# Browsing by Author "Stern, Mark"

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Item Open Access Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics(2016) Andreae, PhillipThe central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

Item Open Access Equivariant Nahm Transforms and Minimal Yang--Mills Connections(2020) Beckett, Matthew James PaulThis dissertation examines two different subjects within the study of instantons: the construction of Nahm transforms for instantons invariant under certain group actions; and a generalization of the proof that Yang--Mills minimizers are instantons.

The first Nahm transform examined is the ADHM construction for $S^1$-invariant instantons on $S^4$, which correspond to singular monopoles on $\RR^3$. In this case, there is a decomposition of the ADHM data in terms of $S^1$-subrepresentations of $\ker \Dir$. The moduli spaces of $S^1$-invariant $SU(2)$-instantons are given up to charge 3, and examples of ADHM data for instantons of charge $4$ are also provided.

The second Nahm transform considered is for instantons on a certain flat quotient of $\RR^4$ with nonabelian fundamental group. Equivalently, one can consider these to be $\ZZ_2$-invariant instantons on $T^4$, and the Nahm transform yields instantons invariant under a crystallographic action.

In our study of minimal Yang--Mills connections, we extend results of Bourguignon--Lawson--Simons and Stern, who showed that connections that minimize $\|F_\nabla\|^2$ on homogeneous manifolds must be instantons or have instanton subbundles. We extend the previous arguments by considering variations constructed using conformal vector fields, and also allow these vector fields to be incomplete. We prove a minimality result over a half-cylinder.

Item Open Access Harmonic Forms, Price Inequalities, and Benjamini-Schramm ConvergenceCerbo, Luca F Di; Stern, MarkWe study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature, we have a convergence result for weakly uniform discrete sequences of closed Riemannian manifolds. In the negative sectional curvature case, we are able to remove the weakly uniform discreteness assumption. This is achieved by combining a refined Thick-Thin decomposition together with a Moser iteration argument for harmonic forms on manifolds with boundary.Item Open Access Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem(Journal of Differential Geometry, 2019-12-06) Cherkis, Sergey A; Larrain-Hubach, Andres; Stern, MarkWe study finite action anti-self-dual Yang-Mills connections on the multi-Taub-NUT space. We establish the curvature and the harmonic spinors decay rates and compute the index of the associated Dirac operator. This is the first in a series of papers proving the completeness of the bow construction of instantons on multi-Taub-NUT spaces and exploring it in detail.Item Open Access Instantons on multi-Taub-NUT Spaces II: Bow ConstructionCherkis, Sergey; Larraín-Hubach, Andrés; Stern, MarkUnitary anti-self-dual connections on Asymptotically Locally Flat (ALF) hyperk\"ahler spaces are constructed in terms of data organized in a bow. Bows generalize quivers, and the relevant bow gives rise to the underlying ALF space as the moduli space of its particular representation -- the small representation. Any other representation of that bow gives rise to anti-self-dual connections on that ALF space. We prove that each resulting connection has finite action, i.e. it is an instanton. Moreover, we derive the asymptotic form of such a connection and compute its topological class.Item Open Access Nonlinear Harmonic Forms and an Indefinite Bochner Formula(2017-06-01) Stern, MarkWe introduce the study of nonlinear harmonic forms. These are forms which minimize the $L_2$ energy in a cohomology class subject to a nonlinear constraint. In this note, we include only motivations and the most basic existence results. We also introduce a variant of the Bochner formula suitable for probing the structure of the intersection form of a 4-manifold.Item Open Access On the Betti Numbers of Compact Rank 2 Locally Symmetric Spaces(2024-04-23) Ong, NathanaelWe obtain upper bounds for the second Betti numbers of compact rank 2 locally symmetric spaces, namely $\Gamma\backslash SL(3)/SO(3)$, $\Gamma\backslash Sp(4)/U(2)$, and $\Gamma\backslash G_{2(2)}/SO(4)$, where $\Gamma$ is a cocompact, torsion free lattice. We use representation theory and directly apply the techniques of Di Cerbo and Stern in \cite{dicerbo2019price}. In the case of $\Gamma\backslash Sp(4)/U(2)$, we also use unitary holonomy, the complex structure operator that arises from it and the (p,q) decomposition of exterior powers to obtain stronger bounds. In particular, the bounds we provide on the Betti numbers of $\Gamma\backslash Sp(4)/U(2)$ and $\Gamma\backslash G_{2(2)}/SO(4)$ are exponential bounds involving injectivity radius. However, the bound we obtained for $\Gamma\backslash SL(3)/SO(3)$ is a weaker polynomial one.Item Open Access On the Betti Numbers of Finite Volume Hyperbolic ManifoldsCerbo, Luca F Di; Stern, MarkWe obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS17]. We also provide effective upper bounds for Betti numbers of compact quaternionic- and Cayley-hyperbolic manifolds in most degrees. More importantly, we extend our techniques to complete finite volume real- and complex-hyperbolic manifolds. In this setting, we develop new monotonicity inequalities for strongly harmonic forms on hyperbolic cusps and employ a new peaking argument to estimate $L^2$-cohomology ranks. Finally, we provide bounds on the de Rham cohomology of such spaces, using a combination of our bounds on $L^2$-cohomology, bounds on the number of cusps in terms of the volume, and the topological interpretation of reduced $L^2$-cohomology on certain rank one locally symmetric spaces.Item Open Access Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points(2017-06-01) Cerbo, Luca F Di; Stern, MarkWe derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points. Finally, we give a vanishing result for $L^{2}$-Betti numbers of closed manifolds without conjugate points.Item Open Access Solitons for the Closed G2 Laplacian Flow(2021-04-19) Dayaprema, AnukGeometric flows are partial differential equations on smooth manifolds which describe the time evolution of some geometric structure on the manifold, such as a Riemannian metric. In the setting of G2 geometry, which is specific to seven dimensions, a natural geometric flow to consider is the (closed) G2 Laplacian flow. Critical points of this flow correspond to torsion-free G2 structures, which satisfy a system of nonlinear partial differential equations. We are interested in such G2 structures since they give rise to Riemannian metrics with exceptional holonomy. Solitons for the Laplacian flow are self-similar solutions which we hope provide models for finite-time singularities of the flow. Since the soliton equation is a nonlinear system of PDEs, to get a feel for concrete solutions we consider a dimensional reduction of the equation by studying cohomogeneity-one solitons. We consider the cohomogeneity-one setting where the group G of isometries is either SU(3) or Sp(2). In these regimes, we show that the soliton equation is equivalent to a (nonlinear) first-order system of real-analytic ODEs. We then describe the space of G-invariant closed G2 cones, identifying it with a smooth surface/curve in R3. We then investigate the uniqueness of a complete shrinking Sp(2)-invariant soliton which possesses an asymptotically conical geometry, and conjecture that it is rigid to first order. We finally discuss ways to study the general closed Laplacian soliton equation, reviewing previous results by others in related directions, with an eye towards asymptotically conical shrinking solitons in particular.