Research

Differential geometry, in one form or another, has been my primary research interest. I use techniques of real analysis, complex variable theory, and Kahler geometry. Here I have decided to present my research in annotated bibliography form. As a public facing webpage, this is not meant to be a professional advertisement, but a summary of my activities that is as readable and public-friendly as possible. Enjoy!

Selected publications and preprints

• “On conformally Kähler, Einstein manifolds” pdf
  Xiuxiong Chen, Claude Lebrun & Brian Weber
  Journal of the American Mathematical Society 21, no. 4

  It is important to understand models for least-energy configurations of manifolds, in part to apply known models to more complex situations—such as when least energy minimization may cause a complicated manifold to break into smaller parts—and in part to understand known invariants and obstructions. In this paper, my co-authors and I came to fully understand the geometry of a particular minimal model: that on CP²#2CP². This was a critical piece of the puzzle in understanding the full class of Hermitian-Einstein metrics, with positive Einstein curvature, on 4-manifolds.

• “Convergence of compact Ricci solitons” pdf
  Brian Weber
  International Mathematics Research Notices 2011, no. 1
  
  A manifold should have a least-energy configuration, an unperturbed state. Such a resting state is know to be as smooth and symmetrical as possible, given the underlying construction of the manifold. One way to find out what this least-energy configuration might be is to use the Ricci flow. In a sense, this is a heat-type flow, and you can think of it as allowing the manifold to dissipate its excess energy. But what are the possible resting configurations? In this paper, I examine one of the main types of “resting configurations,” namely Ricci solitons.

• “Moduli spaces of critical Riemannian metrics with L^n/2 norm curvature bounds” pdf
  Xiuxiong Chen & Brian Weber
  Advances in Mathematics Vol. 226, no. 2

  This was my Ph.D. thesis work from the University of Wisconsin. Like a rod under stress, or the surface tension of a mass of water, manifolds carry energy in the way they flex. One of the best ways to measure this internal energy is by using something called “L^n/2 norm curvature.” The question is, given a certain way a manifold is constructed, what is its minimum-energy configuration? In this paper, methods of geometric analysis are used to study this question for certain important classes of manifolds: extremal Kahler manifolds.

• “The Achilles Heel of O(3,1)?” pdf
  William Floyd, Brian Weber & Jeffrey Weeks
  Experimental Mathematics Vol. 11 , Iss. 1, 2002
  
  Computers routinely use matrices in SO(3) (or E(3)) to process motions in Euclidean space. But what if we want to process motions in hyperbolic space? Unfortunately in that case, floating point errors build up rapidly. When computers manipulate metrics in O(3,1), floating point errors build up exponentially—this is the Achilles heel of O(3,1)—whereas in SO(3) errors only build up arithmetically. Myself and coauthors Bill Floyd and Jeff Weeks discovered that processing in SL(2,C) rather than O(3,1) reduces the exponential constant of the error buildup by half—a huge improvement. Achilles was, in part, the culmination of my senior thesis work at Virginia Tech.