Dr. Andreas Hermann

wissenschaftlicher Mitarbeiter

Kontakt
Raum:
2.09.0.20
Telefon:
+49 331 977-1347
...

Sprechzeiten:
Montag 13-14 Uhr

  • Spin-Geometrie
  • geometrische Analysis
  • mathematische Physik

2018 | Mass functions of a compact manifold | Andreas Hermann, Emmanuel Humbert Link zum Preprint

Mass functions of a compact manifold

Autoren: Andreas Hermann, Emmanuel Humbert (2018)

Let M be a compact manifold of dimension n. In this paper, we introduce the mass function a≥0→XM+(a) (resp. a≥0→XM-(a)) which is defined as the supremum (resp. infimum) of the masses of all metrics on M whose Yamabe constant is larger than a and which are flat on a ball of radius 1 and centered at a point p∈M. Here, the mass of a metric flat around p is the constant term in the expansion of the Green function of the conformal Laplacian at p. We show that these functions are well defined and have many properties which allow to obtain applications to the Yamabe invariant (i.e. the supremum of Yamabe constants over the set of all metrics on M).

2017 | On the positive mass theorem for closed Riemannian manifolds | Andreas Hermann, Emmanuel Humbert Verlag: Springer Buchtitel: Ji, Lizhen, Papadopoulos, Athanase, Yamada, Sumio (Eds.): From Riemann to Differential Geometry and Relativity Seiten: 515-540 Link zur Publikation

On the positive mass theorem for closed Riemannian manifolds

Autoren: Andreas Hermann, Emmanuel Humbert (2017)

The Positive Mass Conjecture for asymptotically flat Riemannian manifolds is a famous open problem in geometric analysis. In this article we consider a variant of this conjecture, namely the Positive Mass Conjecture for closed Riemannian manifolds. We explain why the two positive mass conjectures are equivalent. After that we explain our proof of the following result: If one can prove the Positive Mass Conjecture for one closed simply-connected non-spin manifold of dimension n≥5, then the Positive Mass Conjecture is true for all closed manifolds of dimension n.

Verlag:
Springer
Buchtitel:
Ji, Lizhen, Papadopoulos, Athanase, Yamada, Sumio (Eds.): From Riemann to Differential Geometry and Relativity
Seiten:
515-540

2016 | About the mass of certain second order elliptic operators | Andreas Hermann, Emmanuel Humbert Zeitschrift: Adv. Math. Verlag: Elsevier Seiten: 596-633 Band: 294 Link zur Publikation, Link zum Preprint

About the mass of certain second order elliptic operators

Autoren: Andreas Hermann, Emmanuel Humbert (2016)

Let (M,g) be a closed Riemannian manifold of dimension n ≥ 3 and let f ∈ C(M), such that the operator Pf := Δg + f is positive. If g is flat near some point p and f vanishes around p, we can define the mass of Pf as the constant term in the expansion of the Green function of Pf at p. In this paper, we establish many results on the mass of such operators. In particular, if f = \frac{n-2}{4(n-1)}\scal_g, i.e. if Pf is the Yamabe operator, we show the following result: assume that there exists a closed simply connected non-spin manifold M such that the mass is non-negative for every metric g as above on M, then the mass is non-negative for every such metric on every closed manifold of the same dimension as M.

Zeitschrift:
Adv. Math.
Verlag:
Elsevier
Seiten:
596-633
Band:
294

2014 | Mass endomorphism, surgery and perturbations | Bernd Ammann, Mattias Dahl, Andreas Hermann, Emmanuel Humbert Zeitschrift: Ann. Inst. Fourier Verlag: UJF Grenoble Seiten: 467-487 Band: 64, no. 2 Link zur Publikation, Link zum Preprint

Mass endomorphism, surgery and perturbations

Autoren: Bernd Ammann, Mattias Dahl, Andreas Hermann, Emmanuel Humbert (2014)

We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.

Zeitschrift:
Ann. Inst. Fourier
Verlag:
UJF Grenoble
Seiten:
467-487
Band:
64, no. 2

2014 | Zero sets of eigenspinors for generic metrics | Andreas Hermann Zeitschrift: Comm. Anal. Geom. Verlag: International Press Seiten: 177-218 Band: 22, no. 2 Link zur Publikation, Link zum Preprint

Zero sets of eigenspinors for generic metrics

Autoren: Andreas Hermann (2014)

Let M be a closed connected spin manifold of dimension 2 or 3 with a fixed orientation and a fixed spin structure. We prove that for a generic Riemannian metric on M the non-harmonic eigenspinors of the Dirac operator are nowhere zero. The proof is based on a transversality theorem and the unique continuation property of the Dirac operator.

Zeitschrift:
Comm. Anal. Geom.
Verlag:
International Press
Seiten:
177-218
Band:
22, no. 2

2010 | Generic metrics and the mass endomorphism on spin three-manifolds | Andreas Hermann Zeitschrift: Ann. Global Anal. Geom. Verlag: Springer Seiten: 163-171 Band: 37, no. 2 Link zur Publikation, Link zum Preprint

Generic metrics and the mass endomorphism on spin three-manifolds

Autoren: Andreas Hermann (2010)

Let (M,g) be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point p ∈ M is called the mass endomorphism in p associated to the metric g due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a generic metric on a three-dimensional spin manifold is nonzero. This implies a strict inequality which can be used to avoid bubbling-off phenomena in conformal spin geometry.

Zeitschrift:
Ann. Global Anal. Geom.
Verlag:
Springer
Seiten:
163-171
Band:
37, no. 2