Publications

We consider the evolution of hypersurfaces in R^{n+1} with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature), complete, and given as the graph of a function. Long-time existence of the H^α-flow is established by means of approximation by bounded problems.

Related project(s):
29Curvature flows without singularities

By a symmetric double graph we mean a hypersurface which is mirror-symmetric and the two symmetric parts are graphs over the hyperplane of symmetry. We prove that there is a weak solution of mean curvature flow that preserves these properties and singularities only occur on the hyperplane of symmetry. The result can be used to construct smooth solutions to the free Neumann boundary problem on a supporting hyperplane with singular boundary. For the construction we introduce and investigate a notion named "vanity" and which is similar to convexity. Moreover, we rely on Sáez' and Schnürer's "mean curvature flow without singularities" to approximate weak solutions with smooth graphical solutions in one dimension higher.

Related project(s):
29Curvature flows without singularities

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products.

Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimensions at least five which have odd order abelian fundamental groups, are non-spin and  atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fundamental groups.

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

The unit conjecture, commonly attributed to Kaplansky, predicts that if \(K\) is a field and \(G\) is a torsion-free group then the only units of the group ring \(K[G]\) are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.

Related project(s):
48Profinite and RFRS groups

For a countable group G we construct a small, idempotent complete, symmetric monoidal, stable ∞-category KK^G_sep whose homotopy category recovers the triangulated equivariant Kasparov category of separable G-C*-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable ∞-category KK^G which receives a symmetric monoidal functor kk^G from possibly non-separable G-C*-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying G. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite K-homology theory on proper and locally compact G-topological spaces, allowing for coefficients in arbitrary G-C*-algebras. Finally, we extend the functor kk^G from G-C*-algebras to G-C*-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.

Related project(s):
45Macroscopic invariants of manifolds

We prove that any length metric space homeomorphic to a surface may be decomposed into non-overlapping convex triangles of arbitrarily small diameter. This generalizes a previous result of Alexandrov--Zalgaller for surfaces of bounded curvature.

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

Based on  Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of  spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over  the sphere bundle $\mathbb{S}( \mathbb{R} \oplus V)$  in terms of vector bundles and Clifford module bundles over $X$. After passing to topological K-theory these results imply classical Bott-Thom isomorphism theorems.

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 investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau's problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.

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

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a C^{1,1}-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an α-Hölder continuous Lorentzian metric admit a C^{1,α/4}-parametrization.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We investigate the geometric and topological structure of equidistant decompositions of Riemannian manifolds.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall.

Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We show that a strict, nearly Kähler 6-manifold with either second or third Betti number nonzero is linearly unstable with respect to the \(\nu\)-entropy of Perelman and hence is dynamically unstable for the Ricci flow.

Related project(s):
74Rigidity, stability and deformations in nearly parallel G2-geometry

A Calderón projector for an elliptic operator $P$ on a manifold with boundary $X$ is a projection from general boundary data to the set of boundary data of solutions $u$ of $Pu=0$. Seeley proved in 1966 that for compact $X$ and for $P$ uniformly elliptic up to the boundary there is a Calder\'on projector which is a pseudodifferential operator on $\partial X$. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity.

This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the $\phi$-pseudodifferential calculus introduced by Mazzeo and Melrose.

In our presentation we provide a setting that may  be useful for doing analogous constructions for other types of singularities.

Related project(s):
13Analysis on spaces with fibred cusps49Analysis on spaces with fibred cusps II

We study the low energy resolvent of the Hodge Laplacian on a manifold  equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $\phi$-) pseudodifferential operator when the resolvent parameter tends to zero.

This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic  and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on $\phi$-manifolds.

Related project(s):
13Analysis on spaces with fibred cusps23Spectral geometry, index theory and geometric flows on singular spaces49Analysis on spaces with fibred cusps II

We define a functional \({\cal J}(h)\) for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that \({\cal J}(h)\) is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating \({\cal J}(h)\) and another functional \({\cal I}(h)\), closely related to the Yang-Mills-Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of \({\cal J}(h)\), which is expressed as a certain \(L^{2}\)-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of \({\cal J}(h)\) if and only if the corresponding Hitchin-Simpson mean curvature is parallel with respect to the Hitchin-Simpson connection.

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space69Wall-crossing and hyperkähler geometry of moduli spaces

We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing A-hat-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and \(S^n\times S^n\) or \(S^n\times S^{n+1}\), respectively. We also construct manifolds M whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of 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 Manifolds

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

Related project(s):
36Cohomogeneity, curvature, cohomology

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

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

Let M be a simply connected spin manifold of dimension at least six which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on M has non-trivial higher homotopy groups.

Moreover, denote by \mathcal{M}^+_0(M) the moduli space of positive scalar cuvature metrics on M associated to the group of orientation-preserving diffeomorphisms of M. We show that if M belongs to a certain class of manifolds which includes (2n−2)-connected (4n−2)-dimensional manifolds, then the fundamental group of \mathcal{M}^+_0(M) is non-trivial.

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