publications

2025

1. heat kernel on BIC-Dynkin surfaces

   Batu Güneysu and Maxime Marot

   In preparation , 2025
2. arXiv

   On two notions of curvature on singular surfaces

   Maxime Marot

   2025

   Abs arXiv Bib

   In this paper, we investigate the equivalence of two distinct notions of curvature bounds on singular surfaces. The first notion involves inequalities of the form $\omega \ge \kappa \mu$ (resp. $\omega \le \kappa \mu$) where $\omega$ is the curvature measure and $\mu$ the Hausdorff measure. The second notion is the classical Alexandrov curvature bound CBB (resp. CAT). We demonstrate that these two definitions are, in fact, equivalent. Specifically, we fill an important gap in the theory by showing that the inequalities imply the corresponding Alexandrov CBB (resp. CAT) bound. One striking application of our result is that, in combination with a result of Petrunin, the lower bound $\omega \ge \kappa \mu$ implies $RCD(\kappa ,2)$.

   @misc{marot2025notionscurvaturesingularsurfaces,
     title = {On two notions of curvature on singular surfaces},
     author = {Marot, Maxime},
     year = {2025},
     eprint = {2511.08138},
     archiveprefix = {arXiv},
     primaryclass = {math.DG},
   }

2024

1. arXiv

   A note on the scattering theory of Kato-Ricci manifolds

   Batu Güneysu and Maxime Marot

   To appear in Rocky Mountain Journal of Mathematics , 2024

   Abs arXiv Bib

   In this note we prove a new $L^1$ criterion for the existence and completeness of the wave operators corresponding to the Laplace-Beltrami operators corresponding to two Riemannian metrics on a fixed noncompact manifold. Our result relies on recent estimates on the heat semigroup and its derivative, that are valid if the negative part of the Ricci curvature is in the Kato class - so called Kato-Ricci manifolds.

   @misc{güneysu2024notescatteringtheorykatoricci,
     title = {A note on the scattering theory of Kato-Ricci manifolds},
     author = {Güneysu, Batu and Marot, Maxime},
     year = {2024},
     eprint = {2411.03204},
     archiveprefix = {arXiv},
     primaryclass = {math.SP},
   }