arXiv subjects

Tristan Ozuch

Publications and source records attributed to Tristan Ozuch.

21 records · Page 2Linked to original sources

arXiv2021

Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's $λ$-functional adapted to the ALE situation and denoted $λ_{\operatorname{ALE}}$. This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.

math.DG↗

arXiv2020

A Łojasiewicz inequality for ALE metrics

We introduce a new functional inspired by Perelman's $λ$-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $λ_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that $λ_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to $λ_{\operatorname{ALE}}$ around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations.

math.DG↗

arXiv2017

Perelman's functionals on cones and Construction of type III Ricci flows coming out of cones

In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. In a first part, we study Perelman's $λ$ and $ν$ functionals of cones and characterize their finiteness in terms of the $λ$-functional of the link. As an application, we characterize manifolds with conical singularities on which a $λ$-functional can be defined and get upper bounds on the $ν$-functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman's pseudolocality theorem and prove that cones over some perturbations of the unit sphere can be smoothed out by type III immortal solutions on the Ricci flow.

math.DG↗