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Minimize RSR Award Detail

Research Spending & Results

Award Detail

Awardee:BROWN UNIVERSITY IN PROVIDENCE IN THE STATE OF RHODE ISLAND AND PROVIDENCE PLANTATIONS
Doing Business As Name:Brown University
PD/PI:
  • Christine Breiner
  • (401) 863-1867
  • Christine_Breiner@brown.edu
Award Date:09/07/2021
Estimated Total Award Amount: $ 401,162
Funds Obligated to Date: $ 205,792
  • FY 2020=$56,683
  • FY 2018=$34,498
  • FY 2021=$114,611
Start Date:09/01/2021
End Date:07/31/2023
Transaction Type:Grant
Agency:NSF
Awarding Agency Code:4900
Funding Agency Code:4900
CFDA Number:47.049
Primary Program Source:040100 NSF RESEARCH & RELATED ACTIVIT
Award Title or Description:CAREER: Existence and Regularity of Solutions to Variational Problems in Geometric Analysis
Federal Award ID Number:2147439
DUNS ID:001785542
Parent DUNS ID:001785542
Program:GEOMETRIC ANALYSIS
Program Officer:
  • Christopher Stark
  • (703) 292-4869
  • cstark@nsf.gov

Awardee Location

Street:BOX 1929
City:Providence
State:RI
ZIP:02912-9002
County:Providence
Country:US
Awardee Cong. District:01

Primary Place of Performance

Organization Name:Brown University
Street:
City:Providence
State:RI
ZIP:02912-9002
County:Providence
Country:US
Cong. District:01

Abstract at Time of Award

This project studies optimization questions in geometric analysis, namely constructs that optimize energy or area subject to a constraint. Existence and structural results are of interest in areas such as engineering, physics, and chemistry. Classical examples are minimal surfaces, which locally minimize area subject to fixed boundary conditions, such as soap films supported by wires of various shapes. This project studies constant mean curvature (CMC) and minimal surfaces as well as harmonic maps. CMC surfaces optimize area, but with constraint given by enclosed volume -- CMC surfaces appear in nature as soap bubbles. Harmonic maps optimize energy rather than area and are closely related to minimal surfaces. The objects studied in this project have characterizations in many branches of mathematics; the questions and desired results are of broad interest in mathematics and beyond. This research project primarily studies CMC surfaces immersed in smooth manifolds and harmonic maps into metric spaces. In the work on harmonic maps, the project aims to provide a new direction for resolution of Cannon's conjecture. It is planned to establish the existence of a harmonic homeomorphism from the round unit sphere into a sphere with a metric possessing upper curvature bounds. In a second direction, the project aims to refine techniques that produced a compactness theory for harmonic maps into metric spaces with upper curvature bounds. While the techniques for proving compactness in this setting are necessarily geometric and variational (rather than analytic), the results are analogous to those that establish compactness in the smooth setting. Using the refined techniques, the investigator plans to establish a harmonic replacement argument using energy rather than modulus of continuity methods. Other research directions relate to the study of CMC surfaces. The investigator plans to extend and refine a gluing construction that produced CMC hypersurfaces in Euclidean space. The new construction is expected to produce non-rotational, toroidal drops in Euclidean space and will serve as a model for a subsequent construction to produce CMC tori in three-manifolds. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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