Publications

Related project(s):
22Willmore functional and Lagrangian surfaces25The Willmore energy of degenerating surfaces and singularities of geometric flows

We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant. Examples include the Misner spacetime and the Taub-NUT spacetime. We prove an energy estimate close to the Cauchy horizon for wave equations acting on sections of vector bundles. Using this estimate we prove that if a linear wave equation can be solved up to any order at the Cauchy horizon, then there exists a unique solution on the globally hyperbolic region. As a consequence, we prove several existence and uniqueness results for linear and non-linear wave equations without assuming analyticity or symmetry of the spacetime and without assuming that the generators close. We overcome in particular the essential remaining difficulty in proving that vacuum spacetimes with a compact Cauchy horizon with constant non-zero surface gravity necessarily admits a Killing vector field. This work is therefore related to the strong cosmic censorship conjecture.

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

We prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to Moncrief-Isenberg and Friedrich-Rácz-Wald, where the generators of the Cauchy horizon were closed or densely filled a 2-torus. Consequently, the maximal globally hyperbolic vacuum development of generic initial data cannot be extended across a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant. Our result supports, thereby, the validity of the strong cosmic censorship conjecture in the considered special case. The proof consists of two main steps. First, we show that the Killing equation can be solved up to any order at the Cauchy horizon. Second, by applying a recent result of the first author on wave equations with initial data on a compact Cauchy horizon, we show that this Killing vector field extends to the globally hyperbolic region.

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

We discuss a method to construct Dirac-harmonic maps developed by J. Jost, X. Mo and M. Zhu. The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in the publication by Jost, Mo and Zhu were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions. In several cases we show the existence of solutions.

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

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f from a manifold X to the product of the k-sphere with the n-k dimensional torus, with k=1,2,3. When X is a closed oriented manifold endowed with a metric g of positive scalar curvature and the map f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g and the contracting factor of the map f.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger, and Yu. To take the fundamental groups of the manifold and its boundary into account requires working with maximal C* completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds can not be established.

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

In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor'yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so called refinements.

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

In this expository paper we answer two fundamental questions concerning discrete magnetic Schrödinger operator associated with weighted graphs. We discuss when formal expressions of such operators give rise to self-adjoint operators, i.e., when they have self-adjoint restrictions. If such self-adjoint restrictions exist, we explore when they are unique.

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

We study pairs of Dirichlet forms  related by an intertwining order

isomorphisms between the associated \(L^2\)-spaces. We consider the

measurable, the topological and the geometric setting respectively.

In the measurable setting, we  deal with arbitrary (irreducible)

Dirichlet forms and show that any intertwining order isomorphism is

necessarily unitary (up to a constant). In the topological setting

we deal with quasi-regular forms and show that any intertwining

order isomorphism induces a quasi-homeomorphism between the

underlying spaces. In the geometric setting we deal with both

regular Dirichlet forms as well as resistance forms and essentially

show that the geometry defined by these  forms  is preserved by

intertwining  order isomorphisms. In particular, we prove in the

strongly local regular case that intertwining order isomorphisms

induce isometries with respect to the intrinsic metrics between the

underlying spaces under fairly mild assumptions. This applies to a

wide variety of metric measure spaces including

\(\mathrm{RCD}(K,N)\)-spaces, complete weighted Riemannian manifolds

and  complete quantum graphs. In the non-local regular case our

results cover  in particular graphs as well as fractional Laplacians

as arising in the treatment of \(\alpha\)-stable Lévy processes. For

resistance forms we show that intertwining order isomorphisms are

isometries with respect to the resistance metrics.

Our results can can be understood as saying that  diffusion always

determines the Hilbert space, and -- under natural compatibility

assumptions  -- the topology and the geometry respectively. As special

instances they cover earlier results for manifolds and graphs.

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

We study topological Poincaré type inequalities on generalgraphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory ofLaplacians once a probability measure on the graph is chosen. More specifically,we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures.

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

Related project(s):
25The Willmore energy of degenerating surfaces and singularities of geometric flows

We construct $\eta$- and $\rho$-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

Related project(s):
4Secondary invariants for foliations

For a closed, connected direct product Riemannian manifold (M,g)=(M1×⋯×Ml,g1⊕⋯⊕gl), we define its multiconformal class [[g]] as the totality {f12g1⊕⋯⊕fl2gl} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a function fi2>0 on the total space M. A multiconformal class [[g]] contains not only all warped product type deformations of g but also the whole conformal class [g~] of every g~∈[[g]]. In this article, we prove that [[g]] carries a metric of positive scalar curvature if and only if the conformal class of some factor (Mi,gi) does, under the technical assumption dimMi≥2. We also show that, even in the case where every factor (Mi,gi) has positive scalar curvature, [[g]] carries a metric of scalar curvature constantly equal to −1 and with arbitrarily large volume, provided l≥2 and dimM≥3. In this case, such negative scalar curvature metrics within [[g]] for l=2 cannot be of any warped product type.

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

We study the behavior of the spectrum of the Dirac operator together with a symmetric W1,∞-potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or pin− structure on the limit space N, then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric W1,∞-potential. In the case of an orientable limit space N, D is the spin Dirac operator DN on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then D=DN⊕−DN.

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

We consider pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators, which are defined on the product manifold $M_1 \times M_2$ for closed manifolds $M_1$ and $M_2$. We prove a topological index theorem for Fredholm operators of tensor product type. To this end we construct a suitable double deformation groupoid and prove a Poincaré duality type result in relative $K$-theory.

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

We extend two known existence results to simply connected manifolds with

positive sectional curvature: we show that there exist pairs of simply

connected positively-curved manifolds that are tangentially homotopy equivalent

but not homeomorphic, and we deduce that an open manifold may admit a pair of

non-homeomorphic simply connected and positively-curved souls. Examples of such

pairs are given by explicit pairs of Eschenburg spaces. To deduce the second

statement from the first, we extend our earlier work on the stable converse

soul question and show that it has a positive answer for a class of spaces that

includes all Eschenburg spaces.

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

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen [2016], of Miao and Xie [2016], and of joint work of Miao and the authors [2017], we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

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

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the (n+1)-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for n=2 we obtain a Minkowski-type inequality and for n=3 we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

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

In this paper, by providing the uniform gradient estimates for approximating equations, we prove the existence, uniqueness and regularity of conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we obtain a regularity estimate, that is, any $L^{\infty}$-solution of the conical complex Monge-Ampère equation admits the $C^{2,\alpha,\beta}$-regularity.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.