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Thomas Richard

Publications and source records attributed to Thomas Richard.

10 recordsLinked to original sources
arXiv2026

Balancing The 2-Systoles Of Some K{\"a}hler Manifolds With Positive Scalar Curvature

Theorems by Bray-Brendle-Neves and Zhu about positive scalar curvature metrics on products of a 2-sphere and an n-torus suggests that positive scalar curvature suggests and appropriate topological assumptions should lead to the existence of a topologically non trivial 2-spheres of small area, which can be stated as upper bound on the 2-systole of such manifolds. Recent progress have been made in this direction by Sha and Tsiamis under the additional K{\"a}hler assumption while Checcini-Hirsh-Ziedler, Stryker and Tsiamis showed similar upper bounds on the stable 2-systole using index theoretic methods. We prove here similar inequalities for some K{\"a}hler manifold which control the relative sizes of the representatives of a well chosen set of homology classes. For instance on $(\mathbb{CP}^1x\mathbb{CP}^1 , \omega)$ with a positive scalar curvature K{\"a}hler metric we quantitavely show that largeness of one factor imposes smallness of the other one.

math.DG↗
arXiv2024

Strengthened injectivity radius bounds for manifolds with positive scalar curvature

Green's inequality shows that a compact Riemannian manifold with scalar curvature at least $n(n-1)$ has injectivity radius at most $\pi$, and that equality is achieved only for the radius 1 sphere. In this work we show how extra topological assumptions can lead to stronger upper bounds. The topologies we consider are $\mathbb{S}^2\times\mathbb{T}^{n-k-2}\times\mathbb{R}^k$ for $n\leq 7$ and $0\leq k\leq 2$ and 3-manifolds with positive scalar curvature except lens spaces $L(p,q)$ with $p$ odd. We also prove a strengthened inequality for $3$-manifolds with positive scalar curvature and large diameter. Our proof uses previous results of Gromov and Zhu.

math.DG↗
arXiv2021

Multi-resolution deep learning pipeline for dense large scale point clouds

Recent development of 3D sensors allows the acquisition of extremely dense 3D point clouds of large-scale scenes. The main challenge of processing such large point clouds remains in the size of the data, which induce expensive computational and memory cost. In this context, the full resolution cloud is particularly hard to process, and details it brings are rarely exploited. Although fine-grained details are important for detection of small objects, they can alter the local geometry of large structural parts and mislead deep learning networks. In this paper, we introduce a new generic deep learning pipeline to exploit the full precision of large scale point clouds, but only for objects that require details. The core idea of our approach is to split up the process into multiple sub-networks which operate on different resolutions and with each their specific classes to retrieve. Thus, the pipeline allows each class to benefit either from noise and memory cost reduction of a sub-sampling or from fine-grained details.

cs.CV↗
arXiv2018

Stability of nonnegative isotropic curvature under continuous deformations of the metric

Using a method introduced by R. Bamler to study the behavior of scalar curvature under continuous deformations of Riemannian metrics, we prove that if a sequence of smooth Riemannian metrics gi on a fixed compact manifold M has isotropic curvature bounded from below by a nonnegative function u, and if gi converge in C 0 norm to a smooth metric g, then g has isotropic curvature bounded from below by u. The proof also works for various other bounds from below on the curvature, such has non-negative curvature operator.

math.DG↗
arXiv2014

Curvature cones and the Ricci flow

This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature con-ditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters. The Ricci flow is the following evolution equation:

math.DG↗
arXiv2014

Positive isotropic curvature and self-duality in dimension 4

We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: The half-$PIC$ condition. It is a slight weakening of the positive isotropic curvature ($PIC$) condition introduced by M. Micallef and J. Moore. We observe that the half-$PIC$ condition is preserved by the Ricci flow and satisfies a maximality property among all Ricci flow invariant positivity conditions on the curvature of oriented 4-manifolds. We also study some geometric and topological aspects of half-$PIC$ manifolds.

math.DG↗
arXiv2013

Noncoercive Ricci flow invariant curvature cones

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than "Einstein with nonnegative scalar curvature", then this condition has to be "nonnegative scalar curvature". As a corollary, we obtain that a Ricci flow invariant curvature condition which is stronger than "nonnegative scalar curvature" cannot be (strictly) satisfied by compact Einstein symmetric spaces such as S^2xS^2 or CP^2. We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.

math.DG↗
arXiv2012

Canonical smoothing of compact Alexandrov surfaces via Ricci flow

In this paper, we show existence and uniqueness of Ricci flow whose initial condition is a compact Alexandrov surface with curvature bounded from below. This requires a weakening of the notion of initial condition which is able to deal with a priori non-Riemannian metric spaces. As a by-product, we obtain that the Ricci flow of a surface depends smoothly on Gromov-Hausdorff perturbations of the initial condition.

math.DG↗
arXiv2011

Lower bounds on Ricci flow invariant curvatures and geometric applications

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+εI \in C at the initial time, then it satisfies R +KεI \in C on some time interval depending only on the scalar curvature control. This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and R + I \in C along the sequence of initial conditions. Another application is a stability result for manifolds whose curvature operator is almost in C. Finally, we study the case where C is contained in the cone of operators whose sectional curvature is nonnegative. This allow us to weaken the assumptions of the previously mentioned applications. In particular, we construct a Ricci flow for a class of (not too) singular Alexandrov spaces.

math.DG↗