arXiv research

The paper extends cyclic cocycles and proves relative index theorems for partitioned manifolds.

Extends a theorem for first-order elliptic operators on manifolds.

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.

Constructs index class for manifolds with group action.

The paper solves a general case of the cohomological relative index problem for foliations.

Absolute index theorem for warped product manifolds.

We prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds wit…

Proves a strengthening of Chang, Weinberger, and Yu's theorem on positive scalar curvature.

Study the APS index theorem for even-dimensional manifolds with non-compact boundaries.

We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold…

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac ope…

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes inf…

Proves a quantitative index theorem for positive scalar curvature metrics.

Let $Γ$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $Γ$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (Γ;\mathbb{C})$ which is of polynomial growth wit…

We introduce a notion of cobordism of Callias-type operators over complete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a s…

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external p…

Defines a new calculus for cusp pseudodifferential operators and proves index theorems.

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's…

We show that a complete Euclidean submanifold with minimal index of relative nullity $ν_0>0$ and Ricci curvature with a certain controlled decay must be a $ν_0$-cylinder. This is an extension of the classical Hartman cylindricity theorem.

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

Study eta invariant on non-compact manifolds with positive scalar curvature.

A guide for solving first-order elliptic boundary value problems.

Proves an index theorem for proper group actions on manifolds.

These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomo…

We prove two geometric index theorems for a family of first-order elliptic operators over a manifold with boundary by computing eta form representatives for the Chern character classes of the index bundle. The eta forms occur as relative and regularized traces on infinite-dimensional vector bundles realized as the limi…

This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree'' multiplicative K-theory, the direct image is proved to be functorial for double subme…

This is the sequel of the first part math.DG/0611281. Here, the procedure of transgressing the families index theorem (the so-called $η$-form) is adapted to take in account the case of Dirac type operators with kernels of varying dimension. The constructed form is then used to define the direct image under proper subme…

Extends Dirac operator results to foliations with invariant measures.

Paper proves nonzero foliated Rosenberg index for noncompactly enlargeable foliations.

We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal M,ω)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ a Lagrangian submanifold of $\mathcal M$, $H$ is a smooth time dependent Hamiltonian function on $\mathcal M$ and $Γ:[a,b]\to\mathcal…

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on boundary(M) has the homotopy type of a finite aspherical CW-complex. This answers, for…

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold $M$, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the $C^k$-topology, $k=2,...,\infty$, in the set of metrics of …

Finite index subgroups of relatively hyperbolic groups have equal index.

We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three…

Study submanifolds with relative nullity in space forms using splitting tensor.

Given a Lorentzian manifold $(M,g)$, a geodesic $γ$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $γ$, we introduce a special class of instants along $γ$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate insta…

Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index…

Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.

This text is a revised version of the authors Habilitationsschrift which was submitted to the University of Augsburg, 1993. Fuchs type differential operators are used to model the analysis on manifolds with cone--like singularities, or more general, stratified spaces. This book provides a self--contained treatment of t…

This paper contains the constructions of a real manifold version of relative K-theory, and of an extension of Karoubi's multiplicative K-theory suggested by U. Bunke (which I call ``free multiplicative K-theory'' in the sequel). Chern-Simons-Nadel type classes on relative K-theory are constructed, while it is proved th…

In a previous paper, we showed nonvaninishing of the universal index elements in the K-theory of the maximal C*-algebras of the fundamental groups of enlargeable spin manifolds. The underlying notion of enlargeability was the one from the first relevant paper of Gromov and Lawson, involving contracting maps defined on …

We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the cl…

We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative $K$-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $\ball^m$ be the unit ball in $\mathbb{C}^m$,…

The paper establishes distance estimates for manifolds with lower scalar curvature bounds.

We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative inde…