## 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 compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold $(M,g)$. We conclude under which conditions on $(M,g)$, the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular Einstein manifold

Journal | J. Funct. Anal. |

Volume | 281 |

Pages | 109115 |

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**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**64**Spectral geometry, index theory and geometric flows on singular spaces II

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 present the Laplace operator associated to a hyperbolic surface \(\Gamma\backslash\mathbb{H}\) and a unitary representation of the fundamental group \(\Gamma\), extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of \(\mathbb{C}\) by constructing a parametrix for the Laplacian, following the approach by Guillopé and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.

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

In 2015, Mantoulidis and Schoen constructed 3-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be ``far away'' from being round. The resulting manifolds, called *extensions*, are geometrically not ``close'' to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality. Their construction was later adapted to n+1 dimensions by Cabrera Pacheco and Miao, suggesting instability of the higher dimensional Riemannian Penrose Inequality. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum and by Cabrera Pacheco, Cederbaum, and McCormick, a similar construction was performed for asymptotically Euclidean, electrically charged initial data sets and for asymptotically hyperbolic Riemannian manifolds, respectively, obtaining 3-dimensional extensions that suggest instability of the Riemannian Penrose Inequality with electric charge and of the conjectured asymptotically hyperbolic Riemannian Penrose Inequality in 3 dimensions. This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in n+1 dimensions for n greater or equal to 2.

Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.

**Related project(s):****40**Construction of Riemannian manifolds with scalar curvature constraints and applications to general relativity**41**Geometrically defined asymptotic coordinates in general relativity

We use the theory of Gaiotto, Moore and Neitzke to construct a set of Darboux coordinates on the moduli space \(\mathcal{M}\) of weakly parabolic \(SL(2,\mathbb{C})\)-Higgs bundles. For generic Higgs bundles\((\mathcal{E},R\Phi)\) with \(R\gg 0\) the coordinates are shown to be dominated by a leading term that is given by the coordinates for a corresponding simpler space of limiting configurations and we prove that the deviation from the limiting term is given by a remainder that is exponentially suppressed in \(R\).

We then use this result to solve an associated Riemann-Hilbert problem and construct a twistorial hyperkähler metric \(g_{\text{twist}}\) on \(\mathcal{M}\). Comparing this metric to the simpler semiflat metric \(g_{\text{sf}}\), we show that their difference is \(g_{\text{twist}}-g_{\text{sf}}=O\left(e^{-\mu R}\right)\), where \(\mu\) is a minimal period of the determinant of the Higgs field.

**Related project(s):****77**Asymptotic geometry of the Higgs bundle moduli space II

We prove the existence of a pair $(\Sigma ,\, \Gamma)$, where $\Sigma$ is a compact Riemann surface with $\text{genus}(\Sigma)\, \geq\, 2$, and $\Gamma\, \subset\, {\rm SL}(2, \C)$ is a cocompact lattice, such that there is a generically injective holomorphic map $\Sigma \, \longrightarrow\, {\rm SL}(2, \C)/\Gamma$. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann \cite{HW} and by Ghys \cite{Gh}.

**Related project(s):****55**New hyperkähler spaces from the the self-duality equations**56**Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

Moduli spaces of stable parabolic bundles of parabolic degree \(0\) over the Riemann sphere are stratified according to the Harder-Narasimhan filtration of underlying vector bundles. Over a Zariski open subset \(\mathscr{N}_{0}\) of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function \(\mathscr{S}\) is defined as the regularized critical value of the non-compact Wess-Zumino-Novikov-Witten action functional. The definition of \(\mathscr{S}\) depends on a suitable notion of parabolic bundle 'uniformization map' following from the Mehta-Seshadri and Birkhoff-Grothendieck theorems. It is shown that \(-\mathscr{S}\) is a primitive for a (1,0)-form \(\vartheta\) on \(\mathscr{N}_{0}\) associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that \(-\mathscr{S}\) is a Kähler potential for \((\Omega-\Omega_{\mathrm{T}})|_{\mathscr{N}_{0}}\), where \(\Omega\) is the Narasimhan-Atiyah-Bott Kähler form in \(\mathscr{N}\) and \(\Omega_{\mathrm{T}}\) is a certain linear combination of tautological \((1,1)\)-forms associated with the marked points. These results provide an explicit relation between the cohomology class \([\Omega]\) and tautological classes, which holds globally over certain open chambers of parabolic weights where \(\mathscr{N}_{0} = \mathscr{N}\).

Journal | Commun. Math. Phys. |

Publisher | Springer |

Volume | 387, no. 2 |

Pages | 649–680 |

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**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space**69**Wall-crossing and hyperkähler geometry of moduli spaces

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of such approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations --- such as the so-called thin homotopy --- and the resulting interpretation of gauge fields as group homomorphisms to a Lie group *G* satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal *G*-bundle. We also prove several structural results on thin homotopy, and in particular we clarify the difference between thin equivalence and retrace equivalence for piecewise-smooth based loops on a smooth manifold, which are often used interchangeably in the physics literature. We conclude by listing a set of questions on topological and functional analytic aspects of groups of based loops, which we consider to be fundamental to establish a rigorous differential geometric foundation of the holonomy formulation of gauge theory.

Journal | Contemp. Math. |

Publisher | Amer. Math. Soc. |

Volume | 775 |

Pages | 233–253 |

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**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space**69**Wall-crossing and hyperkähler geometry of moduli spaces

The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions.

In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kähler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.

Journal | Commun. Math. Phys. |

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

Given a manifold $M$, we completely determine which rational $\kappa$-classes are non-trivial for (fiber homotopy trivial) $M$-bundles over the $k$-sphere, provided that the dimension of $M$ is large compared to $k$. We furthermore study the vector space of these spherical $\kappa$-classes and give an upper and a lower bound on its dimension. The proof is based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory and we make use of ideas developed by Krannich--Kupers--Randal-Williams.

As an application, we show the existence of elements of infinite order in the homotopy groups of the spaces $\mathcal R_{\mathrm{Ric}>0}(M)$ and $\mathcal R_{\mathrm{sec}>0}(M)$ of positive Ricci and positive sectional curvature, provided that $M$ is $\mathrm{Spin}$, has a non-trivial rational Pontryagin class and admits such a metric. This is done by showing that the $\kappa$-class associated to the $\hat{\mathcal A}$-class is spherical for such a manifold.

In the appendix co-authored by Jens Reinhold it is (partially) determined which classes of the rational oriented cobordism ring contain an element that fibers over a sphere of a given dimension.

**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds**52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

In this paper we study spaces of Riemannian metrics with lower bounds on intermedi- ate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

Journal | Geometriae Dedicata |

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**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds**52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We construct and study an H-space multiplication on \(\mathcal{R}^+(M)\) for manifolds M which are nullcobordant in their own tangential 2-type. This is applied to give a rigidity criterion for the action of the diffeomorphism group on \(\mathcal{R}^+(M)\) via pullback. We also compare this to other known multiplicative structures on \(\mathcal{R}^+(M)\).

Journal | Transactions of the AMS |

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**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds**52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We present a rigidity theorem for the action of the mapping class group \(\pi_0(\mathrm{Diff}(M))\)on the space \(\mathcal{R}^+(M)\) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional Spin-manifolds.

Journal | Mathematische Annalen |

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**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds**52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

After R. Schoen completed the solution of the Yamabe problem, compact manifolds could be allocated in three classes depending on whether they admit a metric with positive, non-negative or only negative scalar curvature. Here we follow Yamabe's first attempt to solve his problem through variational methods and provide an analogous equivalent classification for manifolds equipped with actions by non-discrete compact Lie groups. Moreover, we apply the method, and the results to classify total spaces of fibre bundles with compact structure groups (concerning scalar curvature), to conclude density results, and compare realizable scalar curvature functions between some exotic manifolds their standard counterpart. We also provide an extended range of prescribed scalar curvature functions of warped products, especially with Calabi--Yau manifolds, providing an upper bound for the first positive eigenvalue of the Laplacian under relatively mild conditions.

**Related project(s):****52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

In this paper we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values \(g\) are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to \(W^{2,2}\)(M) and satisfy \(\frac{1}{a}h \leq g \leq ah\) for some \(1<a<\infty\) and some smooth Riemannian metric \(h\) on M. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci DeTurck solution. Results for a related non-compact setting are also presented. Various \(L^p\) estimates for Ricci flow, which we require for some of the main results, are also derived. As an application we present a possible definition of scalar curvature \(\geq k\) for \(W^{2,2}\)(M) metrics \(g\) on closed four manifolds which are bounded in the \(L^{\infty}\) sense by \(\frac{1}{a}h \leq g \leq ah\) for some \(1<a<\infty\) and some smooth Riemannian metric \(h\) on M.

**Related project(s):****75**Solutions to Ricci flow whose scalar curvature is bounded in L^p II

Let \(\mathcal{E}\) be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold \((M,g)\). We show that if \(\mathcal{E}\) has negative ADM-mass, then there exists a constant \(R > 0\), depending only on \(\mathcal{E}\), such that \(M\) must become incomplete or have a point of negative scalar curvature in the \(R\)-neighborhood around \(\mathcal{E}\) in \(M\). This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions \(\leq 7\) assuming Schwarzschild asymptotics on the end \(\mathcal{E}\). We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end \(\mathcal{E}\). Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.

**Related project(s):****42**Spin obstructions to metrics of positive scalar curvature on nonspin manifolds**78**Duality and the coarse assembly map II

For every $g \gg 1$, we show the existence of a complete and smooth family of closed constant mean curvature surfaces $f_\varphi^g,$ $ \varphi \in [0, \tfrac{\pi}{2}],$ in the round $3$-sphere deforming the Lawson surface $\xi_{1, g}$ to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To do so we use an implicit function theorem argument in the parameter $s= \tfrac{1}{2(g+1)}$. This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of $f_\varphi^g$ at $s= 0$ explicitly. In particular, we obtain for large genus Lawson surfaces $\xi_{1,g}$, due to the real analytic dependence of its area and DPW potential on $s,$ a scheme to explicitly compute the coefficients of the power series in $s$ in terms of multilogarithms. Remarkably, the third order coefficient of the area expansion coincides numerically with $\tfrac{9}{4}\zeta(3),$ where $\zeta$ is the Riemann $\zeta$ function (while the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}).

**Related project(s):****55**New hyperkähler spaces from the the self-duality equations**56**Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. This also answers a question of \cite{CDHL}. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on $\Sigma$, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

**Related project(s):****55**New hyperkähler spaces from the the self-duality equations**56**Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere