Research

My research interests lie in Carnot-Caratheodory geometry.  Recently, I have been focusing on minimal and isoperimetric surface problems in Carnot groups and applications. 

 

Here is a list of papers and preprints.  Most can be found on arxiv.org, but the versions below (if available) are likely up to date and (perhaps) have better quality pictures.

 

This research has been supported, in part, by NSF grants DMS-0306752 and DMS-0503695.  

 

1. “The large scale geometry of nilpotent Lie groups”, Comm. Anal. Geom., 9 (2001), no. 5, 951-982.

2. “A notion of rectifiability modeled on Carnot groups”, Indiana Univ. Math. J. 53 (2004), 49-82.

3. “Minimal surfaces in the Heisenberg group”, Geom. Ded. 104 (2004), 201-231.

4. “The Bernstein problem in the Heisenberg group”, joint with N. Garofalo, submitted.

5. “C¹ hypersurfaces of the Heisenberg group are N-rectifiable”, joint with D. Cole, Houston J. Math. 32, 2006 713-724..

6. “H-minimal graphs of low regularity”, Comm. Math. Helv. 81 (2006), 337-384.

7. “Constant mean curvature surfaces in sub-Riemannain spaces”, joint with R. Hladky, submitted.

8. “Variations of perimeter in sub-Riemannian spaces”, joint with R. Hladky, in preparation.

9. “An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem”, joint with L. Capogna, D. Danielli and J. Tyson, to appear in Birkhauser's Progress in Mathematics Series

10. “Minimal surfaces in the Roto-translation group with applications to a neuro-biological image completion model”, joint with R. Hladky, submitted.

11. “Convexity in Carnot groups and the horizontal second fundamental form”, joint with L. Capogna and J. Tyson, submitted

12. “A positive answer to the Bernstein problem in the Heisenberg group”, joint with D. Danielli, N. Garofalo, and D. M. Nhieu, submitted.

13. “Smooth Stable noncharacteristic complete embedded minimal surfaces in the Heisenberg group are vertical planes”, joint with D. Danielli, N. Garofalo, and D. M. Nhieu, preprint.

14. “Notable examples of H-minimal graphs in the Heisenberg group”, joint with N. Garofalo, in preparation.

16. “A disocclusion algorithm based on a model of the visual cortex”, joint with R. Hladky, in preparation.