Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0305189

Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in K-theory

Y. Kordyukov·V. Mathai·M. Shubin
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study a refined L2 version of the semiclassical approximation of projectively invariant elliptic operators with invariant Morse type potentials on covering spaces of compact manifolds. We work on the level of spectral projections (and not just their traces) and obtain an information about classes of these projections in K-theory in the semiclassical limit as the coupling constant goes to zero. An important corollary is a vanishing theorem for the higher traces in cyclic cohomology for the spectral projections. This result is then applied to the quantum Hall effect. We also give a new proof that there are arbitrarily many gaps in the spectrum of the operators under consideration in the semiclassical limit.

Explore related subjects

math.DGmath-phmath.KTmath.MPmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Y. Kordyukov, V. Mathai, M. Shubin. 2004-04-26. Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in K-theory. https://doi.org/10.1515/crll.2005.2005.581.193

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Shrinkers of the area-preserving curve-shortening flow: Classification and saddle-point property

We consider homothetic evolutions of the area-preserving curve-shortening flow (APCSF), that is, classical curve-shortening flow with an additional non-local forcing term. By using known results on $λ$-curves, we prove the existence of non-circular shrinkers for this flow. In our first main result, we present a full classification of such curves, similar to the well-known Abresch-Langer classification for shrinkers of curve-shortening flow. Finally, we also deduce a saddle-point property for all non-circular (APCSF)-shrinkers analogous to the known saddle-point property of Abresch-Langer curves.

math.DG↗
arXiv2026

A non-trivial index difference on surfaces of genus at least $3$

For every closed surface of genus at least $3$, equipped with any bounding spin structure, we show that the index difference, viewed as a map from the fundamental group of the space of Dirac-invertible Riemannian metrics to $\KO^{-4}(*)$, is non-trivial. If the genus is at least $5$, we also show that the aforementioned index difference is surjective. For products of two closed surfaces of genus at least $3$, equipped with any spin structure, we prove that the corresponding space of Dirac-invertible Riemannian metrics is not contractible. We discuss the relationship of this result to the existence of metrics with harmonic spinors in dimension~$4$.

math.DG↗
arXiv2026

A sharp mass-volume inequality on Einstein manifolds and ADM mass

Let $(M^n,g), n\geq 4$ be a close Einstein manifold with positive scalar curvature and $G$ be the Green function of the conformal Laplacian. The ADM mass of the asymptotically flat metric $\hat g:=G^{4/(n-2)} g$ is an important geometric invariant. When either $n=5$ or $n\geq 6$ with Weyl vanishing condition, we prove a mass formula, generalizing Gursky-Malchiodi's formula in dimension $4$. An important step is an extension of Viaclovsky's Green function expansion in higher dimensions. Next we prove a sharp inequality related to the volume radius and the mass for Einstein manifolds. Finally we present two applications. First, we prove a mass gap theorem for Einstein manifolds. Second, for a volume collapsing family of normalized Einstein $n$-manifolds $(M_i,g_i)$ (satisfying the Weyl vanishing condition when $n\geq 6$), we prove that the mass sequence $m(\widehat{g_i})$ blows up uniformly as $i\to\infty$. This in particular applies to Collins-Székelyhidi's family of Sasaki-Einstein metrics $g_i$ on $\mathbb{S}^5$, and to the recent $4$-dimensional examples of T. Ozuch.

math.DG↗