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The Height of Compact Nonsingular Heisenberg-Like Nilmanifolds

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The Height of Compact Nonsingular Heisenberg-Like Nilmanifolds THE HEIGHT OF COMPACT NONSINGULAR HEISENBERG-LIKE NILMANIFOLDS Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät der Humboldt-Universität zu Berlin von Sebastian Boldt Präsidentin der Humboldt-Universität zu Berlin: Prof. Dr.-Ing. Dr. Sabine Kunst Dekan der Mathematisch-Naturwissenschaftlichen Fakultät: Prof. Dr. Elmar Kulke Gutachter/innen: 1. Prof. Dr. Dorothee Schüth 2. Prof. Dr. Carolyn Gordon 3. Prof. Dr. Werner Ballmann Tag der Verteidigung: 27.10.2017 Abstract This thesis examines the spectral z-function z((M, g), s) and the height z1((M, g), 0) of compact nonsingular Heisenberg-like nilmanifolds (M, g). Heisenberg-like nilmani- folds were introduced by Gornet and Mast as a natural generalisation of nilmanifolds of Heisenberg-type. In case M is a Heisenberg manifold GzHn, every metric induced by a left-invariant metric on Hn is Heisenberg-like. The first result of the main chapter consists of formulas for the spectral z-function and the height which are valid for all Heisenberg-like metrics. Hereafter, by means of these formulas several results concerning the (non-)existence of global lower bounds for the height z1(¨, 0) and the z-function z(¨, s) for positive s not in the pole set are proved. The first of these results is the existence of lower bounds for the height andthe z-function on the moduli space of metrics of Heisenberg-type and with volume 1 on a given nilmanifold M. Next, conditions for the existence of lower bounds for the height and the z-function on the moduli space of Heisenberg-like metrics with volume 1 on M are investigated. In case of the height, lower bounds exist if the dimension of M equals 3 modulo 4. In case of z(¨, s), the existence of lower bounds depends on the position of s. To understand the remaining dimensions resp. positions of s, we introduce a path in the moduli space of Heisenberg-like metrics with volume 1 on M. It is shown that the height is not bounded from below on this path if the dimension of M equals 0 or 1 modulo 4. Analogously, it is shown that for all s not covered by above existence results z(¨, s) is not bounded from below along this path. The final result regarding the existence of lower bounds is that the height and the z-function are always bounded from below if one restrics to the moduli space of Heisenberg-like metrics with volume 1 and sectional curvature bounded from above by a positive constant. In case M is a Heisenberg mani- fold, all of the above results concerning the existence of lower bounds are strengthened. Under the same assumptions the existence of global minima is concluded. The last section treats concrete minima of the height and the z-function of Heisenberg manifolds. First, it is shown that that certain Heisenberg-type nilmanifolds in dimension 3, 5, 9 and 25 are (local) minima under variations of the metric in the space of Heisenberg- type metrics with volume 1. These nilmanifolds are characterised by the fact that the corresponding 2, 4, 8, resp. 24 dimensional base torus is the quotient by the hexagonal, checkerboard, E8-root resp. Leech lattice. The final result is that a global minimum ofthe height or the z-function of three dimensional Heisenberg manifolds is always attained at a metric for which the corresponding base torus is the quotient by a hexagonal lattice. ii Zusammenfassung Die vorliegende Arbeit untersucht die spektrale z-Funktion z((M, g), s) und die Höhe z1((M, g), 0) kompakter nicht-singulärer heisenbergartiger Nilmannigfaltigkeiten (M, g). Heisenbergartige Nilmannigfaltigkeiten wurden von Gornet und Mast als natürliche Ve- rallgemeinerung von Nilmannigfaltigkeiten vom Heisenberg-Typ eingeführt. Ist M eine Heisenbergmannigfaltigkeit GzHn, so ist jede von einer links-invarianten Metrik auf Hn induzierte Metrik heisenbergartig. Das erste Resultat besteht aus Formeln für die spektrale z-Funktion und die Höhe einer heisenbergartigen Nilmannigfaltigkeit. Mithilfe dieser Formeln werden im Weit- eren mehrere Ergebnisse zur (Nicht-)Existenz globaler unterer Schranken der Höhe z1(¨, 0) und der z-Funktion z(¨, s) für positive s bewiesen. Das erste ist die Existenz unterer Schranken für Höhe und z-Funktion auf dem Modulraum der Metriken vom Heisenberg- Typ mit Volumen 1 einer gegebenen Nilmannigfaltigkeit M. Daraufhin wird die Existenz unterer Schranken der Höhe und der z-Funktion auf dem Modulraum der heisenbergar- tigen Metriken mit Volumen 1 über M untersucht. Solche existieren im Falle der Höhe, falls die Dimension von M gleich 3 modulo 4 beträgt. Im Falle von z(¨, s) hängt die Exis- tenz von der Position von s ab. Um die verbleibenden Dimensionen bzw. Positionen von s zu verstehen, wird ein Weg im Modulraum der heisenbergartigen Metriken mit Volumen 1 auf M eingeführt. Es wird gezeigt, dass, wenn die Dimension von M gleich 0 oder 1 modulo 4 ist, die Höhe auf diesem Weg nicht von unten beschränkt ist. Analog wird gezeigt, dass z(¨, s) für alle s, die von obigen Resultaten nicht abgedeckt wurden, auf diesem Weg nicht von unten beschränkt ist. Das finale Ergebnis zur Existenz globaler unterer Schranken ist dann, dass sowohl Höhe als auch z-Funktion stets von unten beschränkt sind, wenn man den Modulraum der heisenbergartigen Metriken mit Volumen 1 und von oben beschränkter Schnittkrümmung betrachtet. Der letzte Abschnitt behandelt konkrete Minima der Höhe und z-Funktion von Heisen- bergmannigfaltigkeiten. Zunächst wird gezeigt, dass in den Dimensionen 3, 5, 9 und 25 gewisse Heisenberg-Typ Mannigfaltigkeiten (lokale) Minima unter Variationen der Metrik im Raum der Heisenberg-Typ Metriken mit Volumen 1 sind. Diese zeichnen sich dadurch aus, dass der entsprechende zugehörige 2, 4, 8, bzw. 24 dimensionale Basistorus Quotient nach dem hexagonalen, Checkerboard-, E8-Wurzel- bzw. Leech-Gitter ist. Das finale Resultat ist, dass ein globales Minimum der Höhe bzw.der z-Funktion einer drei- dimensionalen Heisenbergmannigfaltigkeit stets an einer Metrik angenommen wird, für die der zugehörige Basistorus der Quotient nach einem hexagonalen Gitter ist. iii Contents Introduction 1 Chapter 1. Preliminaries 9 1. Structure of compact 2-step Nilmanifolds 9 2. The Manifold Pn of Symmetric and Positive Definite n ˆ n Matrices 19 3. Schwartz Functions, the Fourier Transform and the Poisson Summation Formula 21 4. Parameter Dependent Integrals 24 5. Moduli Spaces 26 6. A Poisson Type Formula 43 Chapter 2. z-Functions and Heights 63 1. Formulas for the z-Function and the Height 63 2. (Non-)Existence of Global Minima 71 3. Extremal Metrics 105 Bibliography 125 v Introduction Let Mn be a compact n-dimensional manifold and A a 2nd order elliptic self-adjoint op- erator on M. By elliptic PDE theory the spectrum of A is a pure point spectrum with only accumulation point 8. Furthermore, every eigenvalue of A is real and of finite multiplic- ity. We assume further that A is bounded from below so that at most finitely many of its eigenvalues are negative. The spectral z-function of A is defined by ´s (0.1) zA(s) := l , ‰ l¸0 where the sum extends over all nonzero eigenvalues of A and the branch of ls is chosen s so that l ¡ 0 when l ¡ 0 and s P R. The sum in (0.1) converges absolutely for <s ¡ n/2 by known estimates for the growth of the eigenvalues of A. Furthermore, zA extends to a meromorphic function on C which is regular at s = 0 (see [See67]). The value 1 d (0.2) zA(0) = zA(s) ds |s=0 is called the height of A. The associated value ´ 1 ( ) (0.3) det A := e zA 0 is called the determinant of A. This naming can be justified as follows. The derivative of (0.1) for <s large enough is given by d z (s) = ´ (ln l) ¨ l´s . ds A ‰ l¸0 Formally setting s = 0 in this formula and substituting into (0.3) yields ! ´ 1 ( ) e zA 0 = exp ln l = l , ‰ ‰ l¸0 l¹0 which can be seen as a generalisation of the well-known determinant formula from linear algebra. Of course, this calculation was purely formal. Indeed, the product on the right hand side does not converge. But we assign a finite value to it by the process of analytic continuation. Usually, M comes equipped with a Riemannian metric g and A = Ag is chosen to be a canonically given geometric operator such as the Laplace-Beltrami operator Dg, the 1 2 INTRODUCTION Hodge-Laplace operator Dg,p acting on exterior p-forms, the square of the Atiyah-Singer (Dirac) operator Dg or the conformal Laplacian Lg. In case of the Laplace-Beltrami oper- ator we also speak of the height and the determinant of (M, g). Since det Ag is a spectral invariant, an important question is: How is the geometry of (M, g) related to the determinant (of Ag)? One line of work that explores this question examines the behaviour of the determinant under changes of the metric. More specifically, one relates the extrema of the determinant to the degree of symmetry of the metrics at which these are attained. For the determinant to possibly have any local extrema, one has to normalise the volume: a uniform scaling ´1 ´1 of the metric g by a constant t results in Ag as above being scaled by t : At¨g = t ¨ Ag. By formulas (0.2) and (0.3) we then have ´zAg (0) det Atg = t ¨ det Ag . It is very well known that in odd dimensions zAg (0) = ´ dim ker Ag, whereas in even dimensions zAg (0) is related to the heat invariants of Ag.
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