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Bernd Ammann

Publications and source records attributed to Bernd Ammann.

54 records · Page 3Linked to original sources
arXiv2006

Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $λ(M,[g],\si)= \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $λ_1^+(\tilde{g})$ is the smallest positive eigenvalue of the Dirac operator D in the metric $\tilde{g}$. A previous result stated that $λ(M,[g],\si) \leq λ(\mS^n) =\frac{n}{2} \om_n^{1/n}$ where \om_n stands for the volume of the standard n-sphere. In this paper, we study this problem for conformally flat manifolds of dimension n \geq 2 such that D is invertible. E.g. we show that strict inequality holds in dimension $n\equiv 0,1,2\mod 4$ if a certain endomorphism does not vanish. Because of its tight relations to the ADM mass in General Relativity, the endomorphism will be called mass endomorphism. We apply the strict inequality to spin-conformal spectral theory and show that the smallest positive Dirac eigenvalue attains its infimum inside the enlarged volume-1-conformal class of g.

math.DG↗
arXiv2006

The first conformal Dirac eigenvalue on 2-dimensional tori

Let M be a compact manifold with a spin structure χand a Riemannian metric g. Let λ_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ-invariant is defined as τ(M,χ):= sup inf \sqrt{λ_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, and where the infimum runs over all metrics in the given class. We show that τ(T^2,χ)=2\sqrtπ if χis ``the'' non-trivial spin structure on T^2. In order to calculate this invariant, we study the infimum as a function on the spin-conformal moduli space and we show that the infimum converges to 2\sqrtπ at one end of the spin-conformal moduli space.

math.DG↗
arXiv2006

Pseudodifferential operators on manifolds with a Lie structure at infinity

Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset Γ(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra $Ψ_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at infinity $V \subsetΓ(TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to $Ψ_{1,0,V}^\infty(M_0)$. We also consider the algebra $\DiffV{*}(M_0)$ of differential operators on $M_0$ generated by $V$ and $\CI(M)$, and show that $Ψ_{1,0,V}^\infty(M_0)$ is a ``microlocalization'' of $\DiffV{*}(M_0)$. Finally, we introduce and study semi-classical and ``suspended'' versions of the algebra $Ψ_{1,0,V}^\infty(M_0)$. Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto.

math.AP↗
arXiv2006

Surgery and Harmonic Spinors

Let M be a compact manifold with a fixed spin structure χ. The Atiyah-Singer index theorem implies that for any metric g on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and χ. We show that for generic metrics on M this bound is attained.

math.DG↗
arXiv2006

The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions

Let us fix a conformal class $[g_0]$ and a spin structure $σ$ on a compact manifold $M$. For any $g\in [g_0]$, let $λ^+_1(g)$ be the smallest positive eigenvalue of the Dirac operator $D$ on $(M,g,σ)$. In a previous paper we have shown that $$λ_{min}(M,g_0,σ):=\inf_{g\in [g_0]} λ_1^+(g)\vol(M,g)^{1/n}>0.$$ In the present article, we enlarge the conformal class by certain singular metrics. We will show that if $λ_{min}(M,g_0,σ)<λ_{min}(S^n)$, then the infimum is attained on the enlarged conformal class. For proving this, we have to solve a system of semi-linear partial differential equations involving a nonlinearity with critical exponent: $$Dϕ= λ|ϕ|^{2/(n-1)}ϕ.$$ The solution of this problem has many analogies to the solution of the Yamabe problem. However, our reasoning is more involved than in the Yamabe problem as the eigenvalues of the Dirac operator tend to $+\infty$ and $-\infty$. Using the Weierstraß representation, the solution of this equation in dimension 2 provides a tool for constructing new periodic constant mean curvature surfaces.

math.DG↗
arXiv2005

An obstruction for the mean curvature of a conformal immersion S^{n}-> R^{n+1}

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H of a conformal immersion S^{n}-> R^{n+1} satisfies $\int \partial_X H=0$ where X is a conformal vector field on S^{n} and where the integration is carried out with respect to the Euclidean volume measure of the image. This identity is analogous to the Kazdan-Warner obstruction that appears in the problem of prescribing the scalar curvature on S^{n} inside the standard conformal class.

math.DG↗
arXiv2005

The second Yamabe invariant

Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to $g$ and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.

math.DG↗
arXiv2004

Manifolds with small Dirac eigenvalues are nilmanifolds

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let r=1 if n=2,3 and r=2^{[n/2]-1}+1 if n\geq 4. We show that if the square of the Dirac operator on such a manifold has $r$ small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.

math.DG↗
arXiv2003

On the geometry of Riemannian manifolds with a Lie structure at infinity

A manifold with a ``Lie structure at infinity'' is a non-compact manifold $M_0$ whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on $M \smallsetminus M_0$. The Lie structure at infinity on $M_0$ determines a metric on $M_0$ up to bi-Lipschitz equivalence. This leads to the natural problem of understanding the Riemannian geometry of these manifolds. We prove, for example, that on a manifold with a Lie structure at infinity the curvature tensor and its covariant derivatives are bounded. We also study a generalization of the geodesic spray and give conditions for these manifolds to have positive injectivity radius. An important motivation for our work is to study the analysis of geometric operators on manifolds with a Lie structure at infinity. For example, a manifold with cylindrical ends is a manifold with a Lie structure at infinity. The relevant analysis in this case is that of totally characteristic operators on a compact manifold with boundary equipped with a ``b-metric.'' The class of conformally compact manifolds, which was recently proved of interest in the study of Einstein's equation, also consists of manifolds with a Lie structure at infinity.

math.DG↗
arXiv2002

Complex powers and non-compact manifolds

We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, ``extended Weyl algebras,'' whose definition was inspired by Guillemin's paper on the subject. An extended Weyl algebra can be thought of as an algebra of ``abstract pseudodifferential operators.'' Many algebras of pseudodifferential operators are extended Weyl algebras. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between apropriate Sobolev spaces, >...) generalize to extended Weyl algebras. Most important, our results may be used to obtain precise estimates at infinity for $A^{z}$, when $A > 0 $ is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative $Ψ^*$--algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds).

math.OA↗
arXiv2002

Dirac eigenvalue estimates on surfaces

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or, alternatively, by stable norms of certain cohomology classes. In case of the 2-torus we obtain a positive lower bound for all Riemannian metrics and all nontrivial spin structures. The corresponding estimate also holds for the $L^2$-spectrum of the Dirac operator on a noncompact complete surface of finite area. As a corollary we get positive lower bounds on the Willmore integral for all 2-tori embedded in $R^3$.

math.DG↗
arXiv2001

Spectral estimates on 2-tori

We prove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particluar we give a lower bound for the first eigenvalue of the Dirac operator for non-trivial spin structures. It is the only explicit estimate for eigenvalues of the Dirac operator known so far that uses information about the spin structure. As a corollary we obtain lower bounds for the Willmore functional of a torus embedded into S_3. In the final section we compare Dirac spectra for two different spin structures on an arbitrary Riemannian spin manifold.

math.DG↗
arXiv1999

Minimal Geodesics and Nilpotent Fundamental Groups

Hedlund constructed Riemannian metrics on n-tori, $n \geq 3$ for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics are optimal for nilpotent fundamental groups.

dg-ga↗
arXiv1999

Dirac eigenvalues and total scalar curvature

It has recently been conjectured that the eigenvalues $λ$ of the Dirac operator on a closed Riemannian spin manifold $M$ of dimension $n\ge 3$ can be estimated from below by the total scalar curvature: $$ λ^2 \ge \frac{n}{4(n-1)} \cdot \frac{\int_M S}{vol(M)}. $$ We show by example that such an estimate is impossible.

math.DG↗
arXiv1998

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to $\pm\infty$ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities. Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

math.DG↗