Publications

We show that, on a manifold with conical singularities, the asymptotics of the solutions to the porous medium equation near the conical points are determined by the spectrum of the Laplacian on the cross-section of the cone. The key to this result is a precise description of the maximal domain of the cone Laplacian.

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

We study finite abelian covers of the Chamanara surface, an example of a finite-area infinite translation surface with interesting dynamics and a large Veech group. Specifically, the Veech group of the Chamanara surface is a virtually free group on two generators. We characterize when finite abelian covers have large Veech groups themselves, namely when their Veech group has finite index in that of the Chamanara surface. For degree-2 covers, we provide a detailed analysis of these finite-index Veech groups. As an application, we prove that every free group arises as the projective Veech group of a finite-area infinite translation surface.

Related project(s):
72Limits of invariants of translation surfaces

We study how far APS boundary conditions for a Lorentzian Dirac operator may be perturbed without destroying Fredholmness of the Dirac operator. This is done by developing criteria under which the perturbation of a compact pair of projections is a Fredholm pair.

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

We consider Riemannian manifolds $(M^n,g_0)$, $(M^n,h)$, where $(M^n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0 < \infty$, and $(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$ for some small $\varepsilon_0(n)>0$.

It was shown by Simon (2002) that a Ricci--DeTurck flow solution $g(t)_{t \in (0,T)}$ related to $g_0$ exists for some $T=T(n,K_0)>0$.

If $g_0 \in L^2_{\mathrm{loc}}$ or $g_0 \in W^{1,2+2\sigma}_{\mathrm{loc}}$, $\sigma \in (0,\frac{1}{4})$, respectively, we show that $g(t) \to g_0$ in the $L^2_{\mathrm{loc}}$- or $W^{1,2+\sigma}_{\mathrm{loc}}$-sense, respectively.

If $M$ is closed, $g_0 \in W^{1,2+\sigma}(M)$ for some $\sigma>0$, and the distributional scalar curvature of Lee--LeFloch (2015) is not less than $b \in \mathbb{R}$, then we show that $g(t)$ has scalar curvature not less than $b$ in the smooth sense for all $t>0$.

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

We show that for every \(n \geq 2\) and \(D > 0\) there exist a convex domain \(\Omega \subseteq \mathbb{H}^n\) with diameter \(D\) and a convex potential \(V\) on \(\Omega\) such that the fundamental gap of the operator \(-\Delta + V\) is strictly smaller than the fundamental gap of  \(-\Delta\). In comparison to previous work, this result requires more refined control of the eigenfunctions.

Related project(s):
51The geometry of locally symmetric manifolds via natural maps

We show that if a closed manifold of dimension at least four admits a negatively curved metric that is almost Einstein in a suitable sense, then it admits a genuine Einstein metric of negative sectional curvature. Importantly, the pinching constant measuring the almost-Einstein condition neither depends on an upper bound for the diameter or volume, nor on a lower bound for the injectivity radius.

Related project(s):
51The geometry of locally symmetric manifolds via natural maps

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We show the contractibility of spaces of invariant Riemannian metrics of positive scalar curvature on compact connected manifolds of dimension at least two, with and without boundary and equipped with compact Lie group actions. On manifolds without boundary, we assume that the Lie group contains a normal S1-subgroup with fixed-point components of codimension two. In this situation, the existence of invariant metrics of positive scalar curvature was known previously.

For the proof, we combine equivariant Morse theory with conformal deformations near unstable manifolds. On manifolds without boundary, we also use local flexibility properties of positive scalar curvature metrics and the smoothing of mean-convex singularities.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We give a survey of analytic and geometric results on `fibred cusp spaces', a large class of non-compact Riemannian manifolds which include the regular parts of singular spaces with incomplete cusp singularities as well as complete spaces with asymptotically hyperbolic cusp or asymptotically Euclidean structures at infinity. These results cover topics in spectral geometry, in particular analytic torsion and index theory, and boundary value problems. The underlying tools include a careful microlocal analysis of the resolvent and the heat kernel. We include an exposition of the geometric and analytic foundations and sketch the ideas of the proofs of the main theorems. Special emphasis is put on the common features of and the differences between the incomplete and various kinds of complete settings.

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

Using the Schwarzian derivative we construct a sequence \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) of meromorphic differentials on every non-flat oriented minimal surface in Euclidean \(3\)-space. The differentials \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) are invariant under all deformations of the surface arising via the Weierstrass representation and depend on the induced metric and its derivatives only. A minimal surface is said to have degree \(n\) if its \(n\)-th differential is a polynomial expression in the differentials of lower degree. We observe that several well-known minimal surfaces have small degree, including Enneper's surface, the helicoid/catenoid and the Scherk -- as well as the Schwarz family. Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.

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

We introduce the notion of a flat extension of a connection \(\theta\) on a principal bundle. Roughly speaking, \(\theta\) admits a flat extension if it arises as the pull-back of a component of a Maurer--Cartan form. For trivial bundles over closed oriented \(3\)-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated with \(\theta\). As an application, we recover the obstruction of Chern--Simons for the existence of a conformal immersion of a Riemannian \(3\)-manifold into Euclidean \(4\)-space. In addition, we obtain corresponding statements for a Lorentzian \(3\)-manifold, as well as a global obstruction for the existence of an equiaffine immersion into \(\mathbb{R}^4\) of a \(3\)-manifold that is equipped with a torsion-free connection preserving a volume form.

Related project(s):
68Minimal Lagrangian connections and related structures

We study Laplacians on general countable weighted simplicial complexes from a conceptual point of view. These operators will first be introduced formally before showing that those formal operators coincide with self-adjoint realizations of operators arising from quadratic forms. A major conceptual perspective is the correspondence to signed Schrödinger operators unveiling the Forman curvature. The main results are criteria for essential self-adjointness via lower bounded Forman curvature and a Gaffney type result via completeness. Finally, we study spectral relations between these Laplacians.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We study heat kernel convergence of induced subgraphs with Neumann boundary conditions. We first establish convergence of the resulting semigroups to the Neumann semigroup in . While convergence to the Neumann semigroup always holds, convergence to the Dirichlet semigroup in turns out to be equivalent to the coincidence of the Dirichlet and Neumann semigroups while convergence in is equivalent to stochastic completeness. We then investigate the Feller property for the Neumann semigroup via generalized solutions and give applications to graphs satisfying a condition on the edges as well as birth-death chains.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

Given the maximal compact subalgebra k(A) of a split-real Kac–Moody algebra

g(A) of type A, we study certain finite-dimensional representations of k(A) that do not lift to

the maximal compact subgroup K(A) of the minimal Kac–Moody group G(A) associated to

g(A) but only to its spin cover Spin(A) described in [11]. Currently, four elementary of these

so-called spin representations are known. We study their (ir)reducibility, semisimplicity, and

lift to the group level. The interaction of these representations with the spin-extended Weyl

group is used to derive a partial parametrization result of the representation matrices by the

real roots of g(A).

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

We provide a coarse version of the relative index of Gromov and Lawson and thoroughly establish all of its basic properties. As an application, we discuss a general procedure to construct wrong way maps on the \(K\)-theory of the Roe algebra mapping the coarse index class of the Dirac operator of a manifold to the one of a suitably embedded submanifold of arbitrary codimension, thereby establishing an abstract machinery to find obstructions to uniform positive scalar curvature coming from these submanifolds.

Related project(s):
45Macroscopic invariants of manifolds78Duality and the coarse assembly map II

We construct a slant product \(\mathrm{S}^{G\times H}_p(X\times Y)\otimes \mathrm{K}_{-q}(\bar{\mathfrak{c}}^{\mathrm{red}} Y\rtimes H)\to \mathrm{K}_{p-q}(\mathrm{C}^\ast_G X)\) on the analytic structure group of Higson and Roe and the K-theory of the stable Higson compactification taking values in the (equivariant) Roe algebra. This complements the slant products constructed in earlier work of Engel and the authors ( arXiv:1909.03777 [math.KT] ). The distinguishing feature of our new slant product is that it specializes to a duality pairing \(\mathrm{S}^H_p(Y) \otimes \mathrm{K}_{-p}(\bar{\mathfrak{c}}^{\mathrm{red}} (Y)\rtimes H)\to \mathbb{Z}\) which can be used to extract numerical invariants out of elements in the analytic structure group such as rho-invariants associated to positive scalar curvature metrics.

Related project(s):
78Duality and the coarse assembly map II

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\mathrm{dS}_3$, best viewed as a generalized Gauß map. On the gauge theoretic side, it matches $\mathrm{SU}(2)$ and $\mathrm{SU}(1,1)$ solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of \emph{transgressive} harmonic maps $f:M\to \mathbb S^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $\tau$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines.

Related project(s):
55New hyperkähler spaces from the the self-duality equations77Asymptotic geometry of the Higgs bundle moduli space II

Given a non-compact semisimple real Lie group G and an Anosov subgroup Γ, we utilize the correspondence between ℝ-valued additive characters on Levi subgroups L of G and ℝ-affine homogeneous line bundles over G/L to systematically construct families of non-empty domains of proper discontinuity for the Γ-action. If Γ is torsion-free, the analytic dynamical systems on the quotients are Axiom A, and assemble into a single partially hyperbolic multiflow. Each Axiom A system admits global analytic stable/unstable foliations with non-wandering set a single basic set on which the flow is conjugate to Sambarino's refraction flow, establishing that all refraction flows arise in this fashion. Furthermore, the ℝ-valued additive character is regular if and only if the associated Axiom A system admits a compatible pseudo-Riemannian metric and contact structure, which we relate to the Poisson structure on the dual of the Lie algebra of G.

Related project(s):
65Resonances for non-compact locally symmetric spaces

A three-dimensional quasi-Fuchsian Lorentzian manifold M is a globally hyperbolic spacetime diffeomorphic to Σ×(−1,1) for a closed orientable surface Σ of genus ≥2. It is the quotient M=Γ∖ΩΓ of an open set ΩΓ⊂AdS3 by a discrete group Γ of isometries of AdS3 which is a particular example of an Anosov representation of π1(Σ). We first show that the spacelike geodesic flow of M is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for Γ extend meromorphically to ℂ. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian ◻ of M, which is meromorphic on ℂ with poles of finite rank, defining a notion of quantum resonances and quantum resonant states related to the Ruelle resonances and (co-)resonant states by a quantum-classical correspondence. This initiates the spectral study of convex co-compact pseudo-Riemannian locally symmetric spaces.

Related project(s):
65Resonances for non-compact locally symmetric spaces

The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.

Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces