Jeffrey Case
Pennsylvania State University, PA·Mathematics & Statistics
Some of the most basic objects in geometry also describe physical reality: Einstein manifolds are the mathematical form of empty spacetimes in general relativity, and minimal submanifolds are the shapes taken by soap films, which settle into the least possible area. A longstanding goal is to find quantities attached to such spaces that do not depend on how you choose to measure them, because such quantities constrain which shapes can exist at all.
- Fields
Mathematics & Statistics
Physics
- Topics
conformal geometry
Einstein manifolds
minimal submanifolds
- Grants
- 1 known grant, through Jun 2028
What they work on
- Develop the notion of straightening into a full classification of global conformal invariants of Einstein manifolds and of their minimal submanifolds, the mathematical models of vacuum spacetimes and soap films.
- Prove gap theorems that isolate spaceforms and their totally geodesic submanifolds, using higher-order analogues of the L^2-norm of the Weyl tensor and of the Willmore energy of a surface.
- Construct conformal invariants whose positivity obstructs the existence of an Einstein metric on a given smooth manifold, and develop tools for computing them.
- Establish new analytic tools for Q-curvature, including a strong maximum principle for higher-order GJMS operators and an Obata-type classification of conformally Einstein metrics with constant Q-curvature.
More about the research
This project pursues that goal through conformal geometry, the study of geometry where angles are fixed but lengths may be rescaled, and gives particular attention to conformally compact Einstein manifolds and their minimal submanifolds, which arise in string theory.
Why it matters
The work adds to the basic understanding of which geometries are possible: invariants whose positivity obstructs an Einstein metric would tell geometers when a given smooth space cannot carry the kind of metric that models a vacuum spacetime, and gap theorems would sharpen how constant-curvature spaces and soap-film-like surfaces are characterized. Because conformally compact Einstein manifolds are fundamental in string theory, sharper invariants and analytic tools could inform that setting as well as future work in geometric analysis.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Jian Song
Rutgers University-New Brunswick, NJ·Mathematics & Statistics
Study the Yau-Tian-Donaldson conjecture for constant scalar curvature Kahler metrics on projective manifolds, testing uniform K-stability for models as a sufficient condition for existence.
Daniel Stern
Cornell University, NY·Mathematics & Statistics
Use new constructions of extremal metrics for Laplacian and Steklov eigenvalues, the natural vibration frequencies of a shape, to produce many new minimal surfaces of prescribed topology in low-dimensional spheres and balls.
Xin Zhou
Cornell University, NY·Mathematics & Statistics
Prove the existence of closed minimal surfaces with controlled genus, or number of holes, in three-manifolds, with the goal of resolving Yau's Four Minimal Spheres Conjecture.
The school
Pennsylvania State University: 60 labs on PI Pages, $1.3B a year on research, 743 research doctorates a year.
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