Award Abstract # 1309360
Applications of the Convergence of Riemannian Manifolds to General Relativity

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: RESEARCH FOUNDATION OF THE CITY UNIVERSITY OF NEW YORK
Initial Amendment Date: September 11, 2013
Latest Amendment Date: September 11, 2013
Award Number: 1309360
Award Instrument: Standard Grant
Program Manager: Christopher Stark
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: September 1, 2013
End Date: August 31, 2018 (Estimated)
Total Intended Award Amount: $116,000.00
Total Awarded Amount to Date: $116,000.00
Funds Obligated to Date: FY 2013 = $116,000.00
History of Investigator:
  • Christina Sormani (Principal Investigator)
    sormanic@gmail.com
Recipient Sponsored Research Office: Research Foundation Of The City University Of New York (Lehman)
250 BEDFORD PARK BLVD W
BRONX
NY  US  10468-1527
(718)960-8107
Sponsor Congressional District: 13
Primary Place of Performance: CUNY Herbert H Lehman College
250 Bedford Park Blvd West
Bronx
NY  US  10468-1527
Primary Place of Performance
Congressional District:
13
Unique Entity Identifier (UEI): DJ4SM8UQBHT7
Parent UEI:
NSF Program(s): GEOMETRIC ANALYSIS
Primary Program Source: 01001314DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):
Program Element Code(s): 126500
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

Abstract

Award: DMS 1309360, Principal Investigator: Christina A. Sormani

The PI will apply Intrinsic Flat convergence between Riemannian manifolds to better understand how close space-like manifolds studied in Mathematical General Relativity approximate the standard well known models. The Intrinsic Flat distance, first introduced by the PI with Stefan Wenger using methods of Ambrosio-Kirchheim, is particularly well-suited to some questions arising in General Relativity because increasingly thin gravity wells disappear under this convergence. In joint work with Dan Lee, the PI has shown that spherically symmetric Riemannian manifolds with increasingly small ADM mass converge to Euclidean space in the pointed intrinsic flat sense, and here proposes to generalize this result. In addition, the PI proposes to develop two new notions of convergence: the first will allow mathematicians to study Lorentzian manifolds directly, and the second will prevent regions from disappearing due to orientation and cancellation. Both notions are specifically adapted to questions arising in General Relativity.

Einstein's Theory of General Relativity describes how space is curved by gravity. Even within our own solar system, when computing the trajectories of spacecraft heading to Mars, engineers must take into account the curvature caused by the mass of the planets and the sun. Each planet forms a gravity well. If the mass of a planet is small, one would like to know in what sense the space around it is almost flat. In fact, the space around a planet of arbitrarily small mass could be very highly curved (and have a very deep but thin gravity well). In joint work with Dr. Stefan Wenger, the PI has developed a new means of measuring the closeness between curved spaces and, in joint work with Dr. Dan Lee, she has estimated how close the space around a single perfectly spherical planet is to Euclidean space. In this project, she will develop tools allowing one to better understand the space around groups of planets which are not perfect spheres: like the ones in our own solar system.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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C Sormani and C Vega ""Null distance on a spacetime"" Classical and Quantum Gravity , v.33 , 2016
C Sormani and I Stavrov "``Geometrostatic manifolds of small ADM Mass"" CPAM , 2019 , p.https://d
CY Lin and C Sormani ""Bartnik's Mass and Hamilton's Modified Ricci Flow"" Annales Henri Poincare , 2016 10.1007/s00023-016-0483-8
LeFloch, Philippe G. and Sormani, Christina "The nonlinear stability of rotationally symmetric spaces with low regularity." Journal of Functional Analysis , v.268 , 2015 , p.2005-2065 doi:10.1016/j.jfa.2014.12.012
L-H Huang, D A Lee and C. Sormani "Intrinsic flat stability of the Positive Mass Theorem for graphical hypersurfaces in Euclidean space" Journal fur die Riene und Angewandte Mathematik, Crelle's Journal, , v.727 , 2017 , p.269-299 https://doi.org/10.1515/crelle-2015-0051

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

According to Einstein's Theory of General Relativity, space is curved by gravity: forming gravitational wells and black holes.   Schoen and Yau proved that if a time-symmetric space satisfying the positive energy condition has no mass, then it is classical Euclidean space.  However, if it has even a small amount of mass, it can have an arbitrarily deep and thin gravity well.  In fact, it is known that the universe we live in has many such wells.   Nevertheless, using the right notion of convergence of spaces, one can develop an understanding of how close a space with small mass is to classical Euclidean space.

Dr. Christina Sormani, has worked with her doctoral students, Dr. Raquel Perales and Dr. Jorge Basilio, to develop the notion of intrinsic flat convergence of metric spaces.  Note that this notion of convergence was first defined by Sormani and Wenger a decade ago.  During the tenure of this grant, Sormani has investigated how some points may disappear when taking a limit (as seen for example when a point at the bottom of a gravitational well disappears). Perales has studied how tetrahedra can be applied to determine which points disappear in joint work with Nunez-Zimbron.  Basilio and Sormani have investigated how entire regions might merge into a single point when taking a limit of a sequence of spaces.

In joint work with Dr. Dan Lee and Dr. Lan-Hsuan Huang, Dr. Sormani has applied the above results to show that certain spaces (time symmetric graphs that satisfy the positive energy condition) are close in this intrinsic flat sense to Euclidean space when they have small mass. In joint work Dr. Iva Stavrov, Dr. Sormani has shown that geometrostatic spaces (satisfying the Einstein-Maxwell vacuum constraint equations) of small mass are also close to Euclidean space if one cuts off the interiors of black holes.  In this paper they also located the black holes in such spaces (a question first posed by Brill-Lindquist in 1963).  Note that this paper allows for arbitrarily large numbers of charged black holes, but they must be kept apart from one another by a definite amount.  

It also important to understand what happens when the total mass is not arbitrarily close to zero.  In such cases one must understand various ways of defining mass locally, rather than just measuring the total ADM mass at infinity. In joint work with postdoc, Dr. Chen-Yun Lin, Dr. Sormani has analyzed the relationship between Bartnik mass (a quasi local notion of mass) and ADM mass (a total mass measured at infinity), estimating the difference between them using a purely geometric quantity called the asphericity mass.  They prove this asphericity mass of a region is zero if and only if the boundary of the region is a standard round sphere.  In joint work with Dr. Philippe LeFloch, Dr. Sormani has investigated sequences of rotationally symmetric regions of space with uniform bounds on their Hawking mass (another quasilocal mass) and shown subsequences of such spaces converge in the intrinsic flat and Sobolev sense to rotational symmetric regions with the same uniform upper bound on their Hawking mass.  

Einstein's Theory of General Relativity also describes how spacetime is curved by gravity.  Spacetime is not a metric space and so one cannot just apply intrinsic flat convergence to understand how spacetimes converge.  All the results above concern time symmetric spacelike slices of the universe and thus are already metric spaces.  In joint work with postdoc, Dr. Carlos Vega, Dr. Sormani has developed a method of converting a spacetime into a metric space using the cosmological time function.  Once converted into a metric space, one can then apply intrinsic flat convergence to see how a sequence of spacetimes converges to a limiting spacetime.

Dr. Sormani has given lecture series on these topics at the Hausdorff Institute in Bonn, at USTC in Hefei, at the Como School of Advanced Study in Italy, and at the Fields Institute in Toronto.  Jointly with Drs. Yau, Schoen, Wang and Chrusciel, she organized a large workshop for international experts at the Simons Center for Geometry and Physics on Mass in General Relativity.   Jointly with Drs. Khuri and Kazaras, she organized a Spring School on Geometric Aspects of General Relativity for graduate students and postdocs.   She organized small workshops at CUNY with Dr. Lakzian, at McGill with Dr. Jain, and at UNAM with Dr. Perales (where research teams met to apply the techniques developed with this funding in new directions).

Dr. Sormani is dedicated to bringing more minorities and women into Geometric Analysis.  She supervised a research team of young women with Dr. Searle, and organized a small meeting of women in mean curvature at Stanford. At Lehman College, she supervised an undergraduate research team of minority students and ran a seminar this past summer on differential geometry for undergraduates.


Last Modified: 11/24/2018
Modified by: Christina A Sormani

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