint version |

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

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen [2016], of Miao and Xie [2016], and of joint work of Miao and the authors [2017], we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

Journal | J. Geom. Phys. |

Publisher | Elsevier |

Volume | 132 |

Pages | 338--357 |

Link to preprint version | |

Link to published version |

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

We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a covariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau. We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincaré group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau which were described by Cederbaum and Nerz.

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

We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps.

As an application, we prove Fary-Milnor's theorem in the CAT(0) setting.

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

A surface which does not admit a length nonincreasing deformation is called metric minimizing. We show that metric minimizing surfaces in CAT(0) spaces are locally CAT(0) with respect to their intrinsic metric.

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