Publications

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):
55New hyperkähler spaces from the the self-duality equations56Large 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 \(\mathscr{N}_{0} = \mathscr{N}\).

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space69Wall-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.

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space69Wall-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.

Related project(s):
5Index 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):
9Diffeomorphisms and the topology of positive scalar curvature15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces 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.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces 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 \(\mathcal{R}^+(M)\).

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces 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.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces 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):
52Spaces 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):
75Solutions 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):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality 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):
55New hyperkähler spaces from the the self-duality equations56Large 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):
55New hyperkähler spaces from the the self-duality equations56Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space has a meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah--Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and its critical points are studied. Particular emphasis is given to the example of Deligne--Hitchin moduli spaces.

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

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n^{β+1}) where n is the size of the matrix and O(n^β) is the cost of multiplying n×n-matrices, β∈[2,2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $\sigma$ on an oriented surface $M$ equip the total space of a certain disk bundle $Z \to M$ with a pair $(J_{\mathfrak{p}},\mathfrak{J}_{\mathfrak{p},\sigma})$ of almost complex structures. A conformal structure on $M$ corresponds to a section of $Z\to M$ and $\mathfrak{p}$ is metrisable by the metric $g$ if and only if $[g] : M \to Z$ is a pseudo-holomorphic curve with respect to $J_{\mathfrak{p}}$ and $\mathfrak{J}_{\mathfrak{p},dA_g}$.

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

We construct a natural transformation between two versions of G-equivariant K-homology with coefficients in a G-C*-category for a countable discrete group G. Its domain is a coarse geometric K-homology and its target is the usual analytic K-homology. Following classical terminology, we call this transformation the Paschke transformation. We show that under certain finiteness assumptions on a G-space X, the Paschke transformation is an equivalence on X. As an application, we provide a direct comparison of the homotopy theoretic Davis–Lück assembly map with Kasparov’s analytic assembly map appearing in the Baum–Connes conjecture.

Related project(s):
45Macroscopic invariants of manifolds

In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.

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

This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability.

We construct a finitely generated, residually finite, amenable group A and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to A. All of these groups are branch groups.

Moreover, picking up  Grothendieck's problem, the group A embeds in these groups such that the inclusion induces an isomorphism of profinite completions.  

In addition, we review the concept of uniform amenability, a strengthening of amenability introduced in the 70's, and we prove that uniform amenability indeed is detectable from the profinite completion.

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

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing \(\widehat{\mathrm{A}}\)-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II