Publications

We prove large-time Gaussian upper bounds for continuous-time heat kernels of Laplacians on graphs with unbounded geometry. Our estimates hold for centers of large balls satisfying a Sobolev inequality and volume doubling. Distances are measured with respect to an intrinsic metric with finite distance balls and finite jump size. The Gaussian decay is given by Davies’ function which is natural and sharp in the graph setting. Furthermore, we find a new polynomial correction term which does not blow up at zero. Although our main focus is on unbounded Laplacians, the results are new even for the normalized Laplacian. In the case of unbounded vertex degree or degenerating measure, the estimates are affected by new error terms reflecting the unboundedness of the geometry.

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

A singular foliation \(\mathcal{F}\) on a complete Riemannian manifold \(M\) is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of M into orbits of an isometric action. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRF with special properties, e.g. isoparametric foliations, SRF on fibers bundles with Sasaki metric, and orbit-like foliations. More precisely, we prove two results analogous to Palais' Principle of Symmetric Criticality, one is a general principle for \(\mathcal{F}\) symmetric operators on the Hilbert space \(W^{1,2}(M)\), the other one is for \(\mathcal{F}\) symmetric integral operators on the Banach spaces \(W^{1,p}(M)\). These results together with a \(\mathcal{F}\) version of Rellich Kondrachov Hebey Vaugon Embedding Theorem allow us to circumvent difficulties with Sobolev's critical exponents when considering applications of Calculus of Variations to find solutions to PDEs. To exemplify this we prove the existence of weak solutions to a class of variational problems which includes \(p\)-Kirschoff problems.

Related project(s):
43Singular Riemannian foliations and collapse

We present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension on a compact manifold, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that a closed flat regular Riemannian foliation with leaves of positive dimension on a compact simply-connected manifold is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions.

Related project(s):
43Singular Riemannian foliations and collapse

Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and Köhl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple.

Related project(s):
61At infinity of symmetric spaces

We prove an analogue of Kostant's convexity theorem for split real and complex Kac-Moody groups associated to free, cofree and cotorsion-free root data. The result can be seen as a first step towards describing the multiplication map in a Kac-Moody group in terms of Iwasawa coordinates. Our method involves a detailed analysis of the geometry of Weyl group orbits in the Cartan subalgebra of a real Kac-Moody algebra. It provides an alternative proof of Kostant convexity for semisimple Lie groups and also generalizes a linear analogue of Kostant's theorem for Kac-Moody algebras that has been established by Kac and Peterson in 1984.

Related project(s):
61At infinity of symmetric spaces

This is the first of two papers which together prove that the 12-parameter family of parabolic $\mathrm{SU}(2)$-Hitchin moduli spaces on the four-punctured sphere are all ALG gravitational instantons of type D4, and hence are asymptotic to $(\mathbb{C}×T^2_\tau)/\mathbb{Z}_2$ at infinity. The elliptic modulus $\tau$ is determined by the cross-ratio of the four points. In this first paper, we consider each Hitchin moduli space corresponding to an allowable set of parabolic data and compute its Torelli parameters. There is a 12-parameter family of Hitchin moduli spaces corresponding to different parabolic data, and we show that these realize all possible allowable Torelli parameters. In the companion paper, we we will show there that all of the Hitchin moduli spaces studied here are indeed ALG of type D4, and consequently that every ALG-D4 gravitational instanton can be realized as a Hitchin moduli space. Altogether, this will give the first verification of any case of the Modularity Conjecture: that all ALG gravitational instantons with tangent cone $\mathbb{C}/\mathbb{Z}_2$ can be realized as Hitchin moduli spaces with their natural associated $L^2$-metrics.

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

Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove  Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with cone-like singularities and Lipschitz comparison maps to spheres. We use the  language of abstract cone operators which are introduced and studied in a general functional analytic setting and which may be of independent interest. Applying our discussion to spherical suspensions of odd-dimensional closed manifolds, we  generalize a  Lipschitz rigidity  result of the first three named authors from even to odd dimensions. Under stronger conditions, this has already been shown by Lee-Tam using geometric flows and by Bär using an upper estimate for the smallest Dirac eigenvalue.

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 provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class \(W^{2,q}\) with \(q>3\). This requires developing an elliptic theory for the conformal Laplacian for rough metrics and establishing existence, regularity and a delicate blow-up analysis for its Green function. Most of the analytical work is carried out in dimensions \(n\geq 3\) and for \(W^{2,q}\)-Riemannian metrics with \(q>\frac{n}{2}\) and should be of independent interest.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

It is well known in Riemannian geometry that the metric components have the best regularity in harmonic coordinates. These can be used to characterize the most regular element in the isometry class of a rough Riemannian metric. In this work, we study the conformal analogue problem on closed 3-manifolds: given a Riemannian metric \(g\) of class \(W^{2,q}\)  with \(q>3\), we characterize when a more regular representative exists in its conformal class. We highlight a deep link to the Yamabe problem for rough metrics and present some immediate applications to conformally flat, static and Einstein manifolds

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We investigate the Hitchin hyperkähler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the n-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as $\alpha\to 0$. Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of λ-connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperkähler metric on the hyperpolygon space $\mathcal{X}_{\alpha}$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperkähler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.

Related project(s):
69Wall-crossing and hyperkähler geometry of moduli spaces

Starting from the proof of the C^0-inextendibility of Schwarzschild by Sbierski, the past decade has seen renewed interest in showing low-regularity inextendibility for known spacetime models. Specifically, a lot of attention has been paid to FLRW spacetimes and there is an ever-growing array of results in the literature. Apart from hoping to provide a concise summary of the state of the art, we present an extension of work by Galloway and Ling on C^0-inextendibility of certain FLRW spacetimes within a subclass of spherically symmetric spacetimes, Galloway and Ling, to C^0-inextendibility within a subclass of axisymmetric spacetimes. Notably, our result works in the case of flat FLRW spacetimes with $a(t) \to 0$ for $t \to 0^+$, a setting where other known C^0-inextendibility results for FLRW spacetimes due to Sbierski, do not apply.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We present a new proof of the Willmore inequality for an arbitrary bounded domain Ω⊂ℝ^n with smooth boundary. Our proof is based on a parametric geometric inequality involving the electrostatic potential for the domain Ω; this geometric inequality is derived from a geometric differential inequality in divergence form. Our parametric geometric inequality also allows us to give new proofs of the quantitative Willmore-type and the weighted Minkowski inequalities by Agostiniani and Mazzieri.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We present different proofs of the uniqueness of 4-dimensional static vacuum asymptotically flat spacetimes containing a connected equipotential photon surface or in particular a connected photon sphere. We do not assume that the equipotential photon surface is outward directed or non-degenerate and hence cover not only the positive but also the negative and the zero mass case which has not yet been treated in the literature. Our results partially reproduce and extend beyond results by Cederbaum and by Cederbaum and Galloway. In the positive and negative mass cases, we give three proofs which are based on the approaches to proving black hole uniqueness by Israel, Robinson, and Agostiniani--Mazzieri, respectively. In the zero mass case, we give four proofs. One is based on the positive mass theorem, the second one is inspired by Israel's approach and in particular leads to a new proof of the Willmore inequality in (ℝ^3,δ), under a technical assumption. The remaining two proofs are inspired by proofs of the Willmore inequality by Cederbaum and Miehe and by Agostiniani and Mazzieri, respectively. In particular, this suggests to view the Willmore inequality and its rigidity case as a zero mass version of equipotential photon surface uniqueness.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We consider the problem of performing connected sums in the context of positive k^{th} intermediate Ricci curvature. We show that such connected sums are possible if the manifolds involved possess `k-core metrics' for some k. Here, a k-core metric is a generalization of the notion of core metric introduced by Burdick for positive Ricci curvature. Further, we show that connected sums of linear sphere bundles over bases admitting such metrics admit positive k^{th} intermediate Ricci curvature for k in a particular range. This follows from a plumbing result we establish, which generalizes other recent plumbing results in the literature and is possibly of independent interest. As an example of a manifold admitting a k-core metric, we prove that \mathbb{H} P^n admits a (4n-3)-core metric and that \mathbb{O}P^2 admits a 9-core metric, and we show that in both cases these are optimal.

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

We prove a canonical variation-type result for submersion metrics with positive intermediate Ricci curvatures. This can then be used in conjunction with surgery techniques to establish the existence of metrics with positive intermediate Ricci curvatures on a wide range of examples which had previously only been known to admit positive Ricci curvature, such as highly connected manifolds and exotic spheres. Further, we extend results of the second author on the moduli space of metrics with positive Ricci curvature to positive intermediate Ricci curvatures.

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

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

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

In every odd dimension $n\geq 5$ we exhibit large classes of closed n-dimensional manifolds which admit infinitely many different geometries of positive Ricci curvature, i.e., manifolds for which their moduli space of metrics of positive Ricci curvature has infinitely many connected components.

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

On a smooth connected manifold, we consider all possible locally elliptic and locally bounded measurable coefficient Riemannian metrics called rough Riemannian metrics. We equip this set with an extended metric which is connected if and only if the manifold is compact. We prove that this is a complete length extended metric space and the components on which the distance is finite are path-connected. Moreover, we identify the closure of smooth metrics in this space to be continuous metrics.

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

We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the maximal domain. Under additional geometric assumptions, and assumptions on an adapted boundary operator, we obtain a trace theorem on the maximal domain. This allows us to systematically study both local and nonlocal boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition occurs as a special case. Furthermore, we study contexts which induce semi-Fredholm and Fredholm extensions.

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

This is an introduction to the analysis of nonlinear evolution equations on manifolds with conical singularities via maximal regularity techniques. We address the specific difficulties due to the singularities, in particular the choice of extensions of the conic Laplacian that guarantee the existence of a bounded \(H_\infty\)-calculus. We introduce the relevant technical tools and survey, as main examples, applications to the porous medium equation, the fractional porous medium equation, the Yamabe flow, and the Cahn-Hilliard equation.

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