Publications

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal \(L^q\)-regularity space for all times and is instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

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

We construct a codimension 3completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a (4n +3)-dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere S7.

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.

Related project(s):
4Secondary invariants for foliations11Topological and equivariant rigidity in the presence of lower curvature bounds

We decompose locally in time maximal \(L^{q}\)-regular solutions of abstract quasilinear parabolic equations as a sum of a smooth term and an arbitrary small−with respect to the maximal \(L^{q}\)-regularity space norm−remainder. In view of this observation, we next consider the porous medium equation and the Swift-Hohenberg equation on manifolds with conical singularities. We write locally in time each solution as a sum of three terms, namely a term that near the singularity is expressed as a linear combination of complex powers and logarithmic integer powers of the singular variable, a term that decays to zero close to the singularity faster than each of the non-constant summands of the previous term and a remainder that can be chosen arbitrary small with respect e.g. to the \(C^{0}\)-norm. The powers in the first term are time independent and determined explicitly by the local geometry around the singularity, e.g. by the spectrum of the boundary Laplacian in the situation of straight conical tips. The case of the above two problems on closed manifolds is also considered and local space asymptotics for the solutions are provided.

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

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds and construct, in particular, the first classes of manifolds for which these spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist closed (respectively, open) manifolds for which the moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. An analogous statement holds for spaces of non-negative Ricci curvature metrics in every dimension at least eleven (respectively, twelve).

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

Let M be a Milnor sphere or, more generally, the total space of a linear S^3-bundle over S^4 with H^4(M;Q) = 0. We show that the moduli space of metrics of nonnegative sectional curvature on M has infinitely many path components. The same holds true for the moduli space of m etrics of positive Ricci curvature on M.

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

We study immersed tori in 3-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes (0,b) with $b\rightarrow 1$ the homogenous tori are known to be unique constrained Willmore minimizers (up to invariance). In this paper we generalize the result and determine constrained Willmore minimizers in non-rectangular conformal classes (a,b). In a first step we explicitly construct a 2-dimensional family of putative minimizers parametrized by their conformal class (a,b). For $b\rightarrow 1$, b≠1 fixed, this family is then shown to minimize for $a\rightarrow 0^+$. Difficulties arise from the fact that these minimizers are non-degenerate for a≠0 but smoothly converge to the degenerate homogenous tori as a→0. As a byproduct of our arguments, we show that the minimal Willmore energy ω(a,b) is real analytic and concave in a∈(0, a^b) for some b>0 and fixed $b \rightarrow 1$, 

Related project(s):
16Minimizer of the Willmore energy with prescribed rectangular conformal class

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

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

In this paper we prove that any smooth surfaces can be locally isometrically embedded into as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.

Related project(s):
22Willmore functional and Lagrangian surfaces

We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable.

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry

We prove a gluing theorem for solutions \((A_0, \Phi_0)\)  of Hitchin's self-duality equations with logarithmic singularities on a rank-\(2\) vector bundle over a noded Riemann surface \( \Sigma\) representing a boundary point of Teichmüller moduli space. We show that every nearby smooth Riemann surface \( \Sigma_1\) carries a smooth solution \((A_1, \Phi_1)\) of the self-duality equations, which may be viewed as a desingularization of \((A_0, \Phi_0)\).

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We prove that the Teichmüller space of negatively curved metrics on a hyperbolic manifold M has nontrivial i-th rational homotopy groups for some i > dim M. Moreover, some elements of infinite order in the i-th homotopy group of BDiff(M) can be represented by bundles over a sphere with fiberwise negatively curved metrics.

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

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.

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

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of the operators in the algebra on suitable Sobolev spaces is established.

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

The canonical map from the \(\mathbb{Z}/2\)-equivariant Lazard ring to the \(\mathbb{Z}/2\)-equivariant complex bordism ring is an isomorphism. 

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

We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.

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

We prove that the Atiyah-Singer Dirac operator ${\mathrm D}_{\mathrm g}$ in ${\mathrm L}^2$ depends Riesz continuously on ${\mathrm L}^{\infty}$ perturbations of complete metrics ${\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\mathrm g} \to {\mathrm D}_{\mathrm g}(1 + {\mathrm D}_{\mathrm g}^2)^{-\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

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

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces \( \Sigma_R\) converging for \(R\searrow 0\) to a surface \( \Sigma_0\) with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration \(\mathcal M_d \to \Sigma_R\) gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in Mazzeo et al. (Duke Math. J. 165(12):2227–2271, 2016).

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the η-invariant of the Cauchy hypersurfaces.

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

We associate to each stable Higgs pair \((A_0,\Phi_0)\) on a compact Riemann surface X a singular limiting configuration \((A_\infty,\Phi_\infty)\), assuming that \(\det\Phi\) has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions \((A_t,\Phi_t) \) to Hitchin's equations which converge to this limiting configuration as \(t\to\infty\). This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.  

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space