arXiv subjects

Paula Burkhardt-Guim

Publications and source records attributed to Paula Burkhardt-Guim.

4 recordsLinked to original sources

arXiv2025

ADM mass for $C^0$ metrics and distortion under Ricci-DeTurck flow

We show that there exists a quantity, depending only on $C^0$ data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$ sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the $C^0$ mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the $C^0$ local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.

math.DG↗

arXiv2024

Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set

We show that any $L^\infty$ Riemannian metric $g$ on $\mathbb{R}^n$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-α)$-dimensional Minkowski content, for some $α>2$, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that $g$ is sufficiently close in $L^\infty$ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in $C^\infty$ to $g$ away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a $L^\infty$ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times.

math.DG↗

arXiv2020

Defining Pointwise Lower Scalar Curvature Bounds for $C^0$ Metrics with Regularization by Ricci Flow

We survey some recent work using Ricci flow to create a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We discuss several properties of these definitions and explain some applications of this approach to questions regarding uniform convergence of metrics with scalar curvature bounded below. Finally, we consider the relationship between this approach and some other generalized notions of lower scalar curvature bounds.

math.DG↗

arXiv2019

Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$ initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from $C^0$ initial data.

math.DG↗