Nadine Große

• University of Freiburg

By definition, a differential operator T on a connected manifold satisfies the L 2 {L^{2}} -unique continuation property if every L 2 {L^{2}} -solution of T that vanishes on an open subset vanishes identically. We study the L 2 {L^{2}} -unique continuation property of an operator T acting on a manifold with bounded geometry. In particular, we estab...

In this article, we study the spectrum of the magnetic Dirac operator, and the magnetic Dirac operator with potential over complete Riemannian manifolds. We find sufficient conditions on the potentials as well as the manifold so that the spectrum is either maximal, or discrete. We also show that magnetic Dirac operators can have a dense set of eige...

In this article we study the spectrum of the magnetic Dirac operator, and the magnetic Dirac operator with potential over complete Riemannian manifolds. We find sufficient conditions on the potentials as well as the manifold so that the spectrum is either maximal, or discrete. We also show that magnetic Dirac operators can have a dense set of eigen...

A differential operator $T$ satisfies the $L^2$-unique continuation property if every $L^2$-solution of $T$ that vanishes on an open subset vanishes identically. We study the $L^2$-unique continuation property of an operator $T$ acting on a manifold with bounded geometry. In particular, we establish some connections between this property and the re...

Our main goal in the present paper is to expand the known class of noncompact manifolds over which the L2-spectrum of a general Dirac operator and its square is maximal. To achieve this, we first find sufficient conditions on the manifold so that the Lp-spectrum of the Dirac operator and its square is independent of p for p≥1. Using the L1-spectrum...

Our main goal in the present paper is to expand the known class of open manifolds over which the $L^2$-spectrum of a general Dirac operator and its square is maximal. To achieve this, we first find sufficient conditions on the manifold so that the $L^p$-spectrum of the Dirac operator and its square is independent of $p$ for $p\geq 1$. Using the $L^...

We consider the classical Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to APS-boundary conditions. This is achieved by deriving suitable energy estimates, which play a fundamental role in establishing uniqueness and existence of weak solutions. Fi...

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spinc manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spinc case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to 3...

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P,C) is equivalent to the uniform regularity of the natural family (Px,Cx) of associated boundary value problems in local coordinates. We veri...

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then...

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then...

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin$^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin$^c$ case the result of O. Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or eq...

We prove that for cobordant closed spin manifolds of dimension $n\geq 3$ the associated spaces of metrics with invertible Dirac operator are homotopy equivalent. This is the spinorial counterpart of a similar result on positive scalar curvature of Chernysh/Walsh and generalizes the surgery result of Ammann-Dahl-Humbert on the existence of metrics w...

We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains that are conformally equivalent to manifolds with boundary and bounded geometry. Our assumptions are satisfied by the domains with a smooth set of singular cuspidal points, and hence our results apply to the class of domains with isolated...

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel–Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The precise estimates derived in t...

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , , invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nach...

We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with conical singularities. The proo...

Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry, $\partial_D M \subset \partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $\partial M \smallsetminus \partial_D M$. We prove the regularity and well-posedness of the mixed Robin bo...

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then...

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel-Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The precise estimates derived in t...

For a compact subgroup $G$ of the group of isometries acting on a Riemannian manifold $M$ we investigate subspaces of Besov and Triebel-Lizorkin type which are invariant with respect to the group action. Our main aim is to extend the classical Strauss lemma under suitable assumptions on the Riemannian manifold by proving that $G$-invariance of func...

For a compact subgroup $G$ of the group of isometries acting on a Riemannian manifold $M$ we investigate subspaces of Besov and Triebel-Lizorkin type which are invariant with respect to the group action. Our main aim is to extend the classical Strauss lemma under suitable assumptions on the Riemannian manifold by proving that $G$-invariance of func...

For a compact subgroup G of the group of isometries acting on a Riemannian manifold M we investigate subspaces of Besov and Triebel-Lizorkin type which are invariant with respect to the group action. Our main aim is to extend the classical Strauss lemma under suitable assumptions on the Riemannian manifold by proving that G-invariance of functions...

We prove regularity and well-posedness results for the mixed Dirichlet-Neumann problem for a second order, uniformly strongly elliptic differential operator on a manifold $M$ with boundary $\partial M$ and bounded geometry. Our well-posedness result for the Laplacian $\Delta_g := d^*d \ge 0$ associated to the given metric require the additional ass...

We consider the first non-zero eigenvalue $\lambda_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that $8\pi \nabla\log(\lambda_1)$ essentially agrees with the dual of the differential of the degenerating Fenchel-Nielson length coordinate. As a corollary of our analysis, which is based in particu...

We consider the first non-zero eigenvalue $\lambda_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that $8\pi \nabla\log(\lambda_1)$ essentially agrees with the dual of the differential of the degenerating Fenchel-Nielsen length coordinate. As a consequence, we can improve previous results of Scho...

Let M be a manifold with boundary and bounded geometry. We assume that M has "finite width," that is, that the distance $dist(x, \partial M)$ from any point $x \in M$ to the boundary $\partial M$ is bounded uniformly. Under this assumption, we prove that the Poincar\'e inequality for vector valued functions holds on $M$. We also prove a general reg...

We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to products of hyperbolic space with a compact manifold. As conclusion we show the Yamabe inequality for some noncompact manifolds which are important to understand the behaviour of Yamabe invariants under surgeries.

In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescaled hyperbolic spaces. We give...

We study the $L^p$-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on $p$. As a first example where $p$-independence fails we compute explicitly the $L^p$-spectrum for the hyperbolic space and its product with compact spaces.

In this paper, we extend the study of generalized Killing spinors on Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin$^c$ Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing that the dimension of the manifold is greater or equa...

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient condition...

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. B\"ar and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi-Montiel-Zhang, inv...

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe constant at infinity is even...

For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle...

We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal complete metrics. This also g...

We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hijazi inequality for the underlyi...

We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.

We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic ma...

We extend a Yamabe-type invariant of the Dirac operator to noncompact manifolds and show that as in the compact case this invariant is bounded by the corresponding invariant of the standard sphere. Further, this invariant will lead to an obstruction of the conformal compactification of complete noncompact manifolds.