arXiv subjects

Christos Mantoulidis

Publications and source records attributed to Christos Mantoulidis.

22 records · Page 2Linked to original sources

arXiv2018

Positive scalar curvature with skeleton singularities

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean ($L^\infty$) metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles $\leq 2π$ along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles $\leq 2π$) and arbitrary $L^\infty$ isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.

math.DG↗

arXiv2017

A note on the Morse index of 2k-ended phase transitions in R^2

We show that the Morse index of every 2k-ended solution of the Allen-Cahn equation in R^2 is >= k-1. This bound is expected to be sharp.

math.AP↗

arXiv2016

Total mean curvature, scalar curvature, and a variational analog of Brown-York mass

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consists of 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi-Tam and Wang-Yau on the quasi-local mass problem in general relativity. In turn, we define a variational analog of Brown-York quasi-local mass without assuming that the boundary 2-sphere has positive Gauss curvature.

math.DG↗

arXiv2015

On the Bartnik mass of apparent horizons

In this paper we characterize the intrinsic geometry of apparent horizons (outermost marginally outer trapped surfaces) in asymptotically flat spacetimes; that is, the Riemannian metrics on the two sphere which can arise. Furthermore we determine the minimal ADM mass of a spacetime containing such an apparent horizon. The results are conveniently formulated in terms of the quasi-local mass introduced by Bartnik in 1989. The Hawking mass provides a lower bound for Bartnik's quasilocal mass on apparent horizons by way of Penrose's conjecture on time symmetric slices, proven in 1997 by Huisken and Ilmanen and in full generality in 1999 by Bray. We compute Bartnik's mass for all non-degenerate apparent horizons and show that it coincides with the Hawking mass. As a corollary we disprove a conjecture due to Gibbons in the spirit of Thorne's hoop conjecture, and construct a new large class of examples of apparent horizons with the integral of the negative part of the Gauss curvature arbitrarily large.

math.DG↗