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John Lott

Publications and source records attributed to John Lott.

At least 19 recordsLinked to original sources
arXiv2026

Under Ricci flow, a 3-torus goes flat

We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type $\mathbb{R}^3$, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric $g(t)$ converges exponentially fast to a flat metric. The Gromov--Hausdorff limit of $(M,t^{-1}g(t))$ is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, $s^{-1}\widetilde g(s\tau)$, converge, in the pointed Cheeger--Hamilton sense, to an explicit homogeneous expanding Ricci soliton

math.DG↗
arXiv2023

The Ray-Singer torsion

In 1971, Ray and Singer proposed an analytic equivalent of a classical topological invariant, the R-torsion. This Ray-Singer torsion has had many ramifications in mathematics and physics. I will describe the background, the Ray-Singer papers and some subsequent work.

math.DG↗
arXiv2022

A spinorial quasilocal mass

We define a quasilocal energy of a compact manifold-with-boundary, relative to a background manifold. The construction uses spinors on one manifold and the pullback of dual spinors from the other manifold. We prove positivity results for the quasilocal energy, in both the Riemannian and Lorentzian settings.

math.DG↗
arXiv2021

On scalar curvature lower bounds and scalar curvature measure

We relate the (non)existence of lower scalar curvature bounds to the existence of certain distance-decreasing maps. We also give a sufficient condition for the existence of a limiting scalar curvature measure in the backward limit of a Ricci flow solution.

math.DG↗
arXiv2020

Kasner-like regions near crushing singularities

We consider vacuum spacetimes with a crushing singularity. Under some scale-invariant curvature bounds, we relate the existence of Kasner-like regions to the asymptotics of spatial volume densities.

math.DG↗
arXiv2019

On 3-manifolds with pointwise pinched nonnegative Ricci curvature

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.

math.DG↗
arXiv2019

On the initial geometry of a vacuum cosmological spacetime

In the first part of this paper we consider expanding vacuum cosmological spacetimes with a free $T^N$-action. Among them, we give evidence that Gowdy spacetimes have AVTD (asymptotically velocity term dominated) behavior for their initial geometry, in any dimension. We then give sufficient conditions to reach a similar conclusion about a $T^2$-invariant four dimensional nonGowdy spacetime. In the second part of the paper we consider vacuum cosmological spacetimes with crushing singularities. We introduce a monotonic quantity to characterize Kasner spacetimes. Assuming scale-invariant curvature bounds and local volume bounds, we give results about causal pasts.

math.DG↗
arXiv2019

A Dolbeault-Hilbert complex for a variety with isolated singular points

Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.

math.DG↗
arXiv2018

Singular Ricci flows II

We establish several quantitative results about singular Ricci flows, including estimates on the curvature and volume, and the set of singular times.

math.DG↗
arXiv2017

On noncollapsed almost Ricci-flat 4-manifolds

We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.

math.DG↗
arXiv2017

Backreaction in the future behavior of an expanding spacetime

We perform a rescaling analysis to analyze the future behavior of a class of $T^2$-symmetric vacuum spacetimes. We show that on the universal cover, there is $C^0$-convergence to a spatially homogeneous spacetime that does not satisfy the vacuum Einstein equations.

math.DG↗
arXiv2017

The collapsing geometry of almost Ricci-flat 4-manifolds

We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space, with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region is semiflat Kaehler.

math.DG↗