We prove a surgery formula for the smooth Yamabe invariant σ(M ) of a compact manifold M . Assume... more We prove a surgery formula for the smooth Yamabe invariant σ(M ) of a compact manifold M . Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N ) ≥ min{σ(M ), Λn}.
Communications in Mathematical Physics, Apr 16, 2013
The restriction of a parallel spinor on some spin manifold Z to a hypersurface M ⊂ Z is a general... more The restriction of a parallel spinor on some spin manifold Z to a hypersurface M ⊂ Z is a generalized Killing spinor on M. We show, conversely, that in the real analytic category, every spin manifold (M, g) carrying a generalized Killing spinor ψ can be isometrically embedded as a hypersurface in a spin manifold carrying a parallel spinor whose restriction to M is ψ. We also answer negatively the corresponding question in the smooth category.
Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg, Dec 19, 2016
We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to produc... more 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. Then the mass of (Z, g) is nonnegative. Moreover, if the mass of (Z, g) is zero, then
We study the L p-spectrum of the Dirac operator on complete manifolds. One of the main questions ... more 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.
Let (M , g) be a time-and space-oriented Lorentzian spin manifold, and let M be a compact spaceli... more Let (M , g) be a time-and space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of M with induced Riemannian metric g and second fundamental form K. If (M , g) satisfies the dominant energy condition in a strict sense, then the Dirac-Witten operator of M ⊆ M is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial on M satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's α-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac-Witten operator may be non-invertible, and we will study the kernel of this operator in this case. We will show that the kernel may only be non-trivial if π 1 (M) is virtually solvable of derived length at most 2. This allows to extend the index theoretical methods to spaces of initial data, satisfying the dominant energy condition in the weak sense. We will show further that a spinor ϕ is in the kernel of the Dirac-Witten operator on (M, g, K) if and only if (M, g, K, ϕ) admits an extension to a Lorentzian manifold (N , h) with parallel spinorφ such that M is a Cauchy hypersurface of (N , h), such that g and K are the induced metric and second fundamental form of M , respectively, and ϕ is the restriction ofφ to M. Contents DOMINANT ENERGY CONDITION AND SPINORS ON LORENTZIAN MANIFOLDS 1 4.1. The hypersurface spinor bundle 10 4.2. Dirac currents 11 4.3. The Cauchy problem for parallel spinors 12 5. The kernel of the Dirac-Witten operator 15 5.1. Dirac-Witten operators and Schrödinger-Lichnerowicz formula 15 5.2. From the kernel of the Dirac-Witten operator to initial data triples 16 5.3. Examples of lightlike (generalized) initial data triples 20 5.4. Lightlike initial data manifolds and other notation 22 5.5. Lightlike generalized initial data triples with compact leaves 24 5.6. More results for all lightlike generalized initial data triples 25 5.7. Further results in the case of non-compact leaves 29 5.8. Conclusions 32 6. Homotopy groups of I > (M) and I ≥ (M) 32 6.1. Initial data sets and positive scalar curvature 33 6.2. The α-index and index difference for psc metrics 35 6.3. The index difference for initial data sets strictly satisfying DEC 37 6.4. Application to general relativity 40 Concluding remark 42 Appendix A. The Taylor development map for Ricci-flat metrics 42 Appendix B. Proof of Lemma 5.17 45 References 46
Calculus of Variations and Partial Differential Equations, Sep 30, 2016
We study the L p-spectrum of the Dirac operator on complete manifolds. One of the main questions ... more 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.
Dirac-harmonic maps (f, ϕ) consist of a map f ∶ M → N and a twisted spinor ϕ ∈ Γ(ΣM ⊗ f * T N) an... more Dirac-harmonic maps (f, ϕ) consist of a map f ∶ M → N and a twisted spinor ϕ ∈ Γ(ΣM ⊗ f * T N) and they are defined as critical points of the super-symmetric energy functional. A Dirac-harmonic map is called uncoupled, if f is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps defined on a closed domain are uncoupled.
We prove a lower bound for the eigenvalues of the Dirac operator on two-dimensional tori equipped... more We prove a lower bound for the eigenvalues of the Dirac operator on two-dimensional tori equipped with a non-trivial spin structure.
Mitteilungen der Deutschen mathematiker-Vereinigung, Sep 1, 2018
Cédric Villani war der Hauptredner der Gauß-Vorlesung der DMV in Regensburg im Oktober . Da V... more Cédric Villani war der Hauptredner der Gauß-Vorlesung der DMV in Regensburg im Oktober . Da Villani im Juni zuvor für die Partei En Marche! von Emmanuel Macron zum Abgeordneten des französischen Parlaments gewählt wurde, war es eine kleine Sensation, dass er den lange versprochenen Termin einhalten konnte. Obwohl die deutsche Presse die Veranstaltung nahezu ignorierte, war der wunderschöne Neuhaussaal des Regensburger Theaters fast bis auf den letzten Platz gefüllt und zur Vermeidung von Überfüllung mussten sogar kostenlose Platzkarten ausgegeben werden.
Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metricsg co... more Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metricsg conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with respect tog is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension ≥ 2.
Let D be the Dirac operator on a compact spin manifold M. Assume that 0 is in the spectrum of D. ... more Let D be the Dirac operator on a compact spin manifold M. Assume that 0 is in the spectrum of D. We prove the existence of a lower bound on the first positive eigenvalue of D depending only on the spin structure and the conformal type.
One of the fundamental problems in Riemannian geometry is to understand the relation of locally d... more One of the fundamental problems in Riemannian geometry is to understand the relation of locally defined curvature invariants and global properties of smooth manifolds. This workshop was centered around the investigation of scalar curvature, addressing questions in global analysis, geometric topology, relativity and minimal surface theory.
, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds wit... more , 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 $\Psi_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at infinity $V \subset\Gamma(TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to $\Psi_{1,0,V}^\infty(M_0)$. We also consider the algebra $\DiffV{*}(M_0)$ of differential operators on $M_0$ generated by $V$ and $\CI(M)$, and show that $\Psi_{1,0,V}^\infty(M_0)$ is a ``microlocalization'' of $\DiffV{*}(M_0)$. Finally, we introduce and study semi-classical and ``suspended'' versions of the algebra $\Psi_{1,0,V}^\infty(M_0)$. Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto.
We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume t... more We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N) ≥ min{σ(M), Λn}.
Let M be a compact manifold equipped with a Riemannian metric g and a spin structure σ. We let λ ... more Let M be a compact manifold equipped with a Riemannian metric g and a spin structure σ. We let λ + min (M, [g], σ) = infg ∈[g] λ + 1 (g)V ol(M,g) 1/n where λ + 1 (g) is the smallest positive eigenvalue of the Dirac operator D in the metricg. A previous result stated that λ + min (M, [g], σ) ≤ λ + min (S n) = n 2 ω 1/n n where ω n stands for the volume of the standard n-sphere. In this paper, we study this problem for conformally flat manifolds of dimension n ≥ 2 such that D is invertible. E.g., we show that strict inequality holds in dimension n ≡ 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.
Let M be a compact manifold with a spin structure χ and a Riemannian metric g. Let λ 2 g be the s... more Let M be a compact manifold with a spin structure χ and a Riemannian metric g. Let λ 2 g 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 λ 2 g 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 √ π if χ is "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 √ π at one end of the spin-conformal moduli space.
Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metricg conformal t... more Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metricg conformal to g, we denote byλ the first positive eigenvalue of the Dirac operator on (M,g, σ). We show that inf g∈[g]λ Vol(M,g) 1/n ≤ (n/2) Vol(S n) 1/n. This inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D = {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with 2λ 2 ≤μ, whereμ denotes the first positive eigenvalue of the Laplace operator.
Journal of The Mathematical Society of Japan, 2015
Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dim... more Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k ≤ n − 3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N) ≥ min(σ(M), Λ) for a constant Λ > 0 depending only on n and k. We derive explicit positive lower bounds for Λ in dimensions where previous methods failed, namely for (n, k) ∈ {(4, 1), (5, 1), (5, 2), (6, 3), (9, 1), (10, 1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.
We prove a surgery formula for the smooth Yamabe invariant σ(M ) of a compact manifold M . Assume... more We prove a surgery formula for the smooth Yamabe invariant σ(M ) of a compact manifold M . Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N ) ≥ min{σ(M ), Λn}.
Communications in Mathematical Physics, Apr 16, 2013
The restriction of a parallel spinor on some spin manifold Z to a hypersurface M ⊂ Z is a general... more The restriction of a parallel spinor on some spin manifold Z to a hypersurface M ⊂ Z is a generalized Killing spinor on M. We show, conversely, that in the real analytic category, every spin manifold (M, g) carrying a generalized Killing spinor ψ can be isometrically embedded as a hypersurface in a spin manifold carrying a parallel spinor whose restriction to M is ψ. We also answer negatively the corresponding question in the smooth category.
Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg, Dec 19, 2016
We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to produc... more 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. Then the mass of (Z, g) is nonnegative. Moreover, if the mass of (Z, g) is zero, then
We study the L p-spectrum of the Dirac operator on complete manifolds. One of the main questions ... more 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.
Let (M , g) be a time-and space-oriented Lorentzian spin manifold, and let M be a compact spaceli... more Let (M , g) be a time-and space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of M with induced Riemannian metric g and second fundamental form K. If (M , g) satisfies the dominant energy condition in a strict sense, then the Dirac-Witten operator of M ⊆ M is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial on M satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's α-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac-Witten operator may be non-invertible, and we will study the kernel of this operator in this case. We will show that the kernel may only be non-trivial if π 1 (M) is virtually solvable of derived length at most 2. This allows to extend the index theoretical methods to spaces of initial data, satisfying the dominant energy condition in the weak sense. We will show further that a spinor ϕ is in the kernel of the Dirac-Witten operator on (M, g, K) if and only if (M, g, K, ϕ) admits an extension to a Lorentzian manifold (N , h) with parallel spinorφ such that M is a Cauchy hypersurface of (N , h), such that g and K are the induced metric and second fundamental form of M , respectively, and ϕ is the restriction ofφ to M. Contents DOMINANT ENERGY CONDITION AND SPINORS ON LORENTZIAN MANIFOLDS 1 4.1. The hypersurface spinor bundle 10 4.2. Dirac currents 11 4.3. The Cauchy problem for parallel spinors 12 5. The kernel of the Dirac-Witten operator 15 5.1. Dirac-Witten operators and Schrödinger-Lichnerowicz formula 15 5.2. From the kernel of the Dirac-Witten operator to initial data triples 16 5.3. Examples of lightlike (generalized) initial data triples 20 5.4. Lightlike initial data manifolds and other notation 22 5.5. Lightlike generalized initial data triples with compact leaves 24 5.6. More results for all lightlike generalized initial data triples 25 5.7. Further results in the case of non-compact leaves 29 5.8. Conclusions 32 6. Homotopy groups of I > (M) and I ≥ (M) 32 6.1. Initial data sets and positive scalar curvature 33 6.2. The α-index and index difference for psc metrics 35 6.3. The index difference for initial data sets strictly satisfying DEC 37 6.4. Application to general relativity 40 Concluding remark 42 Appendix A. The Taylor development map for Ricci-flat metrics 42 Appendix B. Proof of Lemma 5.17 45 References 46
Calculus of Variations and Partial Differential Equations, Sep 30, 2016
We study the L p-spectrum of the Dirac operator on complete manifolds. One of the main questions ... more 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.
Dirac-harmonic maps (f, ϕ) consist of a map f ∶ M → N and a twisted spinor ϕ ∈ Γ(ΣM ⊗ f * T N) an... more Dirac-harmonic maps (f, ϕ) consist of a map f ∶ M → N and a twisted spinor ϕ ∈ Γ(ΣM ⊗ f * T N) and they are defined as critical points of the super-symmetric energy functional. A Dirac-harmonic map is called uncoupled, if f is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps defined on a closed domain are uncoupled.
We prove a lower bound for the eigenvalues of the Dirac operator on two-dimensional tori equipped... more We prove a lower bound for the eigenvalues of the Dirac operator on two-dimensional tori equipped with a non-trivial spin structure.
Mitteilungen der Deutschen mathematiker-Vereinigung, Sep 1, 2018
Cédric Villani war der Hauptredner der Gauß-Vorlesung der DMV in Regensburg im Oktober . Da V... more Cédric Villani war der Hauptredner der Gauß-Vorlesung der DMV in Regensburg im Oktober . Da Villani im Juni zuvor für die Partei En Marche! von Emmanuel Macron zum Abgeordneten des französischen Parlaments gewählt wurde, war es eine kleine Sensation, dass er den lange versprochenen Termin einhalten konnte. Obwohl die deutsche Presse die Veranstaltung nahezu ignorierte, war der wunderschöne Neuhaussaal des Regensburger Theaters fast bis auf den letzten Platz gefüllt und zur Vermeidung von Überfüllung mussten sogar kostenlose Platzkarten ausgegeben werden.
Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metricsg co... more Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metricsg conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with respect tog is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension ≥ 2.
Let D be the Dirac operator on a compact spin manifold M. Assume that 0 is in the spectrum of D. ... more Let D be the Dirac operator on a compact spin manifold M. Assume that 0 is in the spectrum of D. We prove the existence of a lower bound on the first positive eigenvalue of D depending only on the spin structure and the conformal type.
One of the fundamental problems in Riemannian geometry is to understand the relation of locally d... more One of the fundamental problems in Riemannian geometry is to understand the relation of locally defined curvature invariants and global properties of smooth manifolds. This workshop was centered around the investigation of scalar curvature, addressing questions in global analysis, geometric topology, relativity and minimal surface theory.
, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds wit... more , 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 $\Psi_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at infinity $V \subset\Gamma(TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to $\Psi_{1,0,V}^\infty(M_0)$. We also consider the algebra $\DiffV{*}(M_0)$ of differential operators on $M_0$ generated by $V$ and $\CI(M)$, and show that $\Psi_{1,0,V}^\infty(M_0)$ is a ``microlocalization'' of $\DiffV{*}(M_0)$. Finally, we introduce and study semi-classical and ``suspended'' versions of the algebra $\Psi_{1,0,V}^\infty(M_0)$. Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto.
We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume t... more We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N) ≥ min{σ(M), Λn}.
Let M be a compact manifold equipped with a Riemannian metric g and a spin structure σ. We let λ ... more Let M be a compact manifold equipped with a Riemannian metric g and a spin structure σ. We let λ + min (M, [g], σ) = infg ∈[g] λ + 1 (g)V ol(M,g) 1/n where λ + 1 (g) is the smallest positive eigenvalue of the Dirac operator D in the metricg. A previous result stated that λ + min (M, [g], σ) ≤ λ + min (S n) = n 2 ω 1/n n where ω n stands for the volume of the standard n-sphere. In this paper, we study this problem for conformally flat manifolds of dimension n ≥ 2 such that D is invertible. E.g., we show that strict inequality holds in dimension n ≡ 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.
Let M be a compact manifold with a spin structure χ and a Riemannian metric g. Let λ 2 g be the s... more Let M be a compact manifold with a spin structure χ and a Riemannian metric g. Let λ 2 g 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 λ 2 g 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 √ π if χ is "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 √ π at one end of the spin-conformal moduli space.
Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metricg conformal t... more Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metricg conformal to g, we denote byλ the first positive eigenvalue of the Dirac operator on (M,g, σ). We show that inf g∈[g]λ Vol(M,g) 1/n ≤ (n/2) Vol(S n) 1/n. This inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D = {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with 2λ 2 ≤μ, whereμ denotes the first positive eigenvalue of the Laplace operator.
Journal of The Mathematical Society of Japan, 2015
Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dim... more Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k ≤ n − 3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N) ≥ min(σ(M), Λ) for a constant Λ > 0 depending only on n and k. We derive explicit positive lower bounds for Λ in dimensions where previous methods failed, namely for (n, k) ∈ {(4, 1), (5, 1), (5, 2), (6, 3), (9, 1), (10, 1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.
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