Publications

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

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.

Related project(s):
11Topological and equivariant rigidity in the presence of lower curvature bounds

We present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous bounded cohomology of SL(2,R) in degree 4, establishing a special case of a conjecture of Monod.

Related project(s):
27Invariants and boundaries of spaces

Let \(M^n, n \in \{4,5,6\}\), be a compact, simply connected n-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on \(M^n\) by a torus \(T^{n-2}\) is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

Related project(s):
11Topological and equivariant rigidity in the presence of lower curvature bounds

We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $\mathbb{R}^n$.

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